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Mathlib.Algebra.Module.Basic

Further basic results about modules. #

@[simp]
theorem invOf_two_smul_add_invOf_two_smul {M : Type u_3} (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] [Invertible 2] (x : M) :
2 x + 2 x = x
@[deprecated map_natCast_smul]
theorem map_nat_cast_smul {M : Type u_3} {M₂ : Type u_4} [AddCommMonoid M] [AddCommMonoid M₂] {F : Type u_5} [FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (R : Type u_6) (S : Type u_7) [Semiring R] [Semiring S] [Module R M] [Module S M₂] (x : ) (a : M) :
f (x a) = x f a

Alias of map_natCast_smul.

theorem map_inv_natCast_smul {M : Type u_3} {M₂ : Type u_4} [AddCommMonoid M] [AddCommMonoid M₂] {F : Type u_5} [FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (R : Type u_6) (S : Type u_7) [DivisionSemiring R] [DivisionSemiring S] [Module R M] [Module S M₂] (n : ) (x : M) :
f ((↑n)⁻¹ x) = (↑n)⁻¹ f x
@[deprecated map_inv_natCast_smul]
theorem map_inv_nat_cast_smul {M : Type u_3} {M₂ : Type u_4} [AddCommMonoid M] [AddCommMonoid M₂] {F : Type u_5} [FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (R : Type u_6) (S : Type u_7) [DivisionSemiring R] [DivisionSemiring S] [Module R M] [Module S M₂] (n : ) (x : M) :
f ((↑n)⁻¹ x) = (↑n)⁻¹ f x

Alias of map_inv_natCast_smul.

theorem map_inv_intCast_smul {M : Type u_3} {M₂ : Type u_4} [AddCommGroup M] [AddCommGroup M₂] {F : Type u_5} [FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (R : Type u_6) (S : Type u_7) [DivisionRing R] [DivisionRing S] [Module R M] [Module S M₂] (z : ) (x : M) :
f ((↑z)⁻¹ x) = (↑z)⁻¹ f x
@[deprecated map_inv_intCast_smul]
theorem map_inv_int_cast_smul {M : Type u_3} {M₂ : Type u_4} [AddCommGroup M] [AddCommGroup M₂] {F : Type u_5} [FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (R : Type u_6) (S : Type u_7) [DivisionRing R] [DivisionRing S] [Module R M] [Module S M₂] (z : ) (x : M) :
f ((↑z)⁻¹ x) = (↑z)⁻¹ f x

Alias of map_inv_intCast_smul.

theorem inv_natCast_smul_eq {E : Type u_5} (R : Type u_6) (S : Type u_7) [AddCommMonoid E] [DivisionSemiring R] [DivisionSemiring S] [Module R E] [Module S E] (n : ) (x : E) :
(↑n)⁻¹ x = (↑n)⁻¹ x

If E is a vector space over two division semirings R and S, then scalar multiplications agree on inverses of natural numbers in R and S.

@[deprecated inv_natCast_smul_eq]
theorem inv_nat_cast_smul_eq {E : Type u_5} (R : Type u_6) (S : Type u_7) [AddCommMonoid E] [DivisionSemiring R] [DivisionSemiring S] [Module R E] [Module S E] (n : ) (x : E) :
(↑n)⁻¹ x = (↑n)⁻¹ x

Alias of inv_natCast_smul_eq.


If E is a vector space over two division semirings R and S, then scalar multiplications agree on inverses of natural numbers in R and S.

theorem inv_intCast_smul_eq {E : Type u_5} (R : Type u_6) (S : Type u_7) [AddCommGroup E] [DivisionRing R] [DivisionRing S] [Module R E] [Module S E] (n : ) (x : E) :
(↑n)⁻¹ x = (↑n)⁻¹ x

If E is a vector space over two division rings R and S, then scalar multiplications agree on inverses of integer numbers in R and S.

@[deprecated inv_intCast_smul_eq]
theorem inv_int_cast_smul_eq {E : Type u_5} (R : Type u_6) (S : Type u_7) [AddCommGroup E] [DivisionRing R] [DivisionRing S] [Module R E] [Module S E] (n : ) (x : E) :
(↑n)⁻¹ x = (↑n)⁻¹ x

Alias of inv_intCast_smul_eq.


If E is a vector space over two division rings R and S, then scalar multiplications agree on inverses of integer numbers in R and S.

theorem inv_natCast_smul_comm {α : Type u_5} {E : Type u_6} (R : Type u_7) [AddCommMonoid E] [DivisionSemiring R] [Monoid α] [Module R E] [DistribMulAction α E] (n : ) (s : α) (x : E) :
(↑n)⁻¹ s x = s (↑n)⁻¹ x

If E is a vector space over a division semiring R and has a monoid action by α, then that action commutes by scalar multiplication of inverses of natural numbers in R.

@[deprecated inv_natCast_smul_comm]
theorem inv_nat_cast_smul_comm {α : Type u_5} {E : Type u_6} (R : Type u_7) [AddCommMonoid E] [DivisionSemiring R] [Monoid α] [Module R E] [DistribMulAction α E] (n : ) (s : α) (x : E) :
(↑n)⁻¹ s x = s (↑n)⁻¹ x

Alias of inv_natCast_smul_comm.


If E is a vector space over a division semiring R and has a monoid action by α, then that action commutes by scalar multiplication of inverses of natural numbers in R.

theorem inv_intCast_smul_comm {α : Type u_5} {E : Type u_6} (R : Type u_7) [AddCommGroup E] [DivisionRing R] [Monoid α] [Module R E] [DistribMulAction α E] (n : ) (s : α) (x : E) :
(↑n)⁻¹ s x = s (↑n)⁻¹ x

If E is a vector space over a division ring R and has a monoid action by α, then that action commutes by scalar multiplication of inverses of integers in R

@[deprecated inv_intCast_smul_comm]
theorem inv_int_cast_smul_comm {α : Type u_5} {E : Type u_6} (R : Type u_7) [AddCommGroup E] [DivisionRing R] [Monoid α] [Module R E] [DistribMulAction α E] (n : ) (s : α) (x : E) :
(↑n)⁻¹ s x = s (↑n)⁻¹ x

Alias of inv_intCast_smul_comm.


If E is a vector space over a division ring R and has a monoid action by α, then that action commutes by scalar multiplication of inverses of integers in R

theorem Function.support_smul_subset_left {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] (f : αR) (g : αM) :
theorem Function.support_smul_subset_right {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero M] [SMulZeroClass R M] (f : αR) (g : αM) :
theorem Function.support_const_smul_of_ne_zero {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] [NoZeroSMulDivisors R M] (c : R) (g : αM) (hc : c 0) :
theorem Function.support_smul {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] [NoZeroSMulDivisors R M] (f : αR) (g : αM) :
theorem Function.support_const_smul_subset {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero M] [SMulZeroClass R M] (a : R) (f : αM) :
theorem Set.indicator_smul_apply {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero M] [SMulZeroClass R M] (s : Set α) (r : αR) (f : αM) (a : α) :
s.indicator (fun (a : α) => r a f a) a = r a s.indicator f a
theorem Set.indicator_smul {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero M] [SMulZeroClass R M] (s : Set α) (r : αR) (f : αM) :
(s.indicator fun (a : α) => r a f a) = fun (a : α) => r a s.indicator f a
theorem Set.indicator_const_smul_apply {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero M] [SMulZeroClass R M] (s : Set α) (r : R) (f : αM) (a : α) :
s.indicator (fun (x : α) => r f x) a = r s.indicator f a
theorem Set.indicator_const_smul {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero M] [SMulZeroClass R M] (s : Set α) (r : R) (f : αM) :
(s.indicator fun (x : α) => r f x) = fun (x : α) => r s.indicator f x
theorem Set.indicator_smul_apply_left {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] (s : Set α) (r : αR) (f : αM) (a : α) :
s.indicator (fun (a : α) => r a f a) a = s.indicator r a f a
theorem Set.indicator_smul_left {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] (s : Set α) (r : αR) (f : αM) :
(s.indicator fun (a : α) => r a f a) = fun (a : α) => s.indicator r a f a
theorem Set.indicator_smul_const_apply {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] (s : Set α) (r : αR) (m : M) (a : α) :
s.indicator (fun (x : α) => r x m) a = s.indicator r a m
theorem Set.indicator_smul_const {α : Type u_1} {R : Type u_2} {M : Type u_3} [Zero R] [Zero M] [SMulWithZero R M] (s : Set α) (r : αR) (m : M) :
(s.indicator fun (x : α) => r x m) = fun (x : α) => s.indicator r x m
theorem Set.smul_indicator_one_apply {α : Type u_1} {R : Type u_2} [MulZeroOneClass R] (s : Set α) (r : R) (a : α) :
r s.indicator 1 a = s.indicator (fun (x : α) => r) a