Documentation

Mathlib.LinearAlgebra.LinearPMap

Partially defined linear maps #

A LinearPMap R E F or E →ₗ.[R] F is a linear map from a submodule of E to F. We define a SemilatticeInf with OrderBot instance on this, and define three operations:

Moreover, we define

Partially defined maps are currently used in Mathlib to prove Hahn-Banach theorem and its variations. Namely, LinearPMap.sSup implies that every chain of LinearPMaps is bounded above. They are also the basis for the theory of unbounded operators.

structure LinearPMap (R : Type u) [Ring R] (E : Type v) [AddCommGroup E] [Module R E] (F : Type w) [AddCommGroup F] [Module R F] :
Type (max v w)

A LinearPMap R E F or E →ₗ.[R] F is a linear map from a submodule of E to F.

Instances For
    def LinearPMap.toFun' {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) :
    { x // x f.domain }F
    Instances For
      @[simp]
      theorem LinearPMap.toFun_eq_coe {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (x : { x // x f.domain }) :
      f.toFun x = f x
      theorem LinearPMap.ext {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} (h : f.domain = g.domain) (h' : ∀ ⦃x : { x // x f.domain }⦄ ⦃y : { x // x g.domain }⦄, x = yf x = g y) :
      f = g
      @[simp]
      theorem LinearPMap.map_zero {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) :
      f 0 = 0
      theorem LinearPMap.ext_iff {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} :
      f = g _domain_eq, ∀ ⦃x : { x // x f.domain }⦄ ⦃y : { x // x g.domain }⦄, x = yf x = g y
      theorem LinearPMap.ext' {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {s : Submodule R E} {f : { x // x s } →ₗ[R] F} {g : { x // x s } →ₗ[R] F} (h : f = g) :
      { domain := s, toFun := f } = { domain := s, toFun := g }
      theorem LinearPMap.map_add {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (x : { x // x f.domain }) (y : { x // x f.domain }) :
      f (x + y) = f x + f y
      theorem LinearPMap.map_neg {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (x : { x // x f.domain }) :
      f (-x) = -f x
      theorem LinearPMap.map_sub {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (x : { x // x f.domain }) (y : { x // x f.domain }) :
      f (x - y) = f x - f y
      theorem LinearPMap.map_smul {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (c : R) (x : { x // x f.domain }) :
      f (c x) = c f x
      @[simp]
      theorem LinearPMap.mk_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (p : Submodule R E) (f : { x // x p } →ₗ[R] F) (x : { x // x p }) :
      { domain := p, toFun := f } x = f x
      noncomputable def LinearPMap.mkSpanSingleton' {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (x : E) (y : F) (H : ∀ (c : R), c x = 0c y = 0) :

      The unique LinearPMap on R ∙ x that sends x to y. This version works for modules over rings, and requires a proof of ∀ c, c • x = 0 → c • y = 0.

      Instances For
        @[simp]
        theorem LinearPMap.domain_mkSpanSingleton {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (x : E) (y : F) (H : ∀ (c : R), c x = 0c y = 0) :
        @[simp]
        theorem LinearPMap.mkSpanSingleton'_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (x : E) (y : F) (H : ∀ (c : R), c x = 0c y = 0) (c : R) (h : c x (LinearPMap.mkSpanSingleton' x y H).domain) :
        ↑(LinearPMap.mkSpanSingleton' x y H) { val := c x, property := h } = c y
        @[simp]
        theorem LinearPMap.mkSpanSingleton'_apply_self {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (x : E) (y : F) (H : ∀ (c : R), c x = 0c y = 0) (h : x (LinearPMap.mkSpanSingleton' x y H).domain) :
        ↑(LinearPMap.mkSpanSingleton' x y H) { val := x, property := h } = y
        @[reducible]
        noncomputable def LinearPMap.mkSpanSingleton {K : Type u_5} {E : Type u_6} {F : Type u_7} [DivisionRing K] [AddCommGroup E] [Module K E] [AddCommGroup F] [Module K F] (x : E) (y : F) (hx : x 0) :

        The unique LinearPMap on span R {x} that sends a non-zero vector x to y. This version works for modules over division rings.

        Instances For
          theorem LinearPMap.mkSpanSingleton_apply (K : Type u_5) {E : Type u_6} {F : Type u_7} [DivisionRing K] [AddCommGroup E] [Module K E] [AddCommGroup F] [Module K F] {x : E} (hx : x 0) (y : F) :
          ↑(LinearPMap.mkSpanSingleton x y hx) { val := x, property := (_ : x Submodule.span K {x}) } = y
          def LinearPMap.fst {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (p : Submodule R E) (p' : Submodule R F) :
          E × F →ₗ.[R] E

          Projection to the first coordinate as a LinearPMap

          Instances For
            @[simp]
            theorem LinearPMap.fst_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (p : Submodule R E) (p' : Submodule R F) (x : { x // x Submodule.prod p p' }) :
            ↑(LinearPMap.fst p p') x = (x).fst
            def LinearPMap.snd {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (p : Submodule R E) (p' : Submodule R F) :
            E × F →ₗ.[R] F

            Projection to the second coordinate as a LinearPMap

            Instances For
              @[simp]
              theorem LinearPMap.snd_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (p : Submodule R E) (p' : Submodule R F) (x : { x // x Submodule.prod p p' }) :
              ↑(LinearPMap.snd p p') x = (x).snd
              instance LinearPMap.le {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
              LE (E →ₗ.[R] F)
              theorem LinearPMap.apply_comp_ofLe {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {T : E →ₗ.[R] F} {S : E →ₗ.[R] F} (h : T S) (x : { x // x T.domain }) :
              T x = S (↑(Submodule.ofLe (_ : T.domain S.domain)) x)
              theorem LinearPMap.exists_of_le {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {T : E →ₗ.[R] F} {S : E →ₗ.[R] F} (h : T S) (x : { x // x T.domain }) :
              y, x = y T x = S y
              theorem LinearPMap.eq_of_le_of_domain_eq {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} (hle : f g) (heq : f.domain = g.domain) :
              f = g
              def LinearPMap.eqLocus {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) :

              Given two partial linear maps f, g, the set of points x such that both f and g are defined at x and f x = g x form a submodule.

              Instances For
                instance LinearPMap.inf {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                Inf (E →ₗ.[R] F)
                instance LinearPMap.bot {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                Bot (E →ₗ.[R] F)
                instance LinearPMap.inhabited {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                instance LinearPMap.semilatticeInf {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                instance LinearPMap.orderBot {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                theorem LinearPMap.le_of_eqLocus_ge {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} (H : f.domain LinearPMap.eqLocus f g) :
                f g
                theorem LinearPMap.domain_mono {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                StrictMono LinearPMap.domain
                noncomputable def LinearPMap.sup {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) (h : ∀ (x : { x // x f.domain }) (y : { x // x g.domain }), x = yf x = g y) :

                Given two partial linear maps that agree on the intersection of their domains, f.sup g h is the unique partial linear map on f.domain ⊔ g.domain that agrees with f and g.

                Instances For
                  @[simp]
                  theorem LinearPMap.domain_sup {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) (h : ∀ (x : { x // x f.domain }) (y : { x // x g.domain }), x = yf x = g y) :
                  (LinearPMap.sup f g h).domain = f.domain g.domain
                  theorem LinearPMap.sup_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} (H : ∀ (x : { x // x f.domain }) (y : { x // x g.domain }), x = yf x = g y) (x : { x // x f.domain }) (y : { x // x g.domain }) (z : { x // x f.domain g.domain }) (hz : x + y = z) :
                  ↑(LinearPMap.sup f g H) z = f x + g y
                  theorem LinearPMap.left_le_sup {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) (h : ∀ (x : { x // x f.domain }) (y : { x // x g.domain }), x = yf x = g y) :
                  theorem LinearPMap.right_le_sup {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) (h : ∀ (x : { x // x f.domain }) (y : { x // x g.domain }), x = yf x = g y) :
                  theorem LinearPMap.sup_le {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} {h : E →ₗ.[R] F} (H : ∀ (x : { x // x f.domain }) (y : { x // x g.domain }), x = yf x = g y) (fh : f h) (gh : g h) :
                  theorem LinearPMap.sup_h_of_disjoint {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) (h : Disjoint f.domain g.domain) (x : { x // x f.domain }) (y : { x // x g.domain }) (hxy : x = y) :
                  f x = g y

                  Hypothesis for LinearPMap.sup holds, if f.domain is disjoint with g.domain.

                  Algebraic operations #

                  instance LinearPMap.instZero {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  @[simp]
                  theorem LinearPMap.zero_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  0.domain =
                  @[simp]
                  theorem LinearPMap.zero_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (x : { x // x }) :
                  0 x = 0
                  instance LinearPMap.instSMul {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {M : Type u_5} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F] :
                  SMul M (E →ₗ.[R] F)
                  @[simp]
                  theorem LinearPMap.smul_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {M : Type u_5} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F] (a : M) (f : E →ₗ.[R] F) :
                  (a f).domain = f.domain
                  theorem LinearPMap.smul_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {M : Type u_5} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F] (a : M) (f : E →ₗ.[R] F) (x : { x // x (a f).domain }) :
                  ↑(a f) x = a f x
                  @[simp]
                  theorem LinearPMap.coe_smul {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {M : Type u_5} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F] (a : M) (f : E →ₗ.[R] F) :
                  ↑(a f) = a f
                  instance LinearPMap.instSMulCommClass {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {M : Type u_5} {N : Type u_6} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F] [Monoid N] [DistribMulAction N F] [SMulCommClass R N F] [SMulCommClass M N F] :
                  instance LinearPMap.instIsScalarTower {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {M : Type u_5} {N : Type u_6} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F] [Monoid N] [DistribMulAction N F] [SMulCommClass R N F] [SMul M N] [IsScalarTower M N F] :
                  instance LinearPMap.instMulAction {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {M : Type u_5} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F] :
                  instance LinearPMap.instNeg {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  Neg (E →ₗ.[R] F)
                  @[simp]
                  theorem LinearPMap.neg_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) :
                  (-f).domain = f.domain
                  @[simp]
                  theorem LinearPMap.neg_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (x : { x // x (-f).domain }) :
                  ↑(-f) x = -f x
                  instance LinearPMap.instInvolutiveNeg {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  instance LinearPMap.instAdd {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  Add (E →ₗ.[R] F)
                  theorem LinearPMap.add_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) :
                  (f + g).domain = f.domain g.domain
                  theorem LinearPMap.add_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) (x : { x // x f.domain g.domain }) :
                  ↑(f + g) x = f { val := x, property := (_ : x f.domain) } + g { val := x, property := (_ : x g.domain) }
                  instance LinearPMap.instAddSemigroup {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  instance LinearPMap.instAddZeroClass {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  instance LinearPMap.instAddMonoid {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  instance LinearPMap.instAddCommMonoid {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  instance LinearPMap.instVAdd {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  VAdd (E →ₗ[R] F) (E →ₗ.[R] F)
                  @[simp]
                  theorem LinearPMap.vadd_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ[R] F) (g : E →ₗ.[R] F) :
                  (f +ᵥ g).domain = g.domain
                  theorem LinearPMap.vadd_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ[R] F) (g : E →ₗ.[R] F) (x : { x // x (f +ᵥ g).domain }) :
                  ↑(f +ᵥ g) x = f x + g x
                  @[simp]
                  theorem LinearPMap.coe_vadd {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ[R] F) (g : E →ₗ.[R] F) :
                  ↑(f +ᵥ g) = ↑(LinearMap.comp f (Submodule.subtype g.domain)) + g
                  instance LinearPMap.instAddAction {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  instance LinearPMap.instSub {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  Sub (E →ₗ.[R] F)
                  theorem LinearPMap.sub_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) :
                  (f - g).domain = f.domain g.domain
                  theorem LinearPMap.sub_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) (x : { x // x f.domain g.domain }) :
                  ↑(f - g) x = f { val := x, property := (_ : x f.domain) } - g { val := x, property := (_ : x g.domain) }
                  noncomputable def LinearPMap.supSpanSingleton {E : Type u_2} [AddCommGroup E] {F : Type u_3} [AddCommGroup F] {K : Type u_5} [DivisionRing K] [Module K E] [Module K F] (f : E →ₗ.[K] F) (x : E) (y : F) (hx : ¬x f.domain) :

                  Extend a LinearPMap to f.domain ⊔ K ∙ x.

                  Instances For
                    @[simp]
                    theorem LinearPMap.domain_supSpanSingleton {E : Type u_2} [AddCommGroup E] {F : Type u_3} [AddCommGroup F] {K : Type u_5} [DivisionRing K] [Module K E] [Module K F] (f : E →ₗ.[K] F) (x : E) (y : F) (hx : ¬x f.domain) :
                    (LinearPMap.supSpanSingleton f x y hx).domain = f.domain Submodule.span K {x}
                    @[simp]
                    theorem LinearPMap.supSpanSingleton_apply_mk {E : Type u_2} [AddCommGroup E] {F : Type u_3} [AddCommGroup F] {K : Type u_5} [DivisionRing K] [Module K E] [Module K F] (f : E →ₗ.[K] F) (x : E) (y : F) (hx : ¬x f.domain) (x' : E) (hx' : x' f.domain) (c : K) :
                    ↑(LinearPMap.supSpanSingleton f x y hx) { val := x' + c x, property := (_ : x' + c x f.domain (LinearPMap.mkSpanSingleton x y (_ : x = 0False)).domain) } = f { val := x', property := hx' } + c y
                    noncomputable def LinearPMap.sSup {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (c : Set (E →ₗ.[R] F)) (hc : DirectedOn (fun x x_1 => x x_1) c) :
                    Instances For
                      theorem LinearPMap.le_sSup {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {c : Set (E →ₗ.[R] F)} (hc : DirectedOn (fun x x_1 => x x_1) c) {f : E →ₗ.[R] F} (hf : f c) :
                      theorem LinearPMap.sSup_le {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {c : Set (E →ₗ.[R] F)} (hc : DirectedOn (fun x x_1 => x x_1) c) {g : E →ₗ.[R] F} (hg : ∀ (f : E →ₗ.[R] F), f cf g) :
                      theorem LinearPMap.sSup_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {c : Set (E →ₗ.[R] F)} (hc : DirectedOn (fun x x_1 => x x_1) c) {l : E →ₗ.[R] F} (hl : l c) (x : { x // x l.domain }) :
                      ↑(LinearPMap.sSup c hc) { val := x, property := (_ : x (LinearPMap.sSup c hc).domain) } = l x
                      def LinearMap.toPMap {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ[R] F) (p : Submodule R E) :

                      Restrict a linear map to a submodule, reinterpreting the result as a LinearPMap.

                      Instances For
                        @[simp]
                        theorem LinearMap.toPMap_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ[R] F) (p : Submodule R E) (x : { x // x p }) :
                        ↑(LinearMap.toPMap f p) x = f x
                        @[simp]
                        theorem LinearMap.toPMap_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ[R] F) (p : Submodule R E) :
                        (LinearMap.toPMap f p).domain = p
                        def LinearMap.compPMap {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {G : Type u_4} [AddCommGroup G] [Module R G] (g : F →ₗ[R] G) (f : E →ₗ.[R] F) :

                        Compose a linear map with a LinearPMap

                        Instances For
                          @[simp]
                          theorem LinearMap.compPMap_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {G : Type u_4} [AddCommGroup G] [Module R G] (g : F →ₗ[R] G) (f : E →ₗ.[R] F) (x : { x // x (LinearMap.compPMap g f).domain }) :
                          ↑(LinearMap.compPMap g f) x = g (f x)
                          def LinearPMap.codRestrict {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (p : Submodule R F) (H : ∀ (x : { x // x f.domain }), f x p) :
                          E →ₗ.[R] { x // x p }

                          Restrict codomain of a LinearPMap

                          Instances For
                            def LinearPMap.comp {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {G : Type u_4} [AddCommGroup G] [Module R G] (g : F →ₗ.[R] G) (f : E →ₗ.[R] F) (H : ∀ (x : { x // x f.domain }), f x g.domain) :

                            Compose two LinearPMaps

                            Instances For
                              def LinearPMap.coprod {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {G : Type u_4} [AddCommGroup G] [Module R G] (f : E →ₗ.[R] G) (g : F →ₗ.[R] G) :
                              E × F →ₗ.[R] G

                              f.coprod g is the partially defined linear map defined on f.domain × g.domain, and sending p to f p.1 + g p.2.

                              Instances For
                                @[simp]
                                theorem LinearPMap.coprod_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {G : Type u_4} [AddCommGroup G] [Module R G] (f : E →ₗ.[R] G) (g : F →ₗ.[R] G) (x : { x // x (LinearPMap.coprod f g).domain }) :
                                ↑(LinearPMap.coprod f g) x = f { val := (x).fst, property := (_ : (x).fst f.domain) } + g { val := (x).snd, property := (_ : (x).snd g.domain) }
                                def LinearPMap.domRestrict {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (S : Submodule R E) :

                                Restrict a partially defined linear map to a submodule of E contained in f.domain.

                                Instances For
                                  @[simp]
                                  theorem LinearPMap.domRestrict_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) {S : Submodule R E} :
                                  (LinearPMap.domRestrict f S).domain = S f.domain
                                  theorem LinearPMap.domRestrict_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {S : Submodule R E} ⦃x : { x // x S f.domain } ⦃y : { x // x f.domain } (h : x = y) :
                                  ↑(LinearPMap.domRestrict f S) x = f y
                                  theorem LinearPMap.domRestrict_le {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {S : Submodule R E} :

                                  Graph #

                                  def LinearPMap.graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) :
                                  Submodule R (E × F)

                                  The graph of a LinearPMap viewed as a submodule on E × F.

                                  Instances For
                                    theorem LinearPMap.mem_graph_iff' {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) {x : E × F} :
                                    x LinearPMap.graph f y, (y, f y) = x
                                    @[simp]
                                    theorem LinearPMap.mem_graph_iff {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) {x : E × F} :
                                    x LinearPMap.graph f y, y = x.fst f y = x.snd
                                    theorem LinearPMap.mem_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (x : { x // x f.domain }) :
                                    (x, f x) LinearPMap.graph f

                                    The tuple (x, f x) is contained in the graph of f.

                                    theorem LinearPMap.graph_map_fst_eq_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) :
                                    theorem LinearPMap.graph_map_snd_eq_range {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) :
                                    theorem LinearPMap.smul_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {M : Type u_5} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F] (f : E →ₗ.[R] F) (z : M) :
                                    LinearPMap.graph (z f) = Submodule.map (LinearMap.prodMap LinearMap.id (z LinearMap.id)) (LinearPMap.graph f)

                                    The graph of z • f as a pushforward.

                                    theorem LinearPMap.neg_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) :

                                    The graph of -f as a pushforward.

                                    theorem LinearPMap.mem_graph_snd_inj {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) {x : E} {y : E} {x' : F} {y' : F} (hx : (x, x') LinearPMap.graph f) (hy : (y, y') LinearPMap.graph f) (hxy : x = y) :
                                    x' = y'
                                    theorem LinearPMap.mem_graph_snd_inj' {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) {x : E × F} {y : E × F} (hx : x LinearPMap.graph f) (hy : y LinearPMap.graph f) (hxy : x.fst = y.fst) :
                                    x.snd = y.snd
                                    theorem LinearPMap.graph_fst_eq_zero_snd {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) {x : E} {x' : F} (h : (x, x') LinearPMap.graph f) (hx : x = 0) :
                                    x' = 0

                                    The property that f 0 = 0 in terms of the graph.

                                    theorem LinearPMap.mem_domain_iff {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {x : E} :
                                    x f.domain y, (x, y) LinearPMap.graph f
                                    theorem LinearPMap.mem_domain_of_mem_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {x : E} {y : F} (h : (x, y) LinearPMap.graph f) :
                                    x f.domain
                                    theorem LinearPMap.image_iff {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {x : E} {y : F} (hx : x f.domain) :
                                    y = f { val := x, property := hx } (x, y) LinearPMap.graph f
                                    theorem LinearPMap.mem_range_iff {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {y : F} :
                                    y Set.range f x, (x, y) LinearPMap.graph f
                                    theorem LinearPMap.mem_domain_iff_of_eq_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} (h : LinearPMap.graph f = LinearPMap.graph g) {x : E} :
                                    x f.domain x g.domain
                                    theorem LinearPMap.le_of_le_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} (h : LinearPMap.graph f LinearPMap.graph g) :
                                    f g
                                    theorem LinearPMap.le_graph_of_le {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} (h : f g) :
                                    theorem LinearPMap.le_graph_iff {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} :
                                    theorem LinearPMap.eq_of_eq_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} (h : LinearPMap.graph f = LinearPMap.graph g) :
                                    f = g
                                    theorem Submodule.existsUnique_from_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {g : Submodule R (E × F)} (hg : ∀ {x : E × F}, x gx.fst = 0x.snd = 0) {a : E} (ha : a Submodule.map (LinearMap.fst R E F) g) :
                                    ∃! b, (a, b) g
                                    noncomputable def Submodule.valFromGraph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {g : Submodule R (E × F)} (hg : ∀ (x : E × F), x gx.fst = 0x.snd = 0) {a : E} (ha : a Submodule.map (LinearMap.fst R E F) g) :
                                    F

                                    Auxiliary definition to unfold the existential quantifier.

                                    Instances For
                                      theorem Submodule.valFromGraph_mem {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {g : Submodule R (E × F)} (hg : ∀ (x : E × F), x gx.fst = 0x.snd = 0) {a : E} (ha : a Submodule.map (LinearMap.fst R E F) g) :
                                      noncomputable def Submodule.toLinearPMapAux {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (g : Submodule R (E × F)) (hg : ∀ (x : E × F), x gx.fst = 0x.snd = 0) :
                                      { x // x Submodule.map (LinearMap.fst R E F) g } →ₗ[R] F

                                      Define a LinearMap from its graph.

                                      Helper definition for LinearPMap.

                                      Instances For
                                        noncomputable def Submodule.toLinearPMap {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (g : Submodule R (E × F)) :

                                        Define a LinearPMap from its graph.

                                        In the case that the submodule is not a graph of a LinearPMap then the underlying linear map is just the zero map.

                                        Instances For
                                          theorem Submodule.toLinearPMap_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (g : Submodule R (E × F)) :
                                          theorem Submodule.toLinearPMap_apply_aux {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {g : Submodule R (E × F)} (hg : ∀ (x : E × F), x gx.fst = 0x.snd = 0) (x : { x // x Submodule.map (LinearMap.fst R E F) g }) :
                                          theorem Submodule.mem_graph_toLinearPMap {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {g : Submodule R (E × F)} (hg : ∀ (x : E × F), x gx.fst = 0x.snd = 0) (x : { x // x Submodule.map (LinearMap.fst R E F) g }) :
                                          (x, ↑(Submodule.toLinearPMap g) x) g
                                          @[simp]
                                          theorem Submodule.toLinearPMap_graph_eq {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (g : Submodule R (E × F)) (hg : ∀ (x : E × F), x gx.fst = 0x.snd = 0) :
                                          theorem Submodule.toLinearPMap_range {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (g : Submodule R (E × F)) (hg : ∀ (x : E × F), x gx.fst = 0x.snd = 0) :
                                          noncomputable def LinearPMap.inverse {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) :

                                          The inverse of a LinearPMap.

                                          Instances For
                                            theorem LinearPMap.inverse_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} :
                                            theorem LinearPMap.mem_inverse_graph_snd_eq_zero {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} (hf : LinearMap.ker f.toFun = ) (x : F × E) (hv : x Submodule.map (LinearEquiv.prodComm R E F) (LinearPMap.graph f)) (hv' : x.fst = 0) :
                                            x.snd = 0

                                            The graph of the inverse generates a LinearPMap.

                                            theorem LinearPMap.inverse_range {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} (hf : LinearMap.ker f.toFun = ) :
                                            theorem LinearPMap.mem_inverse_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} (hf : LinearMap.ker f.toFun = ) (x : { x // x f.domain }) :
                                            theorem LinearPMap.inverse_apply_eq {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} (hf : LinearMap.ker f.toFun = ) {y : { x // x (LinearPMap.inverse f).domain }} {x : { x // x f.domain }} (hxy : f x = y) :
                                            ↑(LinearPMap.inverse f) y = x