statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
add_monoid_hom.coe_to_int_linear_map [add_comm_group M] [add_comm_group M₂]
(f : M →+ M₂) :
⇑f.to_int_linear_map = f | rfl | lemma | add_monoid_hom.coe_to_int_linear_map | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_hom.to_rat_linear_map [add_comm_group M] [module ℚ M]
[add_comm_group M₂] [module ℚ M₂] (f : M →+ M₂) :
M →ₗ[ℚ] M₂ | { map_smul' := map_rat_smul f, ..f } | def | add_monoid_hom.to_rat_linear_map | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_group",
"map_rat_smul",
"module"
] | Reinterpret an additive homomorphism as a `ℚ`-linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_hom.to_rat_linear_map_injective
[add_comm_group M] [module ℚ M] [add_comm_group M₂] [module ℚ M₂] :
function.injective (@add_monoid_hom.to_rat_linear_map M M₂ _ _ _ _) | by { intros f g h, ext, exact linear_map.congr_fun h x } | lemma | add_monoid_hom.to_rat_linear_map_injective | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_group",
"add_monoid_hom.to_rat_linear_map",
"linear_map.congr_fun",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_hom.coe_to_rat_linear_map [add_comm_group M] [module ℚ M]
[add_comm_group M₂] [module ℚ M₂] (f : M →+ M₂) :
⇑f.to_rat_linear_map = f | rfl | lemma | add_monoid_hom.coe_to_rat_linear_map | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_group",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_apply (a : S) (f : M →ₛₗ[σ₁₂] M₂) (x : M) : (a • f) x = a • f x | rfl | lemma | linear_map.smul_apply | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (a : S) (f : M →ₛₗ[σ₁₂] M₂) : ⇑(a • f) = a • f | rfl | lemma | linear_map.coe_smul | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (x : M) : (0 : M →ₛₗ[σ₁₂] M₂) x = 0 | rfl | lemma | linear_map.zero_apply | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_zero (g : M₂ →ₛₗ[σ₂₃] M₃) : (g.comp (0 : M →ₛₗ[σ₁₂] M₂) : M →ₛₗ[σ₁₃] M₃) = 0 | ext $ assume c, by rw [comp_apply, zero_apply, zero_apply, g.map_zero] | theorem | linear_map.comp_zero | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_comp (f : M →ₛₗ[σ₁₂] M₂) : ((0 : M₂ →ₛₗ[σ₂₃] M₃).comp f : M →ₛₗ[σ₁₃] M₃) = 0 | rfl | theorem | linear_map.zero_comp | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
default_def : (default : (M →ₛₗ[σ₁₂] M₂)) = 0 | rfl | lemma | linear_map.default_def | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_apply (f g : M →ₛₗ[σ₁₂] M₂) (x : M) : (f + g) x = f x + g x | rfl | lemma | linear_map.add_apply | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comp (f : M →ₛₗ[σ₁₂] M₂) (g h : M₂ →ₛₗ[σ₂₃] M₃) :
((h + g).comp f : M →ₛₗ[σ₁₃] M₃) = h.comp f + g.comp f | rfl | lemma | linear_map.add_comp | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_add (f g : M →ₛₗ[σ₁₂] M₂) (h : M₂ →ₛₗ[σ₂₃] M₃) :
(h.comp (f + g) : M →ₛₗ[σ₁₃] M₃) = h.comp f + h.comp g | ext $ λ _, h.map_add _ _ | lemma | linear_map.comp_add | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_apply (f : M →ₛₗ[σ₁₂] N₂) (x : M) : (- f) x = - f x | rfl | lemma | linear_map.neg_apply | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_comp (f : M →ₛₗ[σ₁₂] M₂) (g : M₂ →ₛₗ[σ₂₃] N₃) : (- g).comp f = - g.comp f | rfl | lemma | linear_map.neg_comp | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_neg (f : M →ₛₗ[σ₁₂] N₂) (g : N₂ →ₛₗ[σ₂₃] N₃) : g.comp (- f) = - g.comp f | ext $ λ _, g.map_neg _ | lemma | linear_map.comp_neg | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_apply (f g : M →ₛₗ[σ₁₂] N₂) (x : M) : (f - g) x = f x - g x | rfl | lemma | linear_map.sub_apply | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_comp (f : M →ₛₗ[σ₁₂] M₂) (g h : M₂ →ₛₗ[σ₂₃] N₃) :
(g - h).comp f = g.comp f - h.comp f | rfl | lemma | linear_map.sub_comp | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_sub (f g : M →ₛₗ[σ₁₂] N₂) (h : N₂ →ₛₗ[σ₂₃] N₃) :
h.comp (g - f) = h.comp g - h.comp f | ext $ λ _, h.map_sub _ _ | lemma | linear_map.comp_sub | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comp (a : S₃) (g : M₂ →ₛₗ[σ₂₃] M₃) (f : M →ₛₗ[σ₁₂] M₂) :
(a • g).comp f = a • (g.comp f) | rfl | theorem | linear_map.smul_comp | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_smul [module R M₂] [module R M₃] [smul_comm_class R S M₂] [distrib_mul_action S M₃]
[smul_comm_class R S M₃] [compatible_smul M₃ M₂ S R]
(g : M₃ →ₗ[R] M₂) (a : S) (f : M →ₗ[R] M₃) : g.comp (a • f) = a • (g.comp f) | ext $ λ x, g.map_smul_of_tower _ _ | theorem | linear_map.comp_smul | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"distrib_mul_action",
"module",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_eq_id : (1 : module.End R M) = id | rfl | lemma | linear_map.one_eq_id | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_comp (f g : module.End R M) : f * g = f.comp g | rfl | lemma | linear_map.mul_eq_comp | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply (x : M) : (1 : module.End R M) x = x | rfl | lemma | linear_map.one_apply | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply (f g : module.End R M) (x : M) : (f * g) x = f (g x) | rfl | lemma | linear_map.mul_apply | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ⇑(1 : module.End R M) = _root_.id | rfl | lemma | linear_map.coe_one | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (f g : module.End R M) : ⇑(f * g) = f ∘ g | rfl | lemma | linear_map.coe_mul | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.module.End.monoid : monoid (module.End R M) | { mul := (*),
one := (1 : M →ₗ[R] M),
mul_assoc := λ f g h, linear_map.ext $ λ x, rfl,
mul_one := comp_id,
one_mul := id_comp } | instance | module.End.monoid | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"linear_map.ext",
"module.End",
"monoid",
"mul_assoc",
"mul_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.module.End.semiring : semiring (module.End R M) | { mul := (*),
one := (1 : M →ₗ[R] M),
zero := 0,
add := (+),
mul_zero := comp_zero,
zero_mul := zero_comp,
left_distrib := λ f g h, comp_add _ _ _,
right_distrib := λ f g h, add_comp _ _ _,
nat_cast := λ n, n • 1,
nat_cast_zero := add_monoid.nsmul_zero' _,
nat_cast_succ := λ n, (add_monoid.nsmul_suc... | instance | module.End.semiring | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_monoid_with_one.unary",
"left_distrib",
"module.End",
"mul_zero",
"right_distrib",
"semiring",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.module.End.nat_cast_apply (n : ℕ) (m : M) :
(↑n : module.End R M) m = n • m | rfl | lemma | module.End.nat_cast_apply | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"module.End"
] | See also `module.End.nat_cast_def`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.module.End.ring : ring (module.End R N₁) | { int_cast := λ z, z • 1,
int_cast_of_nat := of_nat_zsmul _,
int_cast_neg_succ_of_nat := zsmul_neg_succ_of_nat _,
..module.End.semiring, ..linear_map.add_comm_group } | instance | module.End.ring | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"module.End",
"module.End.semiring",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.module.End.int_cast_apply (z : ℤ) (m : N₁) :
(↑z : module.End R N₁) m = z • m | rfl | lemma | module.End.int_cast_apply | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"module.End"
] | See also `module.End.int_cast_def`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.module.End.is_scalar_tower :
is_scalar_tower S (module.End R M) (module.End R M) | ⟨smul_comp⟩ | instance | module.End.is_scalar_tower | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"is_scalar_tower",
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.module.End.smul_comm_class [has_smul S R] [is_scalar_tower S R M] :
smul_comm_class S (module.End R M) (module.End R M) | ⟨λ s _ _, (comp_smul _ s _).symm⟩ | instance | module.End.smul_comm_class | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"has_smul",
"is_scalar_tower",
"module.End",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.module.End.smul_comm_class' [has_smul S R] [is_scalar_tower S R M] :
smul_comm_class (module.End R M) S (module.End R M) | smul_comm_class.symm _ _ _ | instance | module.End.smul_comm_class' | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"has_smul",
"is_scalar_tower",
"module.End",
"smul_comm_class",
"smul_comm_class.symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_module : module (module.End R M) M | { smul := ($),
smul_zero := linear_map.map_zero,
smul_add := linear_map.map_add,
add_smul := linear_map.add_apply,
zero_smul := (linear_map.zero_apply : ∀ m, (0 : M →ₗ[R] M) m = 0),
one_smul := λ _, rfl,
mul_smul := λ _ _ _, rfl } | instance | linear_map.apply_module | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_smul",
"linear_map.add_apply",
"linear_map.map_add",
"linear_map.map_zero",
"linear_map.zero_apply",
"module",
"module.End",
"one_smul",
"smul_add",
"smul_zero",
"zero_smul"
] | The tautological action by `module.End R M` (aka `M →ₗ[R] M`) on `M`.
This generalizes `function.End.apply_mul_action`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_def (f : module.End R M) (a : M) : f • a = f a | rfl | lemma | linear_map.smul_def | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_has_faithful_smul : has_faithful_smul (module.End R M) M | ⟨λ _ _, linear_map.ext⟩ | instance | linear_map.apply_has_faithful_smul | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"has_faithful_smul",
"module.End"
] | `linear_map.apply_module` is faithful. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_smul_comm_class : smul_comm_class R (module.End R M) M | { smul_comm := λ r e m, (e.map_smul r m).symm } | instance | linear_map.apply_smul_comm_class | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"module.End",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_smul_comm_class' : smul_comm_class (module.End R M) R M | { smul_comm := linear_map.map_smul } | instance | linear_map.apply_smul_comm_class' | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"linear_map.map_smul",
"module.End",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_is_scalar_tower {R M : Type*} [comm_semiring R] [add_comm_monoid M] [module R M] :
is_scalar_tower R (module.End R M) M | ⟨λ t f m, rfl⟩ | instance | linear_map.apply_is_scalar_tower | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_monoid",
"comm_semiring",
"is_scalar_tower",
"module",
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_linear_map (s : S) : M →ₗ[R] M | { to_fun := has_smul.smul s,
map_add' := smul_add s,
map_smul' := λ a b, smul_comm _ _ _ } | def | distrib_mul_action.to_linear_map | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"smul_add"
] | Each element of the monoid defines a linear map.
This is a stronger version of `distrib_mul_action.to_add_monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_module_End : S →* module.End R M | { to_fun := to_linear_map R M,
map_one' := linear_map.ext $ one_smul _,
map_mul' := λ a b, linear_map.ext $ mul_smul _ _ } | def | distrib_mul_action.to_module_End | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"linear_map.ext",
"module.End",
"one_smul"
] | Each element of the monoid defines a module endomorphism.
This is a stronger version of `distrib_mul_action.to_add_monoid_End`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_module_End : S →+* module.End R M | { to_fun := distrib_mul_action.to_linear_map R M,
map_zero' := linear_map.ext $ zero_smul _,
map_add' := λ f g, linear_map.ext $ add_smul _ _,
..distrib_mul_action.to_module_End R M } | def | module.to_module_End | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_smul",
"distrib_mul_action.to_linear_map",
"distrib_mul_action.to_module_End",
"linear_map.ext",
"module.End",
"zero_smul"
] | Each element of the semiring defines a module endomorphism.
This is a stronger version of `distrib_mul_action.to_module_End`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
module_End_self : Rᵐᵒᵖ ≃+* module.End R R | { to_fun := distrib_mul_action.to_linear_map R R,
inv_fun := λ f, mul_opposite.op (f 1),
left_inv := mul_one,
right_inv := λ f, linear_map.ext_ring $ one_mul _,
..module.to_module_End R R } | def | module.module_End_self | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"distrib_mul_action.to_linear_map",
"inv_fun",
"linear_map.ext_ring",
"module.End",
"module.to_module_End",
"mul_one",
"mul_opposite.op",
"one_mul"
] | The canonical (semi)ring isomorphism from `Rᵐᵒᵖ` to `module.End R R` induced by the right
multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
module_End_self_op : R ≃+* module.End Rᵐᵒᵖ R | { to_fun := distrib_mul_action.to_linear_map _ _,
inv_fun := λ f, f 1,
left_inv := mul_one,
right_inv := λ f, linear_map.ext_ring_op $ mul_one _,
..module.to_module_End _ _ } | def | module.module_End_self_op | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"distrib_mul_action.to_linear_map",
"inv_fun",
"linear_map.ext_ring_op",
"module.End",
"module.to_module_End",
"mul_one"
] | The canonical (semi)ring isomorphism from `R` to `module.End Rᵐᵒᵖ R` induced by the left
multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
End.nat_cast_def (n : ℕ) [add_comm_monoid N₁] [module R N₁] :
(↑n : module.End R N₁) = module.to_module_End R N₁ n | rfl | lemma | module.End.nat_cast_def | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_monoid",
"module",
"module.End",
"module.to_module_End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
End.int_cast_def (z : ℤ) [add_comm_group N₁] [module R N₁] :
(↑z : module.End R N₁) = module.to_module_End R N₁ z | rfl | lemma | module.End.int_cast_def | algebra.module | src/algebra/module/linear_map.lean | [
"algebra.hom.group_action",
"algebra.module.pi",
"algebra.star.basic",
"data.set.pointwise.smul",
"algebra.ring.comp_typeclasses"
] | [
"add_comm_group",
"module",
"module.End",
"module.to_module_End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
r (a b : M × S) : Prop | ∃ (u : S), u • b.2 • a.1 = u • a.2 • b.1 | def | localized_module.r | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [] | The equivalence relation on `M × S` where `(m1, s1) ≈ (m2, s2)` if and only if
for some (u : S), u * (s2 • m1 - s1 • m2) = 0 | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
r.is_equiv : is_equiv _ (r S M) | { refl := λ ⟨m, s⟩, ⟨1, by rw [one_smul]⟩,
trans := λ ⟨m1, s1⟩ ⟨m2, s2⟩ ⟨m3, s3⟩ ⟨u1, hu1⟩ ⟨u2, hu2⟩, begin
use u1 * u2 * s2,
-- Put everything in the same shape, sorting the terms using `simp`
have hu1' := congr_arg ((•) (u2 * s3)) hu1.symm,
have hu2' := congr_arg ((•) (u1 * s1)) hu2.symm,
simp o... | lemma | localized_module.r.is_equiv | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"mul_assoc",
"mul_comm",
"mul_left_comm",
"one_smul",
"smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
r.setoid : setoid (M × S) | { r := r S M,
iseqv := ⟨(r.is_equiv S M).refl, (r.is_equiv S M).symm, (r.is_equiv S M).trans⟩ } | instance | localized_module.r.setoid | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.localized_module : Type (max u v) | quotient (r.setoid S M) | def | localized_module | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [] | If `S` is a multiplicative subset of a ring `R` and `M` an `R`-module, then
we can localize `M` by `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk (m : M) (s : S) : localized_module S M | quotient.mk ⟨m, s⟩ | def | localized_module.mk | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module"
] | The canonical map sending `(m, s) ↦ m/s` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_eq {m m' : M} {s s' : S} : mk m s = mk m' s' ↔ ∃ (u : S), u • s' • m = u • s • m' | quotient.eq | lemma | localized_module.mk_eq | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"quotient.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on {β : localized_module S M → Prop} (h : ∀ (m : M) (s : S), β (mk m s)) :
∀ (x : localized_module S M), β x | by { rintro ⟨⟨m, s⟩⟩, exact h m s } | lemma | localized_module.induction_on | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on₂ {β : localized_module S M → localized_module S M → Prop}
(h : ∀ (m m' : M) (s s' : S), β (mk m s) (mk m' s')) : ∀ x y, β x y | by { rintro ⟨⟨m, s⟩⟩ ⟨⟨m', s'⟩⟩, exact h m m' s s' } | lemma | localized_module.induction_on₂ | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_on {α : Type*} (x : localized_module S M) (f : M × S → α)
(wd : ∀ (p p' : M × S) (h1 : p ≈ p'), f p = f p') : α | quotient.lift_on x f wd | def | localized_module.lift_on | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module"
] | If `f : M × S → α` respects the equivalence relation `localized_module.r`, then
`f` descents to a map `localized_module M S → α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_on_mk {α : Type*} {f : M × S → α}
(wd : ∀ (p p' : M × S) (h1 : p ≈ p'), f p = f p')
(m : M) (s : S) :
lift_on (mk m s) f wd = f ⟨m, s⟩ | by convert quotient.lift_on_mk f wd ⟨m, s⟩ | lemma | localized_module.lift_on_mk | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"quotient.lift_on_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_on₂ {α : Type*} (x y : localized_module S M) (f : (M × S) → (M × S) → α)
(wd : ∀ (p q p' q' : M × S) (h1 : p ≈ p') (h2 : q ≈ q'), f p q = f p' q') : α | quotient.lift_on₂ x y f wd | def | localized_module.lift_on₂ | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module"
] | If `f : M × S → M × S → α` respects the equivalence relation `localized_module.r`, then
`f` descents to a map `localized_module M S → localized_module M S → α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_on₂_mk {α : Type*} (f : (M × S) → (M × S) → α)
(wd : ∀ (p q p' q' : M × S) (h1 : p ≈ p') (h2 : q ≈ q'), f p q = f p' q')
(m m' : M) (s s' : S) :
lift_on₂ (mk m s) (mk m' s') f wd = f ⟨m, s⟩ ⟨m', s'⟩ | by convert quotient.lift_on₂_mk f wd _ _ | lemma | localized_module.lift_on₂_mk | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"quotient.lift_on₂_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_mk (s : S) : mk (0 : M) s = 0 | mk_eq.mpr ⟨1, by rw [one_smul, smul_zero, smul_zero, one_smul]⟩ | lemma | localized_module.zero_mk | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"one_smul",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_add_mk {m1 m2 : M} {s1 s2 : S} :
mk m1 s1 + mk m2 s2 = mk (s2 • m1 + s1 • m2) (s1 * s2) | mk_eq.mpr $ ⟨1, by dsimp only; rw [one_smul]⟩ | lemma | localized_module.mk_add_mk | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_assoc' (x y z : localized_module S M) :
x + y + z = x + (y + z) | begin
induction x using localized_module.induction_on with mx sx,
induction y using localized_module.induction_on with my sy,
induction z using localized_module.induction_on with mz sz,
simp only [mk_add_mk, smul_add],
refine mk_eq.mpr ⟨1, _⟩,
rw [one_smul, one_smul],
congr' 1,
{ rw [mul_assoc] },
{ r... | lemma | localized_module.add_assoc' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module",
"localized_module.induction_on",
"mul_assoc",
"mul_comm",
"one_smul",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm' (x y : localized_module S M) :
x + y = y + x | localized_module.induction_on₂ (λ m m' s s', by rw [mk_add_mk, mk_add_mk, add_comm, mul_comm]) x y | lemma | localized_module.add_comm' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module",
"localized_module.induction_on₂",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_add' (x : localized_module S M) : 0 + x = x | induction_on (λ m s, by rw [← zero_mk s, mk_add_mk, smul_zero, zero_add, mk_eq];
exact ⟨1, by rw [one_smul, mul_smul, one_smul]⟩) x | lemma | localized_module.zero_add' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module",
"one_smul",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_zero' (x : localized_module S M) : x + 0 = x | induction_on (λ m s, by rw [← zero_mk s, mk_add_mk, smul_zero, add_zero, mk_eq];
exact ⟨1, by rw [one_smul, mul_smul, one_smul]⟩) x | lemma | localized_module.add_zero' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module",
"one_smul",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_nat_smul : has_smul ℕ (localized_module S M) | { smul := λ n, nsmul_rec n } | instance | localized_module.has_nat_smul | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"has_smul",
"localized_module",
"nsmul_rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_zero' (x : localized_module S M) : (0 : ℕ) • x = 0 | localized_module.induction_on (λ _ _, rfl) x | lemma | localized_module.nsmul_zero' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module",
"localized_module.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_succ' (n : ℕ) (x : localized_module S M) :
n.succ • x = x + n • x | localized_module.induction_on (λ _ _, rfl) x | lemma | localized_module.nsmul_succ' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module",
"localized_module.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_neg {M : Type*} [add_comm_group M] [module R M] {m : M} {s : S} :
mk (-m) s = - mk m s | rfl | lemma | localized_module.mk_neg | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"add_comm_group",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_mul_mk {A : Type*} [semiring A] [algebra R A] {a₁ a₂ : A} {s₁ s₂ : S} :
mk a₁ s₁ * mk a₂ s₂ = mk (a₁ * a₂) (s₁ * s₂) | rfl | lemma | localized_module.mk_mul_mk | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_smul_mk (r : R) (m : M) (s t : S) :
localization.mk r s • mk m t = mk (r • m) (s * t) | begin
unfold has_smul.smul,
rw [localization.lift_on_mk, lift_on_mk],
end | lemma | localized_module.mk_smul_mk | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localization.lift_on_mk",
"localization.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_smul' (m : localized_module S M) :
(1 : localization S) • m = m | begin
induction m using localized_module.induction_on with m s,
rw [← localization.mk_one, mk_smul_mk, one_smul, one_mul],
end | lemma | localized_module.one_smul' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localization",
"localization.mk_one",
"localized_module",
"localized_module.induction_on",
"one_mul",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_smul' (x y : localization S) (m : localized_module S M) :
(x * y) • m = x • y • m | begin
induction x using localization.induction_on with data,
induction y using localization.induction_on with data',
rcases ⟨data, data'⟩ with ⟨⟨r, s⟩, ⟨r', s'⟩⟩,
induction m using localized_module.induction_on with m t,
rw [localization.mk_mul, mk_smul_mk, mk_smul_mk, mk_smul_mk, mul_smul, mul_assoc],
end | lemma | localized_module.mul_smul' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localization",
"localization.induction_on",
"localization.mk_mul",
"localized_module",
"localized_module.induction_on",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_add' (x : localization S) (y z : localized_module S M) :
x • (y + z) = x • y + x • z | begin
induction x using localization.induction_on with data,
rcases data with ⟨r, u⟩,
induction y using localized_module.induction_on with m s,
induction z using localized_module.induction_on with n t,
rw [mk_smul_mk, mk_smul_mk, mk_add_mk, mk_smul_mk, mk_add_mk, mk_eq],
use 1,
simp only [one_smul, smul_a... | lemma | localized_module.smul_add' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localization",
"localization.induction_on",
"localized_module",
"localized_module.induction_on",
"one_smul",
"smul_add",
"submonoid.coe_mul",
"submonoid.smul_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_zero' (x : localization S) :
x • (0 : localized_module S M) = 0 | begin
induction x using localization.induction_on with data,
rcases data with ⟨r, s⟩,
rw [←zero_mk s, mk_smul_mk, smul_zero, zero_mk, zero_mk],
end | lemma | localized_module.smul_zero' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localization",
"localization.induction_on",
"localized_module",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_smul' (x y : localization S) (z : localized_module S M) :
(x + y) • z = x • z + y • z | begin
induction x using localization.induction_on with datax,
induction y using localization.induction_on with datay,
induction z using localized_module.induction_on with m t,
rcases ⟨datax, datay⟩ with ⟨⟨r, s⟩, ⟨r', s'⟩⟩,
rw [localization.add_mk, mk_smul_mk, mk_smul_mk, mk_smul_mk, mk_add_mk, mk_eq],
use 1... | lemma | localized_module.add_smul' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"add_smul",
"localization",
"localization.add_mk",
"localization.induction_on",
"localized_module",
"localized_module.induction_on",
"one_smul",
"smul_add",
"submonoid.coe_mul",
"submonoid.coe_one",
"submonoid.smul_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_smul' (x : localized_module S M) :
(0 : localization S) • x = 0 | begin
induction x using localized_module.induction_on with m s,
rw [← localization.mk_zero s, mk_smul_mk, zero_smul, zero_mk],
end | lemma | localized_module.zero_smul' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localization",
"localization.mk_zero",
"localized_module",
"localized_module.induction_on",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_module : module (localization S) (localized_module S M) | { smul := (•),
one_smul := one_smul',
mul_smul := mul_smul',
smul_add := smul_add',
smul_zero := smul_zero',
add_smul := add_smul',
zero_smul := zero_smul' } | instance | localized_module.is_module | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"add_smul",
"localization",
"localized_module",
"module",
"one_smul",
"smul_add",
"smul_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_cancel_common_left (s' s : S) (m : M) : mk (s' • m) (s' * s) = mk m s | mk_eq.mpr ⟨1, by { simp only [mul_smul, one_smul], rw smul_comm }⟩ | lemma | localized_module.mk_cancel_common_left | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_cancel (s : S) (m : M) : mk (s • m) s = mk m 1 | mk_eq.mpr ⟨1, by simp⟩ | lemma | localized_module.mk_cancel | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_cancel_common_right (s s' : S) (m : M) : mk (s' • m) (s * s') = mk m s | mk_eq.mpr ⟨1, by simp [mul_smul]⟩ | lemma | localized_module.mk_cancel_common_right | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_module' : module R (localized_module S M) | { ..module.comp_hom (localized_module S M) $ (algebra_map R (localization S)) } | instance | localized_module.is_module' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra_map",
"localization",
"localized_module",
"module",
"module.comp_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul'_mk (r : R) (s : S) (m : M) : r • mk m s = mk (r • m) s | by erw [mk_smul_mk r m 1 s, one_mul] | lemma | localized_module.smul'_mk | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_mk {A : Type*} [semiring A] [algebra R A] (a : R) (s : S) :
algebra_map _ _ (localization.mk a s) = mk (algebra_map R A a) s | begin
rw [algebra.algebra_map_eq_smul_one],
change _ • mk _ _ = _,
rw [mk_smul_mk, algebra.algebra_map_eq_smul_one, mul_one]
end | lemma | localized_module.algebra_map_mk | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra",
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"localization.mk",
"mul_one",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra' {A : Type*} [semiring A] [algebra R A] :
algebra R (localized_module S A) | { commutes' := begin
intros r x,
obtain ⟨⟨a, s⟩, rfl : mk a s = x⟩ := quotient.exists_rep x,
dsimp,
rw [← localization.mk_one_eq_algebra_map, algebra_map_mk, mk_mul_mk, mk_mul_mk, mul_comm,
algebra.commutes],
end,
smul_def' := begin
intros r x,
obtain ⟨⟨a, s⟩, rfl : mk a s = x⟩ := quot... | instance | localized_module.algebra' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra",
"algebra.commutes",
"algebra.smul_def",
"algebra_map",
"localization",
"localization.mk_one_eq_algebra_map",
"localized_module",
"module",
"mul_comm",
"one_mul",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_linear_map : M →ₗ[R] localized_module S M | { to_fun := λ m, mk m 1,
map_add' := λ x y, by simp [mk_add_mk],
map_smul' := λ r x, (smul'_mk _ _ _).symm } | def | localized_module.mk_linear_map | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module"
] | The function `m ↦ m / 1` as an `R`-linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_by (s : S) : localized_module S M →ₗ[R] localized_module S M | { to_fun := λ p, p.lift_on (λ p, mk p.1 (s * p.2)) $ λ ⟨a, b⟩ ⟨a', b'⟩ ⟨c, eq1⟩, mk_eq.mpr ⟨c,
begin
rw [mul_smul, mul_smul, smul_comm c, eq1, smul_comm s];
apply_instance,
end⟩,
map_add' := λ x y, x.induction_on₂
(begin
intros m₁ m₂ t₁ t₂,
simp only [mk_add_mk, localized_module.lift_on_mk... | def | localized_module.div_by | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"localized_module",
"localized_module.lift_on_mk",
"mul_assoc",
"ring",
"submonoid.coe_mul"
] | For any `s : S`, there is an `R`-linear map given by `a/b ↦ a/(b*s)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_by_mul_by (s : S) (p : localized_module S M) :
div_by s (algebra_map R (module.End R (localized_module S M)) s p) = p | p.induction_on
begin
intros m t,
simp only [localized_module.lift_on_mk, module.algebra_map_End_apply, smul'_mk, div_by_apply],
erw mk_cancel_common_left s t,
end | lemma | localized_module.div_by_mul_by | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra_map",
"localized_module",
"localized_module.lift_on_mk",
"module.End",
"module.algebra_map_End_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_by_div_by (s : S) (p : localized_module S M) :
algebra_map R (module.End R (localized_module S M)) s (div_by s p) = p | p.induction_on
begin
intros m t,
simp only [localized_module.lift_on_mk, div_by_apply, module.algebra_map_End_apply, smul'_mk],
erw mk_cancel_common_left s t,
end | lemma | localized_module.mul_by_div_by | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra_map",
"localized_module",
"localized_module.lift_on_mk",
"module.End",
"module.algebra_map_End_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localized_module : Prop | (map_units [] : ∀ (x : S), is_unit (algebra_map R (module.End R M') x))
(surj [] : ∀ y : M', ∃ (x : M × S), x.2 • y = f x.1)
(eq_iff_exists [] : ∀ {x₁ x₂}, f x₁ = f x₂ ↔ ∃ c : S, c • x₂ = c • x₁) | class | is_localized_module | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra_map",
"is_unit",
"module.End"
] | The characteristic predicate for localized module.
`is_localized_module S f` describes that `f : M ⟶ M'` is the localization map identifying `M'` as
`localized_module S M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift' (g : M →ₗ[R] M'')
(h : ∀ (x : S), is_unit ((algebra_map R (module.End R M'')) x)) :
(localized_module S M) → M'' | λ m, m.lift_on (λ p, (h $ p.2).unit⁻¹ $ g p.1) $ λ ⟨m, s⟩ ⟨m', s'⟩ ⟨c, eq1⟩,
begin
generalize_proofs h1 h2,
erw [module.End_algebra_map_is_unit_inv_apply_eq_iff, ←h2.unit⁻¹.1.map_smul], symmetry,
erw [module.End_algebra_map_is_unit_inv_apply_eq_iff], dsimp,
have : c • s • g m' = c • s' • g m,
{ erw [←g.map_sm... | def | localized_module.lift' | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra_map",
"is_unit",
"is_unit.mul_coe_inv",
"linear_map.one_apply",
"localized_module",
"module.End",
"module.End_algebra_map_is_unit_inv_apply_eq_iff",
"units.inv_eq_coe_inv"
] | If `g` is a linear map `M → M''` such that all scalar multiplication by `s : S` is invertible, then
there is a linear map `localized_module S M → M''`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift'_mk (g : M →ₗ[R] M'')
(h : ∀ (x : S), is_unit ((algebra_map R (module.End R M'')) x)) (m : M) (s : S) :
localized_module.lift' S g h (localized_module.mk m s) =
(h s).unit⁻¹.1 (g m) | rfl | lemma | localized_module.lift'_mk | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra_map",
"is_unit",
"localized_module.lift'",
"localized_module.mk",
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_add (g : M →ₗ[R] M'')
(h : ∀ (x : S), is_unit ((algebra_map R (module.End R M'')) x)) (x y) :
localized_module.lift' S g h (x + y) =
localized_module.lift' S g h x + localized_module.lift' S g h y | localized_module.induction_on₂ begin
intros a a' b b',
erw [localized_module.lift'_mk, localized_module.lift'_mk, localized_module.lift'_mk],
dsimp, generalize_proofs h1 h2 h3,
erw [map_add, module.End_algebra_map_is_unit_inv_apply_eq_iff,
smul_add, ←h2.unit⁻¹.1.map_smul, ←h3.unit⁻¹.1.map_smul],
congr' 1;... | lemma | localized_module.lift'_add | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra_map",
"is_unit",
"localized_module.induction_on₂",
"localized_module.lift'",
"localized_module.lift'_mk",
"module.End",
"module.End_algebra_map_is_unit_inv_apply_eq_iff",
"mul_comm",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift'_smul (g : M →ₗ[R] M'')
(h : ∀ (x : S), is_unit ((algebra_map R (module.End R M'')) x))
(r : R) (m) :
r • localized_module.lift' S g h m = localized_module.lift' S g h (r • m) | m.induction_on begin
intros a b,
rw [localized_module.lift'_mk, localized_module.smul'_mk, localized_module.lift'_mk],
generalize_proofs h1 h2,
erw [←h1.unit⁻¹.1.map_smul, ←g.map_smul],
end | lemma | lift'_smul | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra_map",
"is_unit",
"localized_module.lift'",
"localized_module.lift'_mk",
"localized_module.smul'_mk",
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (g : M →ₗ[R] M'')
(h : ∀ (x : S), is_unit ((algebra_map R (module.End R M'')) x)) :
(localized_module S M) →ₗ[R] M'' | { to_fun := localized_module.lift' S g h,
map_add' := localized_module.lift'_add S g h,
map_smul' := λ r x, by rw [localized_module.lift'_smul, ring_hom.id_apply] } | def | lift | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra_map",
"is_unit",
"localized_module",
"localized_module.lift'",
"localized_module.lift'_add",
"module.End",
"ring_hom.id_apply"
] | If `g` is a linear map `M → M''` such that all scalar multiplication by `s : S` is invertible, then
there is a linear map `localized_module S M → M''`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_mk (g : M →ₗ[R] M'')
(h : ∀ (x : S), is_unit ((algebra_map R (module.End R M'')) x))
(m : M) (s : S) :
localized_module.lift S g h (localized_module.mk m s) = (h s).unit⁻¹.1 (g m) | rfl | lemma | lift_mk | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra_map",
"is_unit",
"localized_module.mk",
"module.End"
] | If `g` is a linear map `M → M''` such that all scalar multiplication by `s : S` is invertible, then
`lift g m s = s⁻¹ • g m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_comp (g : M →ₗ[R] M'')
(h : ∀ (x : S), is_unit ((algebra_map R (module.End R M'')) x)) :
(lift S g h).comp (mk_linear_map S M) = g | begin
ext x, dsimp, rw localized_module.lift_mk,
erw [module.End_algebra_map_is_unit_inv_apply_eq_iff, one_smul],
end | lemma | lift_comp | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra_map",
"is_unit",
"lift",
"module.End",
"module.End_algebra_map_is_unit_inv_apply_eq_iff",
"one_smul"
] | If `g` is a linear map `M → M''` such that all scalar multiplication by `s : S` is invertible, then
there is a linear map `lift g ∘ mk_linear_map = g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_unique (g : M →ₗ[R] M'')
(h : ∀ (x : S), is_unit ((algebra_map R (module.End R M'')) x))
(l : localized_module S M →ₗ[R] M'')
(hl : l.comp (localized_module.mk_linear_map S M) = g) :
localized_module.lift S g h = l | begin
ext x, induction x using localized_module.induction_on with m s,
rw [localized_module.lift_mk],
erw [module.End_algebra_map_is_unit_inv_apply_eq_iff, ←hl, linear_map.coe_comp, function.comp_app,
localized_module.mk_linear_map_apply, ←l.map_smul, localized_module.smul'_mk],
congr' 1, rw localized_modul... | lemma | lift_unique | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"algebra_map",
"is_unit",
"linear_map.coe_comp",
"localized_module",
"localized_module.induction_on",
"localized_module.mk_eq",
"localized_module.mk_linear_map",
"localized_module.smul'_mk",
"module.End",
"module.End_algebra_map_is_unit_inv_apply_eq_iff",
"one_smul"
] | If `g` is a linear map `M → M''` such that all scalar multiplication by `s : S` is invertible and
`l` is another linear map `localized_module S M ⟶ M''` such that `l ∘ mk_linear_map = g` then
`l = lift g` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
localized_module_is_localized_module :
is_localized_module S (localized_module.mk_linear_map S M) | { map_units := λ s, ⟨⟨algebra_map R (module.End R (localized_module S M)) s,
localized_module.div_by s,
fun_like.ext _ _ $ localized_module.mul_by_div_by s,
fun_like.ext _ _ $ localized_module.div_by_mul_by s⟩,
fun_like.ext _ _ $ λ p, p.induction_on $ by { intros, refl }⟩,
surj := λ p, p.induction_on
... | instance | localized_module_is_localized_module | algebra.module | src/algebra/module/localized_module.lean | [
"group_theory.monoid_localization",
"ring_theory.localization.basic",
"algebra.algebra.restrict_scalars"
] | [
"fun_like.ext",
"is_localized_module",
"localized_module",
"localized_module.div_by",
"localized_module.div_by_mul_by",
"localized_module.mk_cancel",
"localized_module.mk_linear_map",
"localized_module.mul_by_div_by",
"localized_module.smul'_mk",
"module.End",
"one_smul",
"submonoid.coe_subtyp... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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