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 |
|---|---|---|---|---|---|---|---|---|---|---|
snd_nat_cast [add_monoid_with_one R] [add_monoid M] (n : ℕ) :
(n : tsze R M).snd = 0 | rfl | lemma | triv_sq_zero_ext.snd_nat_cast | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_monoid",
"add_monoid_with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_nat_cast [add_monoid_with_one R] [add_monoid M] (n : ℕ) :
(inl n : tsze R M) = n | rfl | lemma | triv_sq_zero_ext.inl_nat_cast | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_monoid",
"add_monoid_with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_int_cast [add_group_with_one R] [add_group M] (z : ℤ) :
(z : tsze R M).fst = z | rfl | lemma | triv_sq_zero_ext.fst_int_cast | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_group",
"add_group_with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_int_cast [add_group_with_one R] [add_group M] (z : ℤ) :
(z : tsze R M).snd = 0 | rfl | lemma | triv_sq_zero_ext.snd_int_cast | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_group",
"add_group_with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_int_cast [add_group_with_one R] [add_group M] (z : ℤ) :
(inl z : tsze R M) = z | rfl | lemma | triv_sq_zero_ext.inl_int_cast | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_group",
"add_group_with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_pow [monoid R] [add_monoid M] [distrib_mul_action R M] [distrib_mul_action Rᵐᵒᵖ M]
(x : tsze R M) (n : ℕ) :
fst (x ^ n) = x.fst ^ n | rfl | lemma | triv_sq_zero_ext.fst_pow | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_monoid",
"distrib_mul_action",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_pow_eq_sum [monoid R] [add_monoid M]
[distrib_mul_action R M] [distrib_mul_action Rᵐᵒᵖ M] (x : tsze R M) (n : ℕ) :
snd (x ^ n) = ((list.range n).map (λ i, x.fst ^ (n.pred - i) • op (x.fst ^ i) • x.snd)).sum | rfl | lemma | triv_sq_zero_ext.snd_pow_eq_sum | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_monoid",
"distrib_mul_action",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_pow_of_smul_comm [monoid R] [add_monoid M]
[distrib_mul_action R M] [distrib_mul_action Rᵐᵒᵖ M] [smul_comm_class R Rᵐᵒᵖ M]
(x : tsze R M) (n : ℕ) (h : op x.fst • x.snd = x.fst • x.snd) :
snd (x ^ n) = n • x.fst ^ n.pred • x.snd | begin
have : ∀ n : ℕ, op (x.fst ^ n) • x.snd = x.fst ^ n • x.snd,
{ intro n,
induction n with n ih,
{ simp },
{ rw [pow_succ', mul_opposite.op_mul, mul_smul, mul_smul, ←h,
smul_comm (_ : R) (op x.fst) x.snd, ih] } },
simp_rw [snd_pow_eq_sum, this, smul_smul, ←pow_add],
cases n,
{ rw [nat.p... | lemma | triv_sq_zero_ext.snd_pow_of_smul_comm | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_monoid",
"distrib_mul_action",
"ih",
"list.length_range",
"list.map_nil",
"list.mem_map",
"list.mem_range",
"list.range_zero",
"monoid",
"mul_opposite.op_mul",
"pow_succ'",
"pow_zero",
"smul_comm_class",
"smul_smul",
"tsub_add_cancel_of_le",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_pow [comm_monoid R] [add_monoid M]
[distrib_mul_action R M] [distrib_mul_action Rᵐᵒᵖ M] [is_central_scalar R M]
(x : tsze R M) (n : ℕ) :
snd (x ^ n) = n • x.fst ^ n.pred • x.snd | snd_pow_of_smul_comm _ _ (op_smul_eq_smul _ _) | lemma | triv_sq_zero_ext.snd_pow | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_monoid",
"comm_monoid",
"distrib_mul_action",
"is_central_scalar"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inl_pow [monoid R] [add_monoid M] [distrib_mul_action R M] [distrib_mul_action Rᵐᵒᵖ M]
(r : R) (n : ℕ) :
(inl r ^ n : tsze R M) = inl (r ^ n) | ext rfl $ by simp [snd_pow_eq_sum] | lemma | triv_sq_zero_ext.inl_pow | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_monoid",
"distrib_mul_action",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_list_prod [monoid R] [add_monoid M]
[distrib_mul_action R M] [distrib_mul_action Rᵐᵒᵖ M] [smul_comm_class R Rᵐᵒᵖ M]
(l : list (tsze R M)) :
l.prod.fst = (l.map fst).prod | map_list_prod (⟨fst, fst_one, fst_mul⟩ : tsze R M →* R) _ | lemma | triv_sq_zero_ext.fst_list_prod | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_monoid",
"distrib_mul_action",
"map_list_prod",
"monoid",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_list_prod [semiring R] [add_comm_monoid M]
[module R M] [module Rᵐᵒᵖ M] [smul_comm_class R Rᵐᵒᵖ M]
(l : list (tsze R M)) :
l.prod.snd =
(l.enum.map (λ x : ℕ × tsze R M,
((l.map fst).take x.1).prod • op ((l.map fst).drop x.1.succ).prod • x.snd.snd)).sum | begin
induction l with x xs ih,
{ simp },
{ rw [list.enum_cons, ←list.map_fst_add_enum_eq_enum_from],
simp_rw [list.map_cons, list.map_map, function.comp, prod.map_snd, prod.map_fst, id,
list.take_zero, list.take_cons, list.prod_nil, list.prod_cons, snd_mul, one_smul,
list.drop, mul_smul, list.sum... | lemma | triv_sq_zero_ext.snd_list_prod | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_comm_monoid",
"ih",
"list.enum_cons",
"list.prod_cons",
"list.prod_nil",
"list.smul_sum",
"list.take_cons",
"list.take_zero",
"module",
"one_smul",
"prod.map_fst",
"prod.map_snd",
"semiring",
"smul_comm_class"
] | The second element of a product $\prod_{i=0}^n (r_i + m_i)$ is a sum of terms of the form
$r_0\cdots r_{i-1}m_ir_{i+1}\cdots r_n$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl_hom [semiring R] [add_comm_monoid M] [module R M] [module Rᵐᵒᵖ M] : R →+* tsze R M | { to_fun := inl,
map_one' := inl_one M,
map_mul' := inl_mul M,
map_zero' := inl_zero M,
map_add' := inl_add M } | def | triv_sq_zero_ext.inl_hom | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"add_comm_monoid",
"module",
"semiring"
] | The canonical inclusion of rings `R → triv_sq_zero_ext R M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra' : algebra S (tsze R M) | { smul := (•),
commutes' := λ s x, ext (algebra.commutes _ _) $
show algebra_map S R s • x.snd + op x.fst • 0 = x.fst • 0 + op (algebra_map S R s) • x.snd,
begin
rw [smul_zero, smul_zero, add_zero, zero_add],
rw [algebra.algebra_map_eq_smul_one, mul_opposite.op_smul, mul_opposite.op_one,
smul_asso... | instance | triv_sq_zero_ext.algebra' | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"algebra",
"algebra.algebra_map_eq_smul_one",
"algebra.commutes",
"algebra.smul_def",
"algebra_map",
"algebra_map_smul",
"mul_opposite.op_one",
"mul_opposite.op_smul",
"one_smul",
"smul_assoc",
"smul_zero",
"triv_sq_zero_ext.inl_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eq_inl : ⇑(algebra_map R' (tsze R' M)) = inl | rfl | lemma | triv_sq_zero_ext.algebra_map_eq_inl | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eq_inl_hom : algebra_map R' (tsze R' M) = inl_hom R' M | rfl | lemma | triv_sq_zero_ext.algebra_map_eq_inl_hom | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eq_inl' (s : S) : algebra_map S (tsze R M) s = inl (algebra_map S R s) | rfl | lemma | triv_sq_zero_ext.algebra_map_eq_inl' | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_hom : tsze R M →ₐ[S] R | { to_fun := fst,
map_one' := fst_one,
map_mul' := fst_mul,
map_zero' := fst_zero,
map_add' := fst_add,
commutes' := λ r, fst_inl M _ } | def | triv_sq_zero_ext.fst_hom | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [] | The canonical `R`-algebra projection `triv_sq_zero_ext R M → R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
alg_hom_ext {A} [semiring A] [algebra R' A] ⦃f g : tsze R' M →ₐ[R'] A⦄
(h : ∀ m, f (inr m) = g (inr m)) :
f = g | alg_hom.to_linear_map_injective $ linear_map_ext (λ r, (f.commutes _).trans (g.commutes _).symm) h | lemma | triv_sq_zero_ext.alg_hom_ext | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"alg_hom.to_linear_map_injective",
"algebra",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_hom_ext' {A} [semiring A] [algebra R' A] ⦃f g : tsze R' M →ₐ[R'] A⦄
(h : f.to_linear_map.comp (inr_hom R' M) = g.to_linear_map.comp (inr_hom R' M)) :
f = g | alg_hom_ext $ linear_map.congr_fun h | lemma | triv_sq_zero_ext.alg_hom_ext' | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"algebra",
"linear_map.congr_fun",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_aux (f : M →ₗ[R'] A) (hf : ∀ x y, f x * f y = 0) : tsze R' M →ₐ[R'] A | alg_hom.of_linear_map
((algebra.linear_map _ _).comp (fst_hom R' R' M).to_linear_map + f.comp (snd_hom R' M))
(show algebra_map R' _ 1 + f (0 : M) = 1, by rw [map_zero, map_one, add_zero])
(triv_sq_zero_ext.ind $ λ r₁ m₁, triv_sq_zero_ext.ind $ λ r₂ m₂, begin
dsimp,
simp only [add_zero, zero_add, add_mul,... | def | triv_sq_zero_ext.lift_aux | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"alg_hom.of_linear_map",
"algebra.linear_map",
"algebra_map",
"linear_map.map_add",
"linear_map.map_smul",
"map_one",
"smul_mul_smul",
"smul_zero",
"triv_sq_zero_ext.ind"
] | There is an alg_hom from the trivial square zero extension to any `R`-algebra with a submodule
whose products are all zero.
See `triv_sq_zero_ext.lift` for this as an equiv. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_aux_apply_inr (f : M →ₗ[R'] A) (hf : ∀ x y, f x * f y = 0) (m : M) :
lift_aux f hf (inr m) = f m | show algebra_map R' A 0 + f m = f m, by rw [ring_hom.map_zero, zero_add] | lemma | triv_sq_zero_ext.lift_aux_apply_inr | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"algebra_map",
"ring_hom.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_aux_comp_inr_hom (f : M →ₗ[R'] A) (hf : ∀ x y, f x * f y = 0) :
(lift_aux f hf).to_linear_map.comp (inr_hom R' M) = f | linear_map.ext $ lift_aux_apply_inr f hf | lemma | triv_sq_zero_ext.lift_aux_comp_inr_hom | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"linear_map.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_aux_inr_hom : lift_aux (inr_hom R' M) (inr_mul_inr R') = alg_hom.id R' (tsze R' M) | alg_hom_ext' $ lift_aux_comp_inr_hom _ _ | lemma | triv_sq_zero_ext.lift_aux_inr_hom | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"alg_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift : {f : M →ₗ[R'] A // ∀ x y, f x * f y = 0} ≃ (tsze R' M →ₐ[R'] A) | { to_fun := λ f, lift_aux f f.prop,
inv_fun := λ F, ⟨F.to_linear_map.comp (inr_hom R' M), λ x y,
(F.map_mul _ _).symm.trans $ (F.congr_arg $ inr_mul_inr _ _ _).trans F.map_zero⟩,
left_inv := λ f, subtype.ext $ lift_aux_comp_inr_hom _ _,
right_inv := λ F, alg_hom_ext' $ lift_aux_comp_inr_hom _ _, } | def | triv_sq_zero_ext.lift | algebra | src/algebra/triv_sq_zero_ext.lean | [
"algebra.algebra.basic",
"linear_algebra.prod"
] | [
"inv_fun",
"lift",
"subtype.ext"
] | A universal property of the trivial square-zero extension, providing a unique
`triv_sq_zero_ext R M →ₐ[R] A` for every linear map `M →ₗ[R] A` whose range has no non-zero
products.
This isomorphism is named to match the very similar `complex.lift`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra (R : Type u) (A : Type v) [comm_semiring R] [semiring A]
extends has_smul R A, R →+* A | (commutes' : ∀ r x, to_fun r * x = x * to_fun r)
(smul_def' : ∀ r x, r • x = to_fun r * x) | class | algebra | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"comm_semiring",
"has_smul",
"semiring"
] | An associative unital `R`-algebra is a semiring `A` equipped with a map into its center `R → A`.
See the implementation notes in this file for discussion of the details of this definition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_map (R : Type u) (A : Type v) [comm_semiring R] [semiring A] [algebra R A] : R →+* A | algebra.to_ring_hom | def | algebra_map | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra",
"comm_semiring",
"semiring"
] | Embedding `R →+* A` given by `algebra` structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_lift_t (R A : Type*) [comm_semiring R] [semiring A] [algebra R A] :
has_lift_t R A | ⟨λ r, algebra_map R A r⟩ | def | algebra_map.has_lift_t | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra",
"algebra_map",
"comm_semiring",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : (↑(0 : R) : A) = 0 | map_zero (algebra_map R A) | lemma | algebra_map.coe_zero | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : (↑(1 : R) : A) = 1 | map_one (algebra_map R A) | lemma | algebra_map.coe_one | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (a b : R) : (↑(a + b : R) : A) = ↑a + ↑b | map_add (algebra_map R A) a b | lemma | algebra_map.coe_add | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (a b : R) : (↑(a * b : R) : A) = ↑a * ↑b | map_mul (algebra_map R A) a b | lemma | algebra_map.coe_mul | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow (a : R) (n : ℕ) : (↑(a ^ n : R) : A) = ↑a ^ n | map_pow (algebra_map R A) _ _ | lemma | algebra_map.coe_pow | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg (x : R) : (↑(-x : R) : A) = -↑x | map_neg (algebra_map R A) x | lemma | algebra_map.coe_neg | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_prod {ι : Type*} {s : finset ι} (a : ι → R) :
(↑( ∏ (i : ι) in s, a i : R) : A) = ∏ (i : ι) in s, (↑(a i) : A) | map_prod (algebra_map R A) a s | lemma | algebra_map.coe_prod | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"finset",
"map_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sum {ι : Type*} {s : finset ι} (a : ι → R) :
↑(( ∑ (i : ι) in s, a i)) = ∑ (i : ι) in s, (↑(a i) : A) | map_sum (algebra_map R A) a s | lemma | algebra_map.coe_sum | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inj {a b : R} : (↑a : A) = ↑b ↔ a = b | (algebra_map R A).injective.eq_iff | lemma | algebra_map.coe_inj | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_map_eq_zero_iff (a : R) : (↑a : A) = 0 ↔ a = 0 | map_eq_zero_iff _ (algebra_map R A).injective | lemma | algebra_map.lift_map_eq_zero_iff | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv (r : R) : ↑(r⁻¹) = ((↑r)⁻¹ : A) | map_inv₀ (algebra_map R A) r | lemma | algebra_map.coe_inv | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"map_inv₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_div (r s : R) : ↑(r / s) = (↑r / ↑s : A) | map_div₀ (algebra_map R A) r s | lemma | algebra_map.coe_div | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"map_div₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zpow (r : R) (z : ℤ) : ↑(r ^ z) = (↑r ^ z : A) | map_zpow₀ (algebra_map R A) r z | lemma | algebra_map.coe_zpow | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"map_zpow₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_rat_cast (q : ℚ) : ↑(q : R) = (q : A) | map_rat_cast (algebra_map R A) q | lemma | algebra_map.coe_rat_cast | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"map_rat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.to_algebra' {R S} [comm_semiring R] [semiring S] (i : R →+* S)
(h : ∀ c x, i c * x = x * i c) :
algebra R S | { smul := λ c x, i c * x,
commutes' := h,
smul_def' := λ c x, rfl,
to_ring_hom := i} | def | ring_hom.to_algebra' | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra",
"comm_semiring",
"semiring"
] | Creating an algebra from a morphism to the center of a semiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.to_algebra {R S} [comm_semiring R] [comm_semiring S] (i : R →+* S) :
algebra R S | i.to_algebra' $ λ _, mul_comm _ | def | ring_hom.to_algebra | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra",
"comm_semiring",
"mul_comm"
] | Creating an algebra from a morphism to a commutative semiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.algebra_map_to_algebra {R S} [comm_semiring R] [comm_semiring S]
(i : R →+* S) :
@algebra_map R S _ _ i.to_algebra = i | rfl | lemma | ring_hom.algebra_map_to_algebra | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"comm_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_module' [comm_semiring R] [semiring A] [module R A]
(h₁ : ∀ (r : R) (x : A), (r • 1) * x = r • x)
(h₂ : ∀ (r : R) (x : A), x * (r • 1) = r • x) : algebra R A | { to_fun := λ r, r • 1,
map_one' := one_smul _ _,
map_mul' := λ r₁ r₂, by rw [h₁, mul_smul],
map_zero' := zero_smul _ _,
map_add' := λ r₁ r₂, add_smul r₁ r₂ 1,
commutes' := λ r x, by simp only [h₁, h₂],
smul_def' := λ r x, by simp only [h₁] } | def | algebra.of_module' | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"add_smul",
"algebra",
"comm_semiring",
"module",
"one_smul",
"semiring",
"zero_smul"
] | Let `R` be a commutative semiring, let `A` be a semiring with a `module R` structure.
If `(r • 1) * x = x * (r • 1) = r • x` for all `r : R` and `x : A`, then `A` is an `algebra`
over `R`.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_module [comm_semiring R] [semiring A] [module R A]
(h₁ : ∀ (r : R) (x y : A), (r • x) * y = r • (x * y))
(h₂ : ∀ (r : R) (x y : A), x * (r • y) = r • (x * y)) : algebra R A | of_module' (λ r x, by rw [h₁, one_mul]) (λ r x, by rw [h₂, mul_one]) | def | algebra.of_module | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra",
"comm_semiring",
"module",
"mul_one",
"one_mul",
"semiring"
] | Let `R` be a commutative semiring, let `A` be a semiring with a `module R` structure.
If `(r • x) * y = x * (r • y) = r • (x * y)` for all `r : R` and `x y : A`, then `A`
is an `algebra` over `R`.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_def'' (r : R) (x : A) : r • x = algebra_map R A r * x | algebra.smul_def' r x | lemma | algebra.smul_def'' | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map"
] | We keep this lemma private because it picks up the `algebra.to_has_smul` instance
which we set to priority 0 shortly. See `smul_def` below for the public version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_ext {R : Type*} [comm_semiring R] {A : Type*} [semiring A] (P Q : algebra R A)
(w : ∀ (r : R), by { haveI := P, exact algebra_map R A r } =
by { haveI := Q, exact algebra_map R A r }) :
P = Q | begin
unfreezingI { rcases P with @⟨⟨P⟩⟩, rcases Q with @⟨⟨Q⟩⟩ },
congr,
{ funext r a,
replace w := congr_arg (λ s, s * a) (w r),
simp only [←smul_def''] at w,
apply w, },
{ ext r,
exact w r, },
{ apply proof_irrel_heq, },
{ apply proof_irrel_heq, },
end | lemma | algebra.algebra_ext | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra",
"algebra_map",
"comm_semiring",
"proof_irrel_heq",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_module : module R A | { one_smul := by simp [smul_def''],
mul_smul := by simp [smul_def'', mul_assoc],
smul_add := by simp [smul_def'', mul_add],
smul_zero := by simp [smul_def''],
add_smul := by simp [smul_def'', add_mul],
zero_smul := by simp [smul_def''] } | instance | algebra.to_module | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"add_smul",
"module",
"mul_assoc",
"one_smul",
"smul_add",
"smul_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_def (r : R) (x : A) : r • x = algebra_map R A r * x | algebra.smul_def' r x | lemma | algebra.smul_def | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eq_smul_one (r : R) : algebra_map R A r = r • 1 | calc algebra_map R A r = algebra_map R A r * 1 : (mul_one _).symm
... = r • 1 : (algebra.smul_def r 1).symm | lemma | algebra.algebra_map_eq_smul_one | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra.smul_def",
"algebra_map",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eq_smul_one' : ⇑(algebra_map R A) = λ r, r • (1 : A) | funext algebra_map_eq_smul_one | lemma | algebra.algebra_map_eq_smul_one' | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commutes (r : R) (x : A) : algebra_map R A r * x = x * algebra_map R A r | algebra.commutes' r x | theorem | algebra.commutes | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map"
] | `mul_comm` for `algebra`s when one element is from the base ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_comm (x : A) (r : R) (y : A) :
x * (algebra_map R A r * y) = algebra_map R A r * (x * y) | by rw [← mul_assoc, ← commutes, mul_assoc] | theorem | algebra.left_comm | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"mul_assoc"
] | `mul_left_comm` for `algebra`s when one element is from the base ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_comm (x : A) (r : R) (y : A) :
(x * algebra_map R A r) * y = (x * y) * algebra_map R A r | by rw [mul_assoc, commutes, ←mul_assoc] | theorem | algebra.right_comm | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"mul_assoc"
] | `mul_right_comm` for `algebra`s when one element is from the base ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.is_scalar_tower.right : is_scalar_tower R A A | ⟨λ x y z, by rw [smul_eq_mul, smul_eq_mul, smul_def, smul_def, mul_assoc]⟩ | instance | is_scalar_tower.right | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"is_scalar_tower",
"mul_assoc",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_smul_comm (s : R) (x y : A) :
x * (s • y) = s • (x * y) | -- TODO: set up `is_scalar_tower.smul_comm_class` earlier so that we can actually prove this using
-- `mul_smul_comm s x y`.
by rw [smul_def, smul_def, left_comm] | lemma | algebra.mul_smul_comm | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"mul_smul_comm"
] | This is just a special case of the global `mul_smul_comm` lemma that requires less typeclass
search (and was here first). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_mul_assoc (r : R) (x y : A) :
(r • x) * y = r • (x * y) | smul_mul_assoc r x y | lemma | algebra.smul_mul_assoc | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"smul_mul_assoc"
] | This is just a special case of the global `smul_mul_assoc` lemma that requires less typeclass
search (and was here first). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.smul_algebra_map {α : Type*} [monoid α] [mul_distrib_mul_action α A]
[smul_comm_class α R A] (a : α) (r : R) : a • algebra_map R A r = algebra_map R A r | by rw [algebra_map_eq_smul_one, smul_comm a r (1 : A), smul_one] | lemma | smul_algebra_map | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"monoid",
"mul_distrib_mul_action",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_smul_one : bit0 r • (1 : A) = bit0 (r • (1 : A)) | by simp [bit0, add_smul] | lemma | algebra.bit0_smul_one | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"add_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_smul_one' : bit0 r • (1 : A) = r • 2 | by simp [bit0, add_smul, smul_add] | lemma | algebra.bit0_smul_one' | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"add_smul",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_smul_bit0 : bit0 r • bit0 a = r • (bit0 (bit0 a)) | by simp [bit0, add_smul, smul_add] | lemma | algebra.bit0_smul_bit0 | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"add_smul",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_smul_bit1 : bit0 r • bit1 a = r • (bit0 (bit1 a)) | by simp [bit0, add_smul, smul_add] | lemma | algebra.bit0_smul_bit1 | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"add_smul",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_smul_one : bit1 r • (1 : A) = bit1 (r • (1 : A)) | by simp [bit1, add_smul] | lemma | algebra.bit1_smul_one | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"add_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_smul_one' : bit1 r • (1 : A) = r • 2 + 1 | by simp [bit1, bit0, add_smul, smul_add] | lemma | algebra.bit1_smul_one' | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"add_smul",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_smul_bit0 : bit1 r • bit0 a = r • (bit0 (bit0 a)) + bit0 a | by simp [bit1, add_smul, smul_add] | lemma | algebra.bit1_smul_bit0 | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"add_smul",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_smul_bit1 : bit1 r • bit1 a = r • (bit0 (bit1 a)) + bit1 a | by { simp only [bit0, bit1, add_smul, smul_add, one_smul], abel } | lemma | algebra.bit1_smul_bit1 | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"add_smul",
"one_smul",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_map : R →ₗ[R] A | { map_smul' := λ x y, by simp [algebra.smul_def],
..algebra_map R A } | def | algebra.linear_map | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra.smul_def",
"algebra_map",
"linear_map"
] | The canonical ring homomorphism `algebra_map R A : R →* A` for any `R`-algebra `A`,
packaged as an `R`-linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_map_apply (r : R) : algebra.linear_map R A r = algebra_map R A r | rfl | lemma | algebra.linear_map_apply | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra.linear_map",
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_linear_map : ⇑(algebra.linear_map R A) = algebra_map R A | rfl | lemma | algebra.coe_linear_map | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra.linear_map",
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : algebra R R | (ring_hom.id R).to_algebra | instance | algebra.id | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_id : algebra_map R R = ring_hom.id _ | rfl | lemma | algebra.id.map_eq_id | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_self (x : R) : algebra_map R R x = x | rfl | lemma | algebra.id.map_eq_self | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_eq_mul (x y : R) : x • y = x * y | rfl | lemma | algebra.id.smul_eq_mul | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.punit.algebra : algebra R punit | { to_fun := λ x, punit.star,
map_one' := rfl,
map_mul' := λ _ _, rfl,
map_zero' := rfl,
map_add' := λ _ _, rfl,
commutes' := λ _ _, rfl,
smul_def' := λ _ _, rfl } | instance | punit.algebra | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_punit (r : R) : algebra_map R punit r = punit.star | rfl | lemma | algebra.algebra_map_punit | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.ulift.algebra : algebra R (ulift A) | { to_fun := λ r, ulift.up (algebra_map R A r),
commutes' := λ r x, ulift.down_injective $ algebra.commutes r x.down,
smul_def' := λ r x, ulift.down_injective $ algebra.smul_def' r x.down,
.. ulift.module',
.. (ulift.ring_equiv : ulift A ≃+* A).symm.to_ring_hom.comp (algebra_map R A) } | instance | ulift.algebra | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra",
"algebra.commutes",
"algebra_map",
"ulift.down_injective",
"ulift.module'",
"ulift.ring_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.ulift.algebra_map_eq (r : R) :
algebra_map R (ulift A) r = ulift.up (algebra_map R A r) | rfl | lemma | ulift.algebra_map_eq | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.ulift.down_algebra_map (r : R) :
(algebra_map R (ulift A) r).down = algebra_map R A r | rfl | lemma | ulift.down_algebra_map | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_subsemiring (S : subsemiring R) : algebra S A | { smul := (•),
commutes' := λ r x, algebra.commutes r x,
smul_def' := λ r x, algebra.smul_def r x,
.. (algebra_map R A).comp S.subtype } | instance | algebra.of_subsemiring | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra",
"algebra.commutes",
"algebra.smul_def",
"algebra_map",
"subsemiring"
] | Algebra over a subsemiring. This builds upon `subsemiring.module`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_map_of_subsemiring (S : subsemiring R) :
(algebra_map S R : S →+* R) = subsemiring.subtype S | rfl | lemma | algebra.algebra_map_of_subsemiring | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"subsemiring",
"subsemiring.subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_algebra_map_of_subsemiring (S : subsemiring R) :
(algebra_map S R : S → R) = subtype.val | rfl | lemma | algebra.coe_algebra_map_of_subsemiring | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_of_subsemiring_apply (S : subsemiring R) (x : S) :
algebra_map S R x = x | rfl | lemma | algebra.algebra_map_of_subsemiring_apply | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"subsemiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_subring {R A : Type*} [comm_ring R] [ring A] [algebra R A]
(S : subring R) : algebra S A | { smul := (•),
.. algebra.of_subsemiring S.to_subsemiring,
.. (algebra_map R A).comp S.subtype } | instance | algebra.of_subring | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra",
"algebra.of_subsemiring",
"algebra_map",
"comm_ring",
"ring",
"subring"
] | Algebra over a subring. This builds upon `subring.module`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_map_of_subring {R : Type*} [comm_ring R] (S : subring R) :
(algebra_map S R : S →+* R) = subring.subtype S | rfl | lemma | algebra.algebra_map_of_subring | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"comm_ring",
"subring",
"subring.subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_algebra_map_of_subring {R : Type*} [comm_ring R] (S : subring R) :
(algebra_map S R : S → R) = subtype.val | rfl | lemma | algebra.coe_algebra_map_of_subring | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"comm_ring",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_of_subring_apply {R : Type*} [comm_ring R] (S : subring R) (x : S) :
algebra_map S R x = x | rfl | lemma | algebra.algebra_map_of_subring_apply | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"comm_ring",
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_submonoid (S : Type*) [semiring S] [algebra R S]
(M : submonoid R) : submonoid S | M.map (algebra_map R S) | def | algebra.algebra_map_submonoid | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra",
"algebra_map",
"semiring",
"submonoid"
] | Explicit characterization of the submonoid map in the case of an algebra.
`S` is made explicit to help with type inference | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_algebra_map_submonoid_of_mem {S : Type*} [semiring S] [algebra R S] {M : submonoid R}
(x : M) : (algebra_map R S x) ∈ algebra_map_submonoid S M | set.mem_image_of_mem (algebra_map R S) x.2 | lemma | algebra.mem_algebra_map_submonoid_of_mem | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra",
"algebra_map",
"semiring",
"set.mem_image_of_mem",
"submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_sub_algebra_map_commutes [ring A] [algebra R A] (x : A) (r : R) :
x * (x - algebra_map R A r) = (x - algebra_map R A r) * x | by rw [mul_sub, ←commutes, sub_mul] | lemma | algebra.mul_sub_algebra_map_commutes | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra",
"algebra_map",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_sub_algebra_map_pow_commutes [ring A] [algebra R A] (x : A) (r : R) (n : ℕ) :
x * (x - algebra_map R A r) ^ n = (x - algebra_map R A r) ^ n * x | begin
induction n with n ih,
{ simp },
{ rw [pow_succ, ←mul_assoc, mul_sub_algebra_map_commutes, mul_assoc, ih, ←mul_assoc] }
end | lemma | algebra.mul_sub_algebra_map_pow_commutes | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra",
"algebra_map",
"ih",
"mul_assoc",
"pow_succ",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semiring_to_ring [semiring A] [algebra R A] : ring A | { ..module.add_comm_monoid_to_add_comm_group R,
..(infer_instance : semiring A) } | def | algebra.semiring_to_ring | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra",
"module.add_comm_monoid_to_add_comm_group",
"ring",
"semiring"
] | A `semiring` that is an `algebra` over a commutative ring carries a natural `ring` structure.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_map_apply (c : R) : algebra_map R Aᵐᵒᵖ c = op (algebra_map R A c) | rfl | lemma | mul_opposite.algebra_map_apply | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"algebra_map_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_End_eq_smul_id (a : R) :
(algebra_map R (End R M)) a = a • linear_map.id | rfl | lemma | module.algebra_map_End_eq_smul_id | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map",
"linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_End_apply (a : R) (m : M) :
(algebra_map R (End R M)) a m = a • m | rfl | lemma | module.algebra_map_End_apply | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_algebra_map_End (K : Type u) (V : Type v)
[field K] [add_comm_group V] [module K V] (a : K) (ha : a ≠ 0) :
((algebra_map K (End K V)) a).ker = ⊥ | linear_map.ker_smul _ _ ha | lemma | module.ker_algebra_map_End | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"add_comm_group",
"algebra_map",
"field",
"linear_map.ker_smul",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
End_is_unit_apply_inv_apply_of_is_unit {f : module.End R M} (h : is_unit f) (x : M) :
f (h.unit.inv x) = x | show (f * h.unit.inv) x = x, by simp | lemma | module.End_is_unit_apply_inv_apply_of_is_unit | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"is_unit",
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
End_is_unit_inv_apply_apply_of_is_unit {f : module.End R M} (h : is_unit f) (x : M) :
h.unit.inv (f x) = x | (by simp : (h.unit.inv * f) x = x) | lemma | module.End_is_unit_inv_apply_apply_of_is_unit | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"is_unit",
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
End_is_unit_iff (f : module.End R M) :
is_unit f ↔ function.bijective f | ⟨λ h, function.bijective_iff_has_inverse.mpr $
⟨h.unit.inv, ⟨End_is_unit_inv_apply_apply_of_is_unit h,
End_is_unit_apply_inv_apply_of_is_unit h⟩⟩,
λ H, let e : M ≃ₗ[R] M := { ..f, ..(equiv.of_bijective f H)} in
⟨⟨_, e.symm, linear_map.ext e.right_inv, linear_map.ext e.left_inv⟩, rfl⟩⟩ | lemma | module.End_is_unit_iff | algebra.algebra | src/algebra/algebra/basic.lean | [
"algebra.module.basic",
"algebra.module.ulift",
"algebra.ne_zero",
"algebra.punit_instances",
"algebra.ring.aut",
"algebra.ring.ulift",
"algebra.char_zero.lemmas",
"linear_algebra.basic",
"ring_theory.subring.basic",
"tactic.abel"
] | [
"equiv.of_bijective",
"is_unit",
"linear_map.ext",
"module.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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