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[Merged by Bors] - chore(Algebra/TrivSqZeroExt): use open scoped RightActions #10546

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[Merged by Bors] - chore(Algebra/TrivSqZeroExt): use open scoped RightActions #10546

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a0ebb82
chore(Algebra/TrivSqZeroExt): use `open scoped RightActions`
eric-wieser Feb 14, 2024
44578bb
Rename variable for clarity
eric-wieser Feb 15, 2024
40bd6be
longer proofs in exchange for nicer `show`s
eric-wieser Feb 15, 2024
72c6180
add a second lemma
eric-wieser Feb 15, 2024
981485a
Merge branch 'eric-wieser/use-rightaction-notation' of https://github…
eric-wieser Feb 15, 2024
097ec6c
Update TrivSqZeroExt.lean
eric-wieser Feb 16, 2024
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117 changes: 62 additions & 55 deletions Mathlib/Algebra/TrivSqZeroExt.lean
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Expand Up @@ -44,7 +44,7 @@ Many of the later results in this file are only stated for the commutative `R'`
morphisms `TrivSqZeroExt R M →ₐ[S] A` are uniquely defined by an algebra morphism `f : R →ₐ[S] A`
on `R` and a linear map `g : M →ₗ[S] A` on `M` such that:
* `g x * g y = 0`: the elements of `M` continue to square to zero.
* `g (r • x) = f r * g x` and `g (op r • x) = g x * f r`: left and right actions are preserved by
* `g (r •> x) = f r * g x` and `g (x <• r) = g x * f r`: left and right actions are preserved by
`g`.
* `TrivSqZeroExt.lift`: the universal property of the trivial square-zero extension; algebra
morphisms `TrivSqZeroExt R M →ₐ[R] A` are uniquely defined by linear maps `M →ₗ[R] A` for
Expand All @@ -69,11 +69,11 @@ def TrivSqZeroExt (R : Type u) (M : Type v) :=
-- mathport name: exprtsze
local notation "tsze" => TrivSqZeroExt

open BigOperators
open scoped BigOperators RightActions

namespace TrivSqZeroExt

open MulOpposite (op)
open MulOpposite

section Basic

Expand Down Expand Up @@ -434,7 +434,7 @@ instance one [One R] [Zero M] : One (tsze R M) :=
⟨(1, 0)⟩

instance mul [Mul R] [Add M] [SMul R M] [SMul Rᵐᵒᵖ M] : Mul (tsze R M) :=
⟨fun x y => (x.1 * y.1, x.1 • y.2 + op y.1 • x.2)⟩
⟨fun x y => (x.1 * y.1, x.1 •> y.2 + x.2 <• y.1)⟩

@[simp]
theorem fst_one [One R] [Zero M] : (1 : tsze R M).fst = 1 :=
Expand All @@ -454,7 +454,7 @@ theorem fst_mul [Mul R] [Add M] [SMul R M] [SMul Rᵐᵒᵖ M] (x₁ x₂ : tsze

@[simp]
theorem snd_mul [Mul R] [Add M] [SMul R M] [SMul Rᵐᵒᵖ M] (x₁ x₂ : tsze R M) :
(x₁ * x₂).snd = x₁.fst • x₂.snd + op x₂.fst • x₁.snd :=
(x₁ * x₂).snd = x₁.fst •> x₂.snd + x₁.snd <• x₂.fst :=
rfl
#align triv_sq_zero_ext.snd_mul TrivSqZeroExt.snd_mul

Expand All @@ -470,7 +470,7 @@ theorem inl_one [One R] [Zero M] : (inl 1 : tsze R M) = 1 :=
@[simp]
theorem inl_mul [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M]
(r₁ r₂ : R) : (inl (r₁ * r₂) : tsze R M) = inl r₁ * inl r₂ :=
ext rfl <| show (0 : M) = r₁ • (0 : M) + op r₂ • (0 : M) by rw [smul_zero, zero_add, smul_zero]
ext rfl <| show (0 : M) = r₁ •> (0 : M) + (0 : M) <• r₂ by rw [smul_zero, zero_add, smul_zero]
#align triv_sq_zero_ext.inl_mul TrivSqZeroExt.inl_mul

theorem inl_mul_inl [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M]
Expand All @@ -488,7 +488,7 @@ variable (R)
theorem inr_mul_inr [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] (m₁ m₂ : M) :
(inr m₁ * inr m₂ : tsze R M) = 0 :=
ext (mul_zero _) <|
show (0 : R) • m₂ + (0 : Rᵐᵒᵖ) • m₁ = 0 by rw [zero_smul, zero_add, zero_smul]
show (0 : R) •> m₂ + m₁ <• (0 : R) = 0 by rw [zero_smul, zero_add, op_zero, zero_smul]
#align triv_sq_zero_ext.inr_mul_inr TrivSqZeroExt.inr_mul_inr

end
Expand All @@ -500,30 +500,30 @@ theorem inl_mul_inr [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒ
#align triv_sq_zero_ext.inl_mul_inr TrivSqZeroExt.inl_mul_inr

theorem inr_mul_inl [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] (r : R) (m : M) :
(inr m * inl r : tsze R M) = inr (op r • m) :=
(inr m * inl r : tsze R M) = inr (m <• r) :=
ext (zero_mul r) <|
show (0 : R) • (0 : M) + op r • m = op r • m by rw [smul_zero, zero_add]
show (0 : R) •> (0 : M) + m <• r = m <• r by rw [smul_zero, zero_add]
#align triv_sq_zero_ext.inr_mul_inl TrivSqZeroExt.inr_mul_inl

theorem inl_mul_eq_smul [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M]
(r : R) (x : tsze R M) :
inl r * x = r • x :=
inl r * x = r •> x :=
ext rfl (by dsimp; rw [smul_zero, add_zero])

theorem mul_inl_eq_op_smul [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M]
(x : tsze R M) (r : R) :
x * inl r = op r • x :=
x * inl r = x <• r :=
ext rfl (by dsimp; rw [smul_zero, zero_add])

instance mulOneClass [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] :
MulOneClass (tsze R M) :=
{ TrivSqZeroExt.one, TrivSqZeroExt.mul with
one_mul := fun x =>
ext (one_mul x.1) <|
show (1 : R) • x.2 + op x.1 • (0 : M) = x.2 by rw [one_smul, smul_zero, add_zero]
show (1 : R) •> x.2 + (0 : M) <• x.1 = x.2 by rw [one_smul, smul_zero, add_zero]
mul_one := fun x =>
ext (mul_one x.1) <|
show x.1 • (0 : M) + (1 : Rᵐᵒᵖ) • x.2 = x.2 by rw [smul_zero, zero_add, one_smul] }
show x.1 • (0 : M) + x.2 <• (1 : R) = x.2 by rw [smul_zero, zero_add, op_one, one_smul] }

instance addMonoidWithOne [AddMonoidWithOne R] [AddMonoid M] : AddMonoidWithOne (tsze R M) :=
{ TrivSqZeroExt.addMonoid, TrivSqZeroExt.one with
Expand Down Expand Up @@ -572,21 +572,21 @@ instance nonAssocSemiring [Semiring R] [AddCommMonoid M] [Module R M] [Module R
{ TrivSqZeroExt.addMonoidWithOne, TrivSqZeroExt.mulOneClass, TrivSqZeroExt.addCommMonoid with
zero_mul := fun x =>
ext (zero_mul x.1) <|
show (0 : R) • x.2 + op x.1 • (0 : M) = 0 by rw [zero_smul, zero_add, smul_zero]
show (0 : R) •> x.2 + (0 : M) <• x.1 = 0 by rw [zero_smul, zero_add, smul_zero]
mul_zero := fun x =>
ext (mul_zero x.1) <|
show x.1 • (0 : M) + (0 : Rᵐᵒᵖ) • x.2 = 0 by rw [smul_zero, zero_add, zero_smul]
left_distrib := fun x₁ x₂ x₃ =>
ext (mul_add x₁.1 x₂.1 x₃.1) <|
show
x₁.1 • (x₂.2 + x₃.2) + (op x₂.1 + op x₃.1) • x₁.2 =
x₁.1 • x₂.2 + op x₂.1 • x₁.2 + (x₁.1 • x₃.2 + op x₃.1 • x₁.2)
by simp_rw [smul_add, add_smul, add_add_add_comm]
x₁.1 •> (x₂.2 + x₃.2) + x₁.2 <• (x₂.1 + x₃.1) =
x₁.1 •> x₂.2 + x₁.2 <• x₂.1 + (x₁.1 •> x₃.2 + x₁.2 <• x₃.1)
by simp_rw [smul_add, MulOpposite.op_add, add_smul, add_add_add_comm]
right_distrib := fun x₁ x₂ x₃ =>
ext (add_mul x₁.1 x₂.1 x₃.1) <|
show
(x₁.1 + x₂.1) • x₃.2 + op x₃.1 • (x₁.2 + x₂.2) =
x₁.1 • x₃.2 + op x₃.1 • x₁.2 + (x₂.1 • x₃.2 + op x₃.1 • x₂.2)
(x₁.1 + x₂.1) •> x₃.2 + (x₁.2 + x₂.2) <• x₃.1 =
x₁.1 •> x₃.2 + x₁.2 <• x₃.1 + (x₂.1 •> x₃.2 + x₂.2 <• x₃.1)
by simp_rw [add_smul, smul_add, add_add_add_comm] }

instance nonAssocRing [Ring R] [AddCommGroup M] [Module R M] [Module Rᵐᵒᵖ M] :
Expand All @@ -605,7 +605,7 @@ In the commutative case this becomes the simpler $(r + m)^n = r^n + nr^{n-1}m$.
instance [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] :
Pow (tsze R M) ℕ :=
⟨fun x n =>
⟨x.fst ^ n, ((List.range n).map fun i => x.fst ^ (n.pred - i) • op (x.fst ^ i) • x.snd).sum⟩⟩
⟨x.fst ^ n, ((List.range n).map fun i => x.fst ^ (n.pred - i) •> x.snd <• x.fst ^ i).sum⟩⟩

@[simp]
theorem fst_pow [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M]
Expand All @@ -615,20 +615,14 @@ theorem fst_pow [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulActio

theorem snd_pow_eq_sum [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M]
(x : tsze R M) (n : ℕ) :
snd (x ^ n) = ((List.range n).map fun i => x.fst ^ (n.pred - i) • op (x.fst ^ i) • x.snd).sum :=
snd (x ^ n) = ((List.range n).map fun i => x.fst ^ (n.pred - i) •> x.snd <• x.fst ^ i).sum :=
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rfl
#align triv_sq_zero_ext.snd_pow_eq_sum TrivSqZeroExt.snd_pow_eq_sum

theorem snd_pow_of_smul_comm [Monoid R] [AddMonoid M] [DistribMulAction R M]
[DistribMulAction Rᵐᵒᵖ M] [SMulCommClass 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 := by
have : ∀ n : ℕ, op (x.fst ^ n) • x.snd = x.fst ^ n • x.snd := by
intro n
induction' n with n ih
· simp
· rw [pow_succ', MulOpposite.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]
(h : x.snd <• x.fst = x.fst •> x.snd) : snd (x ^ n) = n • x.fst ^ n.pred •> x.snd := by
simp_rw [snd_pow_eq_sum, ← smul_comm (_ : R) (_ : Rᵐᵒᵖ), aux, smul_smul, ← pow_add]
match n with
| 0 => rw [Nat.pred_zero, pow_zero, List.range_zero, zero_smul, List.map_nil, List.sum_nil]
| (Nat.succ n) =>
Expand All @@ -639,8 +633,19 @@ theorem snd_pow_of_smul_comm [Monoid R] [AddMonoid M] [DistribMulAction R M]
obtain ⟨i, hi, rfl⟩ := hm
rw [tsub_add_cancel_of_le (Nat.lt_succ_iff.mp hi)]
· rw [List.length_map, List.length_range]
where
aux : ∀ n : ℕ, x.snd <• x.fst ^ n = x.fst ^ n •> x.snd := by
intro n
induction' n with n ih
· simp
· rw [pow_succ', op_mul, mul_smul, mul_smul, ← h, smul_comm (_ : R) (op x.fst) x.snd, ih]
#align triv_sq_zero_ext.snd_pow_of_smul_comm TrivSqZeroExt.snd_pow_of_smul_comm

theorem snd_pow_of_smul_comm' [Monoid R] [AddMonoid M] [DistribMulAction R M]
[DistribMulAction Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] (x : tsze R M) (n : ℕ)
(h : x.snd <• x.fst = x.fst •> x.snd) : snd (x ^ n) = n • (x.snd <• x.fst ^ n.pred) := by
rw [snd_pow_of_smul_comm _ _ h, snd_pow_of_smul_comm.aux _ h]

@[simp]
theorem snd_pow [CommMonoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M]
[IsCentralScalar R M] (x : tsze R M) (n : ℕ) : snd (x ^ n) = n • x.fst ^ n.pred • x.snd :=
Expand All @@ -659,9 +664,9 @@ instance monoid [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulActio
mul_assoc := fun x y z =>
ext (mul_assoc x.1 y.1 z.1) <|
show
(x.1 * y.1) • z.2 + op z.1 • (x.1 • y.2 + op y.1 • x.2) =
x.1 • (y.1 • z.2 + op z.1 • y.2) + (op z.1 * op y.1) • x.2
by simp_rw [smul_add, ← mul_smul, add_assoc, smul_comm]
(x.1 * y.1) •> z.2 + (x.1 •> y.2 + x.2 <• y.1) <• z.1 =
x.1 •> (y.1 •> z.2 + y.2 <• z.1) + x.2 <• (y.1 * z.1)
by simp_rw [smul_add, ← mul_smul, add_assoc, smul_comm, op_mul]
npow := fun n x => x ^ n
npow_zero := fun x => ext (pow_zero x.fst) (by simp [snd_pow_eq_sum])
npow_succ := fun n x =>
Expand All @@ -673,12 +678,13 @@ instance monoid [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulActio
simp_rw [Nat.pred_succ]
rw [List.range_succ, List.map_append, List.sum_append, List.map_singleton,
List.sum_singleton, Nat.sub_self, pow_zero, one_smul, List.smul_sum, List.map_map,
fst_pow] --porting note: `Function.comp` no longer works in `rw` so move to `simp_rw`
simp_rw [Function.comp, smul_smul, ← pow_succ, Nat.succ_eq_add_one]
fst_pow, Function.comp]
simp_rw [← smul_comm (_ : R) (_ : Rᵐᵒᵖ), smul_smul, ← pow_succ, Nat.succ_eq_add_one]
congr 2
refine' List.map_congr fun i hi => _
rw [List.mem_range, Nat.lt_succ_iff] at hi
rw [Nat.sub_add_comm hi]) }

theorem fst_list_prod [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M]
[SMulCommClass R Rᵐᵒᵖ M] (l : List (tsze R M)) : l.prod.fst = (l.map fst).prod :=
map_list_prod ({ toFun := fst, map_one' := fst_one, map_mul' := fst_mul } : tsze R M →* R) _
Expand All @@ -694,13 +700,14 @@ theorem snd_list_prod [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐ
[SMulCommClass R Rᵐᵒᵖ M] (l : List (tsze R M)) :
l.prod.snd =
(l.enum.map fun 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 := by
((l.map fst).take x.1).prod •> x.snd.snd <• ((l.map fst).drop x.1.succ).prod).sum := by
induction' l with x xs ih
· simp
· rw [List.enum_cons, ← List.map_fst_add_enum_eq_enumFrom]
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_cons, fst_list_prod, ih, List.smul_sum, List.map_map]
mul_smul, List.sum_cons, fst_list_prod, ih, List.smul_sum, List.map_map,
← smul_comm (_ : R) (_ : Rᵐᵒᵖ)]
exact add_comm _ _
#align triv_sq_zero_ext.snd_list_prod TrivSqZeroExt.snd_list_prod

Expand All @@ -713,7 +720,7 @@ instance commMonoid [CommMonoid R] [AddCommMonoid M] [DistribMulAction R M]
{ TrivSqZeroExt.monoid with
mul_comm := fun x₁ x₂ =>
ext (mul_comm x₁.1 x₂.1) <|
show x₁.1 • x₂.2 + op x₂.1 • x₁.2 = x₂.1 • x₁.2 + op x₁.1 • x₂.2 by
show x₁.1 •> x₂.2 + x₁.2 <• x₂.1 = x₂.1 •> x₁.2 + x₂.2 <• x₁.1 by
rw [op_smul_eq_smul, op_smul_eq_smul, add_comm] }

instance commSemiring [CommSemiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M]
Expand Down Expand Up @@ -757,14 +764,14 @@ instance algebra' : Algebra S (tsze R M) :=
smul := (· • ·)
commutes' := fun s x =>
ext (Algebra.commutes _ _) <|
show algebraMap S R s • x.snd + op x.fst • (0 : M)
= x.fst • (0 : M) + op (algebraMap S R s) • x.snd by
show algebraMap S R s •> x.snd + (0 : M) <• x.fst
= x.fst •> (0 : M) + x.snd <• algebraMap S R s by
rw [smul_zero, smul_zero, add_zero, zero_add]
rw [Algebra.algebraMap_eq_smul_one, MulOpposite.op_smul, MulOpposite.op_one, smul_assoc,
rw [Algebra.algebraMap_eq_smul_one, MulOpposite.op_smul, op_one, smul_assoc,
one_smul, smul_assoc, one_smul]
smul_def' := fun r x =>
smul_def' := fun s x =>
ext (Algebra.smul_def _ _) <|
show r • x.2 = algebraMap S R r • x.2 + op x.1 • (0 : M) by
show s • x.snd = algebraMap S R s •> x.snd + (0 : M) <• x.fst by
rw [smul_zero, add_zero, algebraMap_smul] }
#align triv_sq_zero_ext.algebra' TrivSqZeroExt.algebra'

Expand Down Expand Up @@ -831,7 +838,7 @@ Namely, we require that for an algebra morphism `f : R →ₐ[S] A` and a linear
we have:

* `g x * g y = 0`: the elements of `M` continue to square to zero.
* `g (r • x) = f r * g x` and `g (op r • x) = g x * f r`: scalar multiplication on the left and
* `g (r •> x) = f r * g x` and `g (x <• r) = g x * f r`: scalar multiplication on the left and
right is sent to left- and right- multiplication by the image under `f`.

See `TrivSqZeroExt.liftEquiv` for this as an equiv; namely that any such algebra morphism can be
Expand All @@ -841,8 +848,8 @@ When `R` is commutative, this can be invoked with `f = Algebra.ofId R A`, which
`hgf`. This version is captured as an equiv by `TrivSqZeroExt.liftEquivOfComm`. -/
def lift (f : R →ₐ[S] A) (g : M →ₗ[S] A)
(hg : ∀ x y, g x * g y = 0)
(hfg : ∀ r x, g (r • x) = f r * g x)
(hgf : ∀ r x, g (op r • x) = g x * f r) : tsze R M →ₐ[S] A :=
(hfg : ∀ r x, g (r •> x) = f r * g x)
(hgf : ∀ r x, g (x <• r) = g x * f r) : tsze R M →ₐ[S] A :=
AlgHom.ofLinearMap
((f.comp <| fstHom S R M).toLinearMap + g ∘ₗ (sndHom R M |>.restrictScalars S))
(show f 1 + g (0 : M) = 1 by rw [map_zero, map_one, add_zero])
Expand All @@ -857,17 +864,17 @@ def lift (f : R →ₐ[S] A) (g : M →ₗ[S] A)
@[simp]
theorem lift_apply_inl (f : R →ₐ[S] A) (g : M →ₗ[S] A)
(hg : ∀ x y, g x * g y = 0)
(hfg : ∀ r x, g (r • x) = f r * g x)
(hgf : ∀ r x, g (op r • x) = g x * f r)
(hfg : ∀ r x, g (r •> x) = f r * g x)
(hgf : ∀ r x, g (x <• r) = g x * f r)
(r : R) :
lift f g hg hfg hgf (inl r) = f r :=
show f r + g 0 = f r by rw [map_zero, add_zero]

@[simp]
theorem lift_apply_inr (f : R →ₐ[S] A) (g : M →ₗ[S] A)
(hg : ∀ x y, g x * g y = 0)
(hfg : ∀ r x, g (r • x) = f r * g x)
(hgf : ∀ r x, g (op r • x) = g x * f r)
(hfg : ∀ r x, g (r •> x) = f r * g x)
(hgf : ∀ r x, g (x <• r) = g x * f r)
(m : M) :
lift f g hg hfg hgf (inr m) = g m :=
show f 0 + g m = g m by rw [map_zero, zero_add]
Expand All @@ -876,16 +883,16 @@ theorem lift_apply_inr (f : R →ₐ[S] A) (g : M →ₗ[S] A)
@[simp]
theorem lift_comp_inlHom (f : R →ₐ[S] A) (g : M →ₗ[S] A)
(hg : ∀ x y, g x * g y = 0)
(hfg : ∀ r x, g (r • x) = f r * g x)
(hgf : ∀ r x, g (op r • x) = g x * f r) :
(hfg : ∀ r x, g (r •> x) = f r * g x)
(hgf : ∀ r x, g (x <• r) = g x * f r) :
(lift f g hg hfg hgf).comp (inlAlgHom S R M) = f :=
AlgHom.ext <| lift_apply_inl f g hg hfg hgf

@[simp]
theorem lift_comp_inrHom (f : R →ₐ[S] A) (g : M →ₗ[S] A)
(hg : ∀ x y, g x * g y = 0)
(hfg : ∀ r x, g (r • x) = f r * g x)
(hgf : ∀ r x, g (op r • x) = g x * f r) :
(hfg : ∀ r x, g (r •> x) = f r * g x)
(hgf : ∀ r x, g (x <• r) = g x * f r) :
(lift f g hg hfg hgf).toLinearMap.comp (inrHom R M |>.restrictScalars S) = g :=
LinearMap.ext <| lift_apply_inr f g hg hfg hgf
#align triv_sq_zero_ext.lift_aux_comp_inr_hom TrivSqZeroExt.lift_comp_inrHom
Expand All @@ -909,8 +916,8 @@ This isomorphism is named to match the very similar `Complex.lift`. -/
def liftEquiv :
{fg : (R →ₐ[S] A) × (M →ₗ[S] A) //
(∀ x y, fg.2 x * fg.2 y = 0) ∧
(∀ r x, fg.2 (r • x) = fg.1 r * fg.2 x) ∧
(∀ r x, fg.2 (op r • x) = fg.2 x * fg.1 r)} ≃ (tsze R M →ₐ[S] A) where
(∀ r x, fg.2 (r •> x) = fg.1 r * fg.2 x) ∧
(∀ r x, fg.2 (x <• r) = fg.2 x * fg.1 r)} ≃ (tsze R M →ₐ[S] A) where
toFun fg := lift fg.val.1 fg.val.2 fg.prop.1 fg.prop.2.1 fg.prop.2.2
invFun F :=
⟨(F.comp (inlAlgHom _ _ _), F.toLinearMap ∘ₗ (inrHom _ _ |>.restrictScalars _)),
Expand Down
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