# modular equivalence for submodule #

def SModEq {R : Type u_1} [Ring R] {M : Type u_3} [] [Module R M] (U : ) (x : M) (y : M) :

A predicate saying two elements of a module are equivalent modulo a submodule.

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• One or more equations did not get rendered due to their size.
Instances For
theorem SModEq.def {R : Type u_1} [Ring R] {M : Type u_3} [] [Module R M] {U : } {x : M} {y : M} :
x y [SMOD U]
theorem SModEq.sub_mem {R : Type u_1} [Ring R] {M : Type u_3} [] [Module R M] {U : } {x : M} {y : M} :
x y [SMOD U] x - y U
@[simp]
theorem SModEq.top {R : Type u_1} [Ring R] {M : Type u_3} [] [Module R M] {x : M} {y : M} :
@[simp]
theorem SModEq.bot {R : Type u_1} [Ring R] {M : Type u_3} [] [Module R M] {x : M} {y : M} :
x y [SMOD ] x = y
theorem SModEq.mono {R : Type u_1} [Ring R] {M : Type u_3} [] [Module R M] {U₁ : } {U₂ : } {x : M} {y : M} (HU : U₁ U₂) (hxy : x y [SMOD U₁]) :
x y [SMOD U₂]
theorem SModEq.refl {R : Type u_1} [Ring R] {M : Type u_3} [] [Module R M] {U : } (x : M) :
x x [SMOD U]
theorem SModEq.rfl {R : Type u_1} [Ring R] {M : Type u_3} [] [Module R M] {U : } {x : M} :
x x [SMOD U]
instance SModEq.instIsRefl {R : Type u_1} [Ring R] {M : Type u_3} [] [Module R M] {U : } :
IsRefl M ()
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• =
theorem SModEq.symm {R : Type u_1} [Ring R] {M : Type u_3} [] [Module R M] {U : } {x : M} {y : M} (hxy : x y [SMOD U]) :
y x [SMOD U]
theorem SModEq.trans {R : Type u_1} [Ring R] {M : Type u_3} [] [Module R M] {U : } {x : M} {y : M} {z : M} (hxy : x y [SMOD U]) (hyz : y z [SMOD U]) :
x z [SMOD U]
instance SModEq.instTrans {R : Type u_1} [Ring R] {M : Type u_3} [] [Module R M] {U : } :
Trans () () ()
Equations
• SModEq.instTrans = { trans := }
theorem SModEq.add {R : Type u_1} [Ring R] {M : Type u_3} [] [Module R M] {U : } {x₁ : M} {x₂ : M} {y₁ : M} {y₂ : M} (hxy₁ : x₁ y₁ [SMOD U]) (hxy₂ : x₂ y₂ [SMOD U]) :
x₁ + x₂ y₁ + y₂ [SMOD U]
theorem SModEq.smul {R : Type u_1} [Ring R] {M : Type u_3} [] [Module R M] {U : } {x : M} {y : M} (hxy : x y [SMOD U]) (c : R) :
c x c y [SMOD U]
theorem SModEq.mul {A : Type u_2} [] {I : } {x₁ : A} {x₂ : A} {y₁ : A} {y₂ : A} (hxy₁ : x₁ y₁ [SMOD I]) (hxy₂ : x₂ y₂ [SMOD I]) :
x₁ * x₂ y₁ * y₂ [SMOD I]
theorem SModEq.zero {R : Type u_1} [Ring R] {M : Type u_3} [] [Module R M] {U : } {x : M} :
x 0 [SMOD U] x U
theorem SModEq.map {R : Type u_1} [Ring R] {M : Type u_3} [] [Module R M] {U : } {x : M} {y : M} {N : Type u_4} [] [Module R N] (hxy : x y [SMOD U]) (f : M →ₗ[R] N) :
f x f y [SMOD ]
theorem SModEq.comap {R : Type u_1} [Ring R] {M : Type u_3} [] [Module R M] {x : M} {y : M} {N : Type u_4} [] [Module R N] (V : ) {f : M →ₗ[R] N} (hxy : f x f y [SMOD V]) :
x y [SMOD ]
theorem SModEq.eval {R : Type u_5} [] {I : } {x : R} {y : R} (h : x y [SMOD I]) (f : ) :