phanerozoic/Lean4-Mathlib · Datasets at Hugging Face

fact stringlengths 6 3.84k	type stringclasses 11 values	library stringclasses 32 values	imports listlengths 1 14	filename stringlengths 20 95	symbolic_name stringlengths 1 90	docstring stringlengths 7 20k ⌀
addCommSemigroup [AddCommSemigroup R] [AddCommSemigroup M] : AddCommSemigroup (tsze R M) := Prod.instAddCommSemigroup	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	addCommSemigroup	null
addCommMonoid [AddCommMonoid R] [AddCommMonoid M] : AddCommMonoid (tsze R M) := Prod.instAddCommMonoid	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	addCommMonoid	null
addCommGroup [AddCommGroup R] [AddCommGroup M] : AddCommGroup (tsze R M) := Prod.instAddCommGroup	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	addCommGroup	null
smul [SMul S R] [SMul S M] : SMul S (tsze R M) := Prod.instSMul	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	smul	null
isScalarTower [SMul T R] [SMul T M] [SMul S R] [SMul S M] [SMul T S] [IsScalarTower T S R] [IsScalarTower T S M] : IsScalarTower T S (tsze R M) := Prod.isScalarTower	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	isScalarTower	null
smulCommClass [SMul T R] [SMul T M] [SMul S R] [SMul S M] [SMulCommClass T S R] [SMulCommClass T S M] : SMulCommClass T S (tsze R M) := Prod.smulCommClass	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	smulCommClass	null
isCentralScalar [SMul S R] [SMul S M] [SMul Sᵐᵒᵖ R] [SMul Sᵐᵒᵖ M] [IsCentralScalar S R] [IsCentralScalar S M] : IsCentralScalar S (tsze R M) := Prod.isCentralScalar	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	isCentralScalar	null
mulAction [Monoid S] [MulAction S R] [MulAction S M] : MulAction S (tsze R M) := Prod.mulAction	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	mulAction	null
distribMulAction [Monoid S] [AddMonoid R] [AddMonoid M] [DistribMulAction S R] [DistribMulAction S M] : DistribMulAction S (tsze R M) := Prod.distribMulAction	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	distribMulAction	null
module [Semiring S] [AddCommMonoid R] [AddCommMonoid M] [Module S R] [Module S M] : Module S (tsze R M) := Prod.instModule	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	module	null
instNontrivial_of_left {R M : Type*} [Nontrivial R] [Nonempty M] : Nontrivial (TrivSqZeroExt R M) := fst_surjective.nontrivial	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	instNontrivial_of_left	The trivial square-zero extension is nontrivial if it is over a nontrivial ring.
instNontrivial_of_right {R M : Type*} [Nonempty R] [Nontrivial M] : Nontrivial (TrivSqZeroExt R M) := snd_surjective.nontrivial @[simp]	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	instNontrivial_of_right	The trivial square-zero extension is nontrivial if it is over a nontrivial module.
fst_zero [Zero R] [Zero M] : (0 : tsze R M).fst = 0 := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	fst_zero	null
snd_zero [Zero R] [Zero M] : (0 : tsze R M).snd = 0 := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	snd_zero	null
fst_add [Add R] [Add M] (x₁ x₂ : tsze R M) : (x₁ + x₂).fst = x₁.fst + x₂.fst := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	fst_add	null
snd_add [Add R] [Add M] (x₁ x₂ : tsze R M) : (x₁ + x₂).snd = x₁.snd + x₂.snd := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	snd_add	null
fst_neg [Neg R] [Neg M] (x : tsze R M) : (-x).fst = -x.fst := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	fst_neg	null
snd_neg [Neg R] [Neg M] (x : tsze R M) : (-x).snd = -x.snd := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	snd_neg	null
fst_sub [Sub R] [Sub M] (x₁ x₂ : tsze R M) : (x₁ - x₂).fst = x₁.fst - x₂.fst := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	fst_sub	null
snd_sub [Sub R] [Sub M] (x₁ x₂ : tsze R M) : (x₁ - x₂).snd = x₁.snd - x₂.snd := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	snd_sub	null
fst_smul [SMul S R] [SMul S M] (s : S) (x : tsze R M) : (s • x).fst = s • x.fst := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	fst_smul	null
snd_smul [SMul S R] [SMul S M] (s : S) (x : tsze R M) : (s • x).snd = s • x.snd := rfl	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	snd_smul	null
fst_sum {ι} [AddCommMonoid R] [AddCommMonoid M] (s : Finset ι) (f : ι → tsze R M) : (∑ i ∈ s, f i).fst = ∑ i ∈ s, (f i).fst := Prod.fst_sum	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	fst_sum	null
snd_sum {ι} [AddCommMonoid R] [AddCommMonoid M] (s : Finset ι) (f : ι → tsze R M) : (∑ i ∈ s, f i).snd = ∑ i ∈ s, (f i).snd := Prod.snd_sum	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	snd_sum	null
@[simp] inl_zero [Zero R] [Zero M] : (inl 0 : tsze R M) = 0 := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inl_zero	null
inl_add [Add R] [AddZeroClass M] (r₁ r₂ : R) : (inl (r₁ + r₂) : tsze R M) = inl r₁ + inl r₂ := ext rfl (add_zero 0).symm @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inl_add	null
inl_neg [Neg R] [NegZeroClass M] (r : R) : (inl (-r) : tsze R M) = -inl r := ext rfl neg_zero.symm @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inl_neg	null
inl_sub [Sub R] [SubNegZeroMonoid M] (r₁ r₂ : R) : (inl (r₁ - r₂) : tsze R M) = inl r₁ - inl r₂ := ext rfl (sub_zero _).symm @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inl_sub	null
inl_smul [Monoid S] [AddMonoid M] [SMul S R] [DistribMulAction S M] (s : S) (r : R) : (inl (s • r) : tsze R M) = s • inl r := ext rfl (smul_zero s).symm	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inl_smul	null
inl_sum {ι} [AddCommMonoid R] [AddCommMonoid M] (s : Finset ι) (f : ι → R) : (inl (∑ i ∈ s, f i) : tsze R M) = ∑ i ∈ s, inl (f i) := map_sum (LinearMap.inl ℕ _ _) _ _	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inl_sum	null
@[simp] inr_zero [Zero R] [Zero M] : (inr 0 : tsze R M) = 0 := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inr_zero	null
inr_add [AddZeroClass R] [Add M] (m₁ m₂ : M) : (inr (m₁ + m₂) : tsze R M) = inr m₁ + inr m₂ := ext (add_zero 0).symm rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inr_add	null
inr_neg [NegZeroClass R] [Neg M] (m : M) : (inr (-m) : tsze R M) = -inr m := ext neg_zero.symm rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inr_neg	null
inr_sub [SubNegZeroMonoid R] [Sub M] (m₁ m₂ : M) : (inr (m₁ - m₂) : tsze R M) = inr m₁ - inr m₂ := ext (sub_zero _).symm rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inr_sub	null
inr_smul [Zero R] [SMulZeroClass S R] [SMul S M] (r : S) (m : M) : (inr (r • m) : tsze R M) = r • inr m := ext (smul_zero _).symm rfl	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inr_smul	null
inr_sum {ι} [AddCommMonoid R] [AddCommMonoid M] (s : Finset ι) (f : ι → M) : (inr (∑ i ∈ s, f i) : tsze R M) = ∑ i ∈ s, inr (f i) := map_sum (LinearMap.inr ℕ _ _) _ _	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inr_sum	null
inl_fst_add_inr_snd_eq [AddZeroClass R] [AddZeroClass M] (x : tsze R M) : inl x.fst + inr x.snd = x := ext (add_zero x.1) (zero_add x.2)	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inl_fst_add_inr_snd_eq	null
@[elab_as_elim, induction_eliminator, cases_eliminator] ind {R M} [AddZeroClass R] [AddZeroClass M] {P : TrivSqZeroExt R M → Prop} (inl_add_inr : ∀ r m, P (inl r + inr m)) (x) : P x := inl_fst_add_inr_snd_eq x ▸ inl_add_inr x.1 x.2	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	ind	To show a property hold on all `TrivSqZeroExt R M` it suffices to show it holds on terms of the form `inl r + inr m`.
linearMap_ext {N} [Semiring S] [AddCommMonoid R] [AddCommMonoid M] [AddCommMonoid N] [Module S R] [Module S M] [Module S N] ⦃f g : tsze R M →ₗ[S] N⦄ (hl : ∀ r, f (inl r) = g (inl r)) (hr : ∀ m, f (inr m) = g (inr m)) : f = g := LinearMap.prod_ext (LinearMap.ext hl) (LinearMap.ext hr) variable (R M)	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	linearMap_ext	This cannot be marked `@[ext]` as it ends up being used instead of `LinearMap.prod_ext` when working with `R × M`.
@[simps apply] inrHom [Semiring R] [AddCommMonoid M] [Module R M] : M →ₗ[R] tsze R M := { LinearMap.inr R R M with toFun := inr }	def	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inrHom	The canonical `R`-linear inclusion `M → TrivSqZeroExt R M`.
@[simps apply] sndHom [Semiring R] [AddCommMonoid M] [Module R M] : tsze R M →ₗ[R] M := { LinearMap.snd _ _ _ with toFun := snd }	def	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	sndHom	The canonical `R`-linear projection `TrivSqZeroExt R M → M`.
one [One R] [Zero M] : One (tsze R M) := ⟨(1, 0)⟩	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	one	null
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 + x.2 <• y.1)⟩ @[simp]	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	mul	null
fst_one [One R] [Zero M] : (1 : tsze R M).fst = 1 := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	fst_one	null
snd_one [One R] [Zero M] : (1 : tsze R M).snd = 0 := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	snd_one	null
fst_mul [Mul R] [Add M] [SMul R M] [SMul Rᵐᵒᵖ M] (x₁ x₂ : tsze R M) : (x₁ * x₂).fst = x₁.fst * x₂.fst := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	fst_mul	null
snd_mul [Mul R] [Add M] [SMul R M] [SMul Rᵐᵒᵖ M] (x₁ x₂ : tsze R M) : (x₁ * x₂).snd = x₁.fst •> x₂.snd + x₁.snd <• x₂.fst := rfl	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	snd_mul	null
@[simp] inl_one [One R] [Zero M] : (inl 1 : tsze R M) = 1 := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inl_one	null
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) + (0 : M) <• r₂ by rw [smul_zero, zero_add, smul_zero]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inl_mul	null
inl_mul_inl [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (r₁ r₂ : R) : (inl r₁ * inl r₂ : tsze R M) = inl (r₁ * r₂) := (inl_mul M r₁ r₂).symm	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inl_mul_inl	null
@[simp] 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 + m₁ <• (0 : R) = 0 by rw [zero_smul, zero_add, op_zero, zero_smul]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inr_mul_inr	null
inl_mul_inr [MonoidWithZero R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (r : R) (m : M) : (inl r * inr m : tsze R M) = inr (r • m) := ext (mul_zero r) <\| show r • m + (0 : Rᵐᵒᵖ) • (0 : M) = r • m by rw [smul_zero, add_zero]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inl_mul_inr	null
inr_mul_inl [MonoidWithZero R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (r : R) (m : M) : (inr m * inl r : tsze R M) = inr (m <• r) := ext (zero_mul r) <\| show (0 : R) •> (0 : M) + m <• r = m <• r by rw [smul_zero, zero_add]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inr_mul_inl	null
inl_mul_eq_smul [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (r : R) (x : tsze R M) : inl r * x = r •> x := ext rfl (by dsimp; rw [smul_zero, add_zero])	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inl_mul_eq_smul	null
mul_inl_eq_op_smul [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (x : tsze R M) (r : R) : x * inl r = x <• r := ext rfl (by dsimp; rw [smul_zero, zero_add])	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	mul_inl_eq_op_smul	null
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 + (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) + x.2 <• (1 : R) = x.2 by rw [smul_zero, zero_add, op_one, one_smul] }	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	mulOneClass	null
addMonoidWithOne [AddMonoidWithOne R] [AddMonoid M] : AddMonoidWithOne (tsze R M) := { TrivSqZeroExt.addMonoid, TrivSqZeroExt.one with natCast := fun n => inl n natCast_zero := by simp [Nat.cast] natCast_succ := fun _ => by ext <;> simp [Nat.cast] } @[simp]	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	addMonoidWithOne	null
fst_natCast [AddMonoidWithOne R] [AddMonoid M] (n : ℕ) : (n : tsze R M).fst = n := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	fst_natCast	null
snd_natCast [AddMonoidWithOne R] [AddMonoid M] (n : ℕ) : (n : tsze R M).snd = 0 := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	snd_natCast	null
inl_natCast [AddMonoidWithOne R] [AddMonoid M] (n : ℕ) : (inl n : tsze R M) = n := rfl	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inl_natCast	null
addGroupWithOne [AddGroupWithOne R] [AddGroup M] : AddGroupWithOne (tsze R M) := { TrivSqZeroExt.addGroup, TrivSqZeroExt.addMonoidWithOne with intCast := fun z => inl z intCast_ofNat := fun _n => ext (Int.cast_natCast _) rfl intCast_negSucc := fun _n => ext (Int.cast_negSucc _) neg_zero.symm } @[simp]	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	addGroupWithOne	null
fst_intCast [AddGroupWithOne R] [AddGroup M] (z : ℤ) : (z : tsze R M).fst = z := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	fst_intCast	null
snd_intCast [AddGroupWithOne R] [AddGroup M] (z : ℤ) : (z : tsze R M).snd = 0 := rfl @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	snd_intCast	null
inl_intCast [AddGroupWithOne R] [AddGroup M] (z : ℤ) : (inl z : tsze R M) = z := rfl	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inl_intCast	null
nonAssocSemiring [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] : NonAssocSemiring (tsze R M) := { TrivSqZeroExt.addMonoidWithOne, TrivSqZeroExt.mulOneClass, TrivSqZeroExt.addCommMonoid with zero_mul := fun x => ext (zero_mul x.1) <\| 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) + 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 + (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	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	nonAssocSemiring	null
nonAssocRing [Ring R] [AddCommGroup M] [Module R M] [Module Rᵐᵒᵖ M] : NonAssocRing (tsze R M) := { TrivSqZeroExt.addGroupWithOne, TrivSqZeroExt.nonAssocSemiring with }	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	nonAssocRing	null
snd_list_prod [Monoid R] [AddCommMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] (l : List (tsze R M)) : l.prod.snd = (l.zipIdx.map fun x : tsze R M × ℕ => ((l.map fst).take x.2).prod •> x.fst.snd <• ((l.map fst).drop x.2.succ).prod).sum := by induction l with \| nil => simp \| cons x xs ih => rw [List.zipIdx_cons'] simp_rw [List.map_cons, List.map_map, Function.comp_def, Prod.map_snd, Prod.map_fst, id, List.take_zero, List.take_succ_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, ← smul_comm (_ : R) (_ : Rᵐᵒᵖ)] exact add_comm _ _	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	snd_list_prod	In the general non-commutative case, the power operator is $$\begin{align} (r + m)^n &= r^n + r^{n-1}m + r^{n-2}mr + \cdots + rmr^{n-2} + mr^{n-1} \\ & =r^n + \sum_{i = 0}^{n - 1} r^{(n - 1) - i} m r^{i} \end{align}$$ 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) •> x.snd <• x.fst ^ i).sum⟩⟩ @[simp] theorem fst_pow [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (x : tsze R M) (n : ℕ) : fst (x ^ n) = x.fst ^ n := rfl 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) •> x.snd <• x.fst ^ i).sum := rfl 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.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) => simp_rw [Nat.pred_succ] exact (List.sum_eq_card_nsmul _ (x.fst ^ n • x.snd) (by grind)).trans (by 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 \| zero => simp \| succ n ih => rw [pow_succ, op_mul, mul_smul, mul_smul, ← h, smul_comm (_ : R) (op x.fst) x.snd, ih] 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 := snd_pow_of_smul_comm _ _ (op_smul_eq_smul _ _) @[simp] theorem inl_pow [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (r : R) (n : ℕ) : (inl r ^ n : tsze R M) = inl (r ^ n) := ext rfl <\| by simp [snd_pow_eq_sum, List.map_const'] instance monoid [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] : Monoid (tsze R M) := { TrivSqZeroExt.mulOneClass with mul_assoc := fun x y z => ext (mul_assoc x.1 y.1 z.1) <\| show (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 => ext (pow_succ _ _) (by simp_rw [snd_mul, snd_pow_eq_sum, Nat.pred_succ] cases n · simp [List.range_succ] rw [List.sum_range_succ'] simp only [pow_zero, op_one, Nat.sub_zero, one_smul, Nat.succ_sub_succ_eq_sub, fst_pow, Nat.pred_succ, List.smul_sum, List.map_map, Function.comp_def] simp_rw [← smul_comm (_ : R) (_ : Rᵐᵒᵖ), smul_smul, pow_succ] rfl) } 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) _ instance semiring [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] : Semiring (tsze R M) := { TrivSqZeroExt.monoid, TrivSqZeroExt.nonAssocSemiring with } /-- The second element of a product $\prod_{i=0}^n (r_i + m_i)$ is a sum of terms of the form $r_0\cdots r_{i-1}m_ir_{i+1}\cdots r_n$.
ring [Ring R] [AddCommGroup M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] : Ring (tsze R M) := { TrivSqZeroExt.semiring, TrivSqZeroExt.nonAssocRing with }	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	ring	null
commMonoid [CommMonoid R] [AddCommMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] [IsCentralScalar R M] : CommMonoid (tsze R M) := { TrivSqZeroExt.monoid with mul_comm := fun x₁ x₂ => ext (mul_comm x₁.1 x₂.1) <\| 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	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	commMonoid	null
commSemiring [CommSemiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] : CommSemiring (tsze R M) := { TrivSqZeroExt.commMonoid, TrivSqZeroExt.nonAssocSemiring with }	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	commSemiring	null
commRing [CommRing R] [AddCommGroup M] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] : CommRing (tsze R M) := { TrivSqZeroExt.nonAssocRing, TrivSqZeroExt.commSemiring with } variable (R M)	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	commRing	null
@[simps apply] inlHom [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] : R →+* tsze R M where toFun := inl map_one' := inl_one M map_mul' := inl_mul M map_zero' := inl_zero M map_add' := inl_add M	def	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inlHom	The canonical inclusion of rings `R → TrivSqZeroExt R M`.
instInv : Inv (tsze R M) := ⟨fun b => (b.1⁻¹, -(b.1⁻¹ •> b.2 <• b.1⁻¹))⟩ @[simp] theorem fst_inv (x : tsze R M) : fst x⁻¹ = (fst x)⁻¹ := rfl @[simp] theorem snd_inv (x : tsze R M) : snd x⁻¹ = -((fst x)⁻¹ •> snd x <• (fst x)⁻¹) := rfl	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	instInv	Inversion of the trivial-square-zero extension, sending $r + m$ to $r^{-1} - r^{-1}mr^{-1}$. Strictly this is only a _two_-sided inverse when the left and right actions associate.
invertibleFstOfInvertible (x : tsze R M) [Invertible x] : Invertible x.fst where invOf := (⅟x).fst invOf_mul_self := by rw [← fst_mul, invOf_mul_self, fst_one] mul_invOf_self := by rw [← fst_mul, mul_invOf_self, fst_one]	abbrev	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	invertibleFstOfInvertible	`x.fst : R` is invertible when `x : tzre R M` is.
fst_invOf (x : tsze R M) [Invertible x] [Invertible x.fst] : (⅟x).fst = ⅟(x.fst) := by letI := invertibleFstOfInvertible x convert (rfl : _ = ⅟x.fst)	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	fst_invOf	null
mul_left_eq_one (r : R) (x : tsze R M) (h : r * x.fst = 1) : (inl r + inr (-((r •> x.snd) <• r))) * x = 1 := by ext <;> dsimp · rw [add_zero, h] · rw [add_zero, zero_add, smul_neg, op_smul_op_smul, h, op_one, one_smul, add_neg_cancel]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	mul_left_eq_one	null
mul_right_eq_one (x : tsze R M) (r : R) (h : x.fst * r = 1) : x * (inl r + inr (-(r •> (x.snd <• r)))) = 1 := by ext <;> dsimp · rw [add_zero, h] · rw [add_zero, zero_add, smul_neg, smul_smul, h, one_smul, neg_add_cancel] variable [SMulCommClass R Rᵐᵒᵖ M]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	mul_right_eq_one	null
invertibleOfInvertibleFst (x : tsze R M) [Invertible x.fst] : Invertible x where invOf := (⅟x.fst, -(⅟x.fst •> x.snd <• ⅟x.fst)) invOf_mul_self := by convert mul_left_eq_one _ _ (invOf_mul_self x.fst) ext <;> simp mul_invOf_self := by convert mul_right_eq_one _ _ (mul_invOf_self x.fst) ext <;> simp [smul_comm]	abbrev	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	invertibleOfInvertibleFst	`x : tzre R M` is invertible when `x.fst : R` is.
snd_invOf (x : tsze R M) [Invertible x] [Invertible x.fst] : (⅟x).snd = -(⅟x.fst •> x.snd <• ⅟x.fst) := by letI := invertibleOfInvertibleFst x convert congr_arg (TrivSqZeroExt.snd (R := R) (M := M)) (_ : _ = ⅟x) convert rfl	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	snd_invOf	null
@[simps] invertibleEquivInvertibleFst (x : tsze R M) : Invertible x ≃ Invertible x.fst where toFun _ := invertibleFstOfInvertible x invFun _ := invertibleOfInvertibleFst x left_inv _ := Subsingleton.elim _ _ right_inv _ := Subsingleton.elim _ _	def	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	invertibleEquivInvertibleFst	Together `TrivSqZeroExt.detInvertibleOfInvertible` and `TrivSqZeroExt.invertibleOfDetInvertible` form an equivalence, although both sides of the equiv are subsingleton anyway.
isUnit_iff_isUnit_fst {x : tsze R M} : IsUnit x ↔ IsUnit x.fst := by simp only [← nonempty_invertible_iff_isUnit, (invertibleEquivInvertibleFst x).nonempty_congr] @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	isUnit_iff_isUnit_fst	When lowered to a prop, `Matrix.invertibleEquivInvertibleFst` forms an `iff`.
isUnit_inl_iff {r : R} : IsUnit (inl r : tsze R M) ↔ IsUnit r := by rw [isUnit_iff_isUnit_fst, fst_inl] @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	isUnit_inl_iff	null
isUnit_inr_iff {m : M} : IsUnit (inr m : tsze R M) ↔ Subsingleton R := by simp_rw [isUnit_iff_isUnit_fst, fst_inr, isUnit_zero_iff, subsingleton_iff_zero_eq_one]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	isUnit_inr_iff	null
protected inv_inl (r : R) : (inl r)⁻¹ = (inl (r⁻¹ : R) : tsze R M) := by ext · rw [fst_inv, fst_inl, fst_inl] · rw [snd_inv, fst_inl, snd_inl, snd_inl, smul_zero, smul_zero, neg_zero] @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inv_inl	null
inv_inr (m : M) : (inr m)⁻¹ = (0 : tsze R M) := by ext · rw [fst_inv, fst_inr, fst_zero, inv_zero] · rw [snd_inv, snd_inr, fst_inr, inv_zero, op_zero, zero_smul, snd_zero, neg_zero] @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inv_inr	null
protected inv_zero : (0 : tsze R M)⁻¹ = (0 : tsze R M) := by rw [← inl_zero, TrivSqZeroExt.inv_inl, inv_zero] @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inv_zero	null
protected inv_one : (1 : tsze R M)⁻¹ = (1 : tsze R M) := by rw [← inl_one, TrivSqZeroExt.inv_inl, inv_one]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inv_one	null
protected inv_mul_cancel {x : tsze R M} (hx : fst x ≠ 0) : x⁻¹ * x = 1 := by convert mul_left_eq_one _ _ (_root_.inv_mul_cancel₀ hx) using 2 ext <;> simp variable [SMulCommClass R Rᵐᵒᵖ M] @[simp] theorem invOf_eq_inv (x : tsze R M) [Invertible x] : ⅟x = x⁻¹ := by letI := invertibleFstOfInvertible x ext <;> simp [fst_invOf, snd_invOf]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inv_mul_cancel	null
protected mul_inv_cancel {x : tsze R M} (hx : fst x ≠ 0) : x * x⁻¹ = 1 := by have : Invertible x.fst := Units.invertible (.mk0 _ hx) have := invertibleOfInvertibleFst x rw [← invOf_eq_inv, mul_invOf_self]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	mul_inv_cancel	null
protected mul_inv_rev (a b : tsze R M) : (a * b)⁻¹ = b⁻¹ * a⁻¹ := by ext · rw [fst_inv, fst_mul, fst_mul, mul_inv_rev, fst_inv, fst_inv] · simp only [snd_inv, snd_mul, fst_mul, fst_inv] simp only [smul_neg, smul_add] simp_rw [mul_inv_rev, smul_comm (_ : R), op_smul_op_smul, smul_smul, add_comm, neg_add] obtain ha0 \| ha := eq_or_ne (fst a) 0 · simp [ha0] obtain hb0 \| hb := eq_or_ne (fst b) 0 · simp [hb0] rw [inv_mul_cancel_right₀ ha, mul_inv_cancel_left₀ hb]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	mul_inv_rev	null
protected inv_inv {x : tsze R M} (hx : fst x ≠ 0) : x⁻¹⁻¹ = x := calc x⁻¹⁻¹ = 1 * x⁻¹⁻¹ := by rw [one_mul] _ = x * x⁻¹ * x⁻¹⁻¹ := by rw [TrivSqZeroExt.mul_inv_cancel hx] _ = x := by rw [mul_assoc, TrivSqZeroExt.mul_inv_cancel, mul_one] rw [fst_inv] apply inv_ne_zero hx @[simp]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inv_inv	null
isUnit_inv_iff {x : tsze R M} : IsUnit x⁻¹ ↔ IsUnit x := by simp_rw [isUnit_iff_isUnit_fst, fst_inv, isUnit_iff_ne_zero, ne_eq, inv_eq_zero]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	isUnit_inv_iff	null
protected inv_neg {x : tsze R M} : (-x)⁻¹ = -(x⁻¹) := by ext <;> simp [inv_neg]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inv_neg	null
algebra' : Algebra S (tsze R M) where algebraMap := (TrivSqZeroExt.inlHom R M).comp (algebraMap S R) smul := (· • ·) commutes' := fun s x => ext (Algebra.commutes _ _) <\| 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, op_one, smul_assoc, one_smul, smul_assoc, one_smul] smul_def' := fun s x => ext (Algebra.smul_def _ _) <\| show s • x.snd = algebraMap S R s •> x.snd + (0 : M) <• x.fst by rw [smul_zero, add_zero, algebraMap_smul]	instance	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	algebra'	null
algebraMap_eq_inl : ⇑(algebraMap R' (tsze R' M)) = inl := rfl	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	algebraMap_eq_inl	null
algebraMap_eq_inlHom : algebraMap R' (tsze R' M) = inlHom R' M := rfl	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	algebraMap_eq_inlHom	null
algebraMap_eq_inl' (s : S) : algebraMap S (tsze R M) s = inl (algebraMap S R s) := rfl	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	algebraMap_eq_inl'	null
@[simps] fstHom : tsze R M →ₐ[S] R where toFun := fst map_one' := fst_one map_mul' := fst_mul map_zero' := fst_zero (M := M) map_add' := fst_add commutes' _r := fst_inl M _	def	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	fstHom	The canonical `S`-algebra projection `TrivSqZeroExt R M → R`.
@[simps] inlAlgHom : R →ₐ[S] tsze R M where toFun := inl map_one' := inl_one _ map_mul' := inl_mul _ map_zero' := inl_zero (M := M) map_add' := inl_add _ commutes' _r := (algebraMap_eq_inl' _ _ _ _).symm variable {R R' S M}	def	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	inlAlgHom	The canonical `S`-algebra inclusion `R → TrivSqZeroExt R M`.
algHom_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 := AlgHom.toLinearMap_injective <\| linearMap_ext (fun _r => (f.commutes _).trans (g.commutes _).symm) h @[ext]	theorem	Algebra	[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]	Mathlib/Algebra/TrivSqZeroExt.lean	algHom_ext	null

addCommSemigroup [AddCommSemigroup R] [AddCommSemigroup M] : AddCommSemigroup (tsze R M) := Prod.instAddCommSemigroup

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

addCommSemigroup

null

addCommMonoid [AddCommMonoid R] [AddCommMonoid M] : AddCommMonoid (tsze R M) := Prod.instAddCommMonoid

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

addCommMonoid

null

addCommGroup [AddCommGroup R] [AddCommGroup M] : AddCommGroup (tsze R M) := Prod.instAddCommGroup

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

addCommGroup

null

smul [SMul S R] [SMul S M] : SMul S (tsze R M) := Prod.instSMul

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

smul

null

isScalarTower [SMul T R] [SMul T M] [SMul S R] [SMul S M] [SMul T S] [IsScalarTower T S R] [IsScalarTower T S M] : IsScalarTower T S (tsze R M) := Prod.isScalarTower

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

isScalarTower

null

smulCommClass [SMul T R] [SMul T M] [SMul S R] [SMul S M] [SMulCommClass T S R] [SMulCommClass T S M] : SMulCommClass T S (tsze R M) := Prod.smulCommClass

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

smulCommClass

null

isCentralScalar [SMul S R] [SMul S M] [SMul Sᵐᵒᵖ R] [SMul Sᵐᵒᵖ M] [IsCentralScalar S R] [IsCentralScalar S M] : IsCentralScalar S (tsze R M) := Prod.isCentralScalar

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

isCentralScalar

null

mulAction [Monoid S] [MulAction S R] [MulAction S M] : MulAction S (tsze R M) := Prod.mulAction

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

mulAction

null

distribMulAction [Monoid S] [AddMonoid R] [AddMonoid M] [DistribMulAction S R] [DistribMulAction S M] : DistribMulAction S (tsze R M) := Prod.distribMulAction

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

distribMulAction

null

module [Semiring S] [AddCommMonoid R] [AddCommMonoid M] [Module S R] [Module S M] : Module S (tsze R M) := Prod.instModule

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

module

null

instNontrivial_of_left {R M : Type*} [Nontrivial R] [Nonempty M] : Nontrivial (TrivSqZeroExt R M) := fst_surjective.nontrivial

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

instNontrivial_of_left

The trivial square-zero extension is nontrivial if it is over a nontrivial ring.

instNontrivial_of_right {R M : Type*} [Nonempty R] [Nontrivial M] : Nontrivial (TrivSqZeroExt R M) := snd_surjective.nontrivial @[simp]

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

instNontrivial_of_right

The trivial square-zero extension is nontrivial if it is over a nontrivial module.

fst_zero [Zero R] [Zero M] : (0 : tsze R M).fst = 0 := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

fst_zero

null

snd_zero [Zero R] [Zero M] : (0 : tsze R M).snd = 0 := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

snd_zero

null

fst_add [Add R] [Add M] (x₁ x₂ : tsze R M) : (x₁ + x₂).fst = x₁.fst + x₂.fst := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

fst_add

null

snd_add [Add R] [Add M] (x₁ x₂ : tsze R M) : (x₁ + x₂).snd = x₁.snd + x₂.snd := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

snd_add

null

fst_neg [Neg R] [Neg M] (x : tsze R M) : (-x).fst = -x.fst := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

fst_neg

null

snd_neg [Neg R] [Neg M] (x : tsze R M) : (-x).snd = -x.snd := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

snd_neg

null

fst_sub [Sub R] [Sub M] (x₁ x₂ : tsze R M) : (x₁ - x₂).fst = x₁.fst - x₂.fst := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

fst_sub

null

snd_sub [Sub R] [Sub M] (x₁ x₂ : tsze R M) : (x₁ - x₂).snd = x₁.snd - x₂.snd := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

snd_sub

null

fst_smul [SMul S R] [SMul S M] (s : S) (x : tsze R M) : (s • x).fst = s • x.fst := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

fst_smul

null

snd_smul [SMul S R] [SMul S M] (s : S) (x : tsze R M) : (s • x).snd = s • x.snd := rfl

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

snd_smul

null

fst_sum {ι} [AddCommMonoid R] [AddCommMonoid M] (s : Finset ι) (f : ι → tsze R M) : (∑ i ∈ s, f i).fst = ∑ i ∈ s, (f i).fst := Prod.fst_sum

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

fst_sum

null

snd_sum {ι} [AddCommMonoid R] [AddCommMonoid M] (s : Finset ι) (f : ι → tsze R M) : (∑ i ∈ s, f i).snd = ∑ i ∈ s, (f i).snd := Prod.snd_sum

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

snd_sum

null

@[simp] inl_zero [Zero R] [Zero M] : (inl 0 : tsze R M) = 0 := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inl_zero

null

inl_add [Add R] [AddZeroClass M] (r₁ r₂ : R) : (inl (r₁ + r₂) : tsze R M) = inl r₁ + inl r₂ := ext rfl (add_zero 0).symm @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inl_add

null

inl_neg [Neg R] [NegZeroClass M] (r : R) : (inl (-r) : tsze R M) = -inl r := ext rfl neg_zero.symm @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inl_neg

null

inl_sub [Sub R] [SubNegZeroMonoid M] (r₁ r₂ : R) : (inl (r₁ - r₂) : tsze R M) = inl r₁ - inl r₂ := ext rfl (sub_zero _).symm @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inl_sub

null

inl_smul [Monoid S] [AddMonoid M] [SMul S R] [DistribMulAction S M] (s : S) (r : R) : (inl (s • r) : tsze R M) = s • inl r := ext rfl (smul_zero s).symm

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inl_smul

null

inl_sum {ι} [AddCommMonoid R] [AddCommMonoid M] (s : Finset ι) (f : ι → R) : (inl (∑ i ∈ s, f i) : tsze R M) = ∑ i ∈ s, inl (f i) := map_sum (LinearMap.inl ℕ _ _) _ _

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inl_sum

null

@[simp] inr_zero [Zero R] [Zero M] : (inr 0 : tsze R M) = 0 := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inr_zero

null

inr_add [AddZeroClass R] [Add M] (m₁ m₂ : M) : (inr (m₁ + m₂) : tsze R M) = inr m₁ + inr m₂ := ext (add_zero 0).symm rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inr_add

null

inr_neg [NegZeroClass R] [Neg M] (m : M) : (inr (-m) : tsze R M) = -inr m := ext neg_zero.symm rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inr_neg

null

inr_sub [SubNegZeroMonoid R] [Sub M] (m₁ m₂ : M) : (inr (m₁ - m₂) : tsze R M) = inr m₁ - inr m₂ := ext (sub_zero _).symm rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inr_sub

null

inr_smul [Zero R] [SMulZeroClass S R] [SMul S M] (r : S) (m : M) : (inr (r • m) : tsze R M) = r • inr m := ext (smul_zero _).symm rfl

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inr_smul

null

inr_sum {ι} [AddCommMonoid R] [AddCommMonoid M] (s : Finset ι) (f : ι → M) : (inr (∑ i ∈ s, f i) : tsze R M) = ∑ i ∈ s, inr (f i) := map_sum (LinearMap.inr ℕ _ _) _ _

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inr_sum

null

inl_fst_add_inr_snd_eq [AddZeroClass R] [AddZeroClass M] (x : tsze R M) : inl x.fst + inr x.snd = x := ext (add_zero x.1) (zero_add x.2)

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inl_fst_add_inr_snd_eq

null

@[elab_as_elim, induction_eliminator, cases_eliminator] ind {R M} [AddZeroClass R] [AddZeroClass M] {P : TrivSqZeroExt R M → Prop} (inl_add_inr : ∀ r m, P (inl r + inr m)) (x) : P x := inl_fst_add_inr_snd_eq x ▸ inl_add_inr x.1 x.2

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

ind

To show a property hold on all `TrivSqZeroExt R M` it suffices to show it holds on terms of the form `inl r + inr m`.

linearMap_ext {N} [Semiring S] [AddCommMonoid R] [AddCommMonoid M] [AddCommMonoid N] [Module S R] [Module S M] [Module S N] ⦃f g : tsze R M →ₗ[S] N⦄ (hl : ∀ r, f (inl r) = g (inl r)) (hr : ∀ m, f (inr m) = g (inr m)) : f = g := LinearMap.prod_ext (LinearMap.ext hl) (LinearMap.ext hr) variable (R M)

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

linearMap_ext

This cannot be marked `@[ext]` as it ends up being used instead of `LinearMap.prod_ext` when working with `R × M`.

@[simps apply] inrHom [Semiring R] [AddCommMonoid M] [Module R M] : M →ₗ[R] tsze R M := { LinearMap.inr R R M with toFun := inr }

def

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inrHom

The canonical `R`-linear inclusion `M → TrivSqZeroExt R M`.

@[simps apply] sndHom [Semiring R] [AddCommMonoid M] [Module R M] : tsze R M →ₗ[R] M := { LinearMap.snd _ _ _ with toFun := snd }

def

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

sndHom

The canonical `R`-linear projection `TrivSqZeroExt R M → M`.

one [One R] [Zero M] : One (tsze R M) := ⟨(1, 0)⟩

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

one

null

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 + x.2 <• y.1)⟩ @[simp]

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

mul

null

fst_one [One R] [Zero M] : (1 : tsze R M).fst = 1 := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

fst_one

null

snd_one [One R] [Zero M] : (1 : tsze R M).snd = 0 := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

snd_one

null

fst_mul [Mul R] [Add M] [SMul R M] [SMul Rᵐᵒᵖ M] (x₁ x₂ : tsze R M) : (x₁ * x₂).fst = x₁.fst * x₂.fst := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

fst_mul

null

snd_mul [Mul R] [Add M] [SMul R M] [SMul Rᵐᵒᵖ M] (x₁ x₂ : tsze R M) : (x₁ * x₂).snd = x₁.fst •> x₂.snd + x₁.snd <• x₂.fst := rfl

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

snd_mul

null

@[simp] inl_one [One R] [Zero M] : (inl 1 : tsze R M) = 1 := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inl_one

null

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) + (0 : M) <• r₂ by rw [smul_zero, zero_add, smul_zero]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inl_mul

null

inl_mul_inl [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (r₁ r₂ : R) : (inl r₁ * inl r₂ : tsze R M) = inl (r₁ * r₂) := (inl_mul M r₁ r₂).symm

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inl_mul_inl

null

@[simp] 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 + m₁ <• (0 : R) = 0 by rw [zero_smul, zero_add, op_zero, zero_smul]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inr_mul_inr

null

inl_mul_inr [MonoidWithZero R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (r : R) (m : M) : (inl r * inr m : tsze R M) = inr (r • m) := ext (mul_zero r) <| show r • m + (0 : Rᵐᵒᵖ) • (0 : M) = r • m by rw [smul_zero, add_zero]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inl_mul_inr

null

inr_mul_inl [MonoidWithZero R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (r : R) (m : M) : (inr m * inl r : tsze R M) = inr (m <• r) := ext (zero_mul r) <| show (0 : R) •> (0 : M) + m <• r = m <• r by rw [smul_zero, zero_add]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inr_mul_inl

null

inl_mul_eq_smul [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (r : R) (x : tsze R M) : inl r * x = r •> x := ext rfl (by dsimp; rw [smul_zero, add_zero])

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inl_mul_eq_smul

null

mul_inl_eq_op_smul [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (x : tsze R M) (r : R) : x * inl r = x <• r := ext rfl (by dsimp; rw [smul_zero, zero_add])

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

mul_inl_eq_op_smul

null

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 + (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) + x.2 <• (1 : R) = x.2 by rw [smul_zero, zero_add, op_one, one_smul] }

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

mulOneClass

null

addMonoidWithOne [AddMonoidWithOne R] [AddMonoid M] : AddMonoidWithOne (tsze R M) := { TrivSqZeroExt.addMonoid, TrivSqZeroExt.one with natCast := fun n => inl n natCast_zero := by simp [Nat.cast] natCast_succ := fun _ => by ext <;> simp [Nat.cast] } @[simp]

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

addMonoidWithOne

null

fst_natCast [AddMonoidWithOne R] [AddMonoid M] (n : ℕ) : (n : tsze R M).fst = n := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

fst_natCast

null

snd_natCast [AddMonoidWithOne R] [AddMonoid M] (n : ℕ) : (n : tsze R M).snd = 0 := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

snd_natCast

null

inl_natCast [AddMonoidWithOne R] [AddMonoid M] (n : ℕ) : (inl n : tsze R M) = n := rfl

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inl_natCast

null

addGroupWithOne [AddGroupWithOne R] [AddGroup M] : AddGroupWithOne (tsze R M) := { TrivSqZeroExt.addGroup, TrivSqZeroExt.addMonoidWithOne with intCast := fun z => inl z intCast_ofNat := fun _n => ext (Int.cast_natCast _) rfl intCast_negSucc := fun _n => ext (Int.cast_negSucc _) neg_zero.symm } @[simp]

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

addGroupWithOne

null

fst_intCast [AddGroupWithOne R] [AddGroup M] (z : ℤ) : (z : tsze R M).fst = z := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

fst_intCast

null

snd_intCast [AddGroupWithOne R] [AddGroup M] (z : ℤ) : (z : tsze R M).snd = 0 := rfl @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

snd_intCast

null

inl_intCast [AddGroupWithOne R] [AddGroup M] (z : ℤ) : (inl z : tsze R M) = z := rfl

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inl_intCast

null

nonAssocSemiring [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] : NonAssocSemiring (tsze R M) := { TrivSqZeroExt.addMonoidWithOne, TrivSqZeroExt.mulOneClass, TrivSqZeroExt.addCommMonoid with zero_mul := fun x => ext (zero_mul x.1) <| 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) + 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 + (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

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

nonAssocSemiring

null

nonAssocRing [Ring R] [AddCommGroup M] [Module R M] [Module Rᵐᵒᵖ M] : NonAssocRing (tsze R M) := { TrivSqZeroExt.addGroupWithOne, TrivSqZeroExt.nonAssocSemiring with }

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

nonAssocRing

null

snd_list_prod [Monoid R] [AddCommMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] (l : List (tsze R M)) : l.prod.snd = (l.zipIdx.map fun x : tsze R M × ℕ => ((l.map fst).take x.2).prod •> x.fst.snd <• ((l.map fst).drop x.2.succ).prod).sum := by induction l with | nil => simp | cons x xs ih => rw [List.zipIdx_cons'] simp_rw [List.map_cons, List.map_map, Function.comp_def, Prod.map_snd, Prod.map_fst, id, List.take_zero, List.take_succ_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, ← smul_comm (_ : R) (_ : Rᵐᵒᵖ)] exact add_comm _ _

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

snd_list_prod

In the general non-commutative case, the power operator is $$\begin{align} (r + m)^n &= r^n + r^{n-1}m + r^{n-2}mr + \cdots + rmr^{n-2} + mr^{n-1} \\ & =r^n + \sum_{i = 0}^{n - 1} r^{(n - 1) - i} m r^{i} \end{align}$$ 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) •> x.snd <• x.fst ^ i).sum⟩⟩ @[simp] theorem fst_pow [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (x : tsze R M) (n : ℕ) : fst (x ^ n) = x.fst ^ n := rfl 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) •> x.snd <• x.fst ^ i).sum := rfl 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.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) => simp_rw [Nat.pred_succ] exact (List.sum_eq_card_nsmul _ (x.fst ^ n • x.snd) (by grind)).trans (by 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 | zero => simp | succ n ih => rw [pow_succ, op_mul, mul_smul, mul_smul, ← h, smul_comm (_ : R) (op x.fst) x.snd, ih] 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 := snd_pow_of_smul_comm _ _ (op_smul_eq_smul _ _) @[simp] theorem inl_pow [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] (r : R) (n : ℕ) : (inl r ^ n : tsze R M) = inl (r ^ n) := ext rfl <| by simp [snd_pow_eq_sum, List.map_const'] instance monoid [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] : Monoid (tsze R M) := { TrivSqZeroExt.mulOneClass with mul_assoc := fun x y z => ext (mul_assoc x.1 y.1 z.1) <| show (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 => ext (pow_succ _ _) (by simp_rw [snd_mul, snd_pow_eq_sum, Nat.pred_succ] cases n · simp [List.range_succ] rw [List.sum_range_succ'] simp only [pow_zero, op_one, Nat.sub_zero, one_smul, Nat.succ_sub_succ_eq_sub, fst_pow, Nat.pred_succ, List.smul_sum, List.map_map, Function.comp_def] simp_rw [← smul_comm (_ : R) (_ : Rᵐᵒᵖ), smul_smul, pow_succ] rfl) } 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) _ instance semiring [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] : Semiring (tsze R M) := { TrivSqZeroExt.monoid, TrivSqZeroExt.nonAssocSemiring with } /-- The second element of a product $\prod_{i=0}^n (r_i + m_i)$ is a sum of terms of the form $r_0\cdots r_{i-1}m_ir_{i+1}\cdots r_n$.

ring [Ring R] [AddCommGroup M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] : Ring (tsze R M) := { TrivSqZeroExt.semiring, TrivSqZeroExt.nonAssocRing with }

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

ring

null

commMonoid [CommMonoid R] [AddCommMonoid M] [DistribMulAction R M] [DistribMulAction Rᵐᵒᵖ M] [IsCentralScalar R M] : CommMonoid (tsze R M) := { TrivSqZeroExt.monoid with mul_comm := fun x₁ x₂ => ext (mul_comm x₁.1 x₂.1) <| 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

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

commMonoid

null

commSemiring [CommSemiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] : CommSemiring (tsze R M) := { TrivSqZeroExt.commMonoid, TrivSqZeroExt.nonAssocSemiring with }

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

commSemiring

null

commRing [CommRing R] [AddCommGroup M] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] : CommRing (tsze R M) := { TrivSqZeroExt.nonAssocRing, TrivSqZeroExt.commSemiring with } variable (R M)

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

commRing

null

@[simps apply] inlHom [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] : R →+* tsze R M where toFun := inl map_one' := inl_one M map_mul' := inl_mul M map_zero' := inl_zero M map_add' := inl_add M

def

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inlHom

The canonical inclusion of rings `R → TrivSqZeroExt R M`.

instInv : Inv (tsze R M) := ⟨fun b => (b.1⁻¹, -(b.1⁻¹ •> b.2 <• b.1⁻¹))⟩ @[simp] theorem fst_inv (x : tsze R M) : fst x⁻¹ = (fst x)⁻¹ := rfl @[simp] theorem snd_inv (x : tsze R M) : snd x⁻¹ = -((fst x)⁻¹ •> snd x <• (fst x)⁻¹) := rfl

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

instInv

Inversion of the trivial-square-zero extension, sending $r + m$ to $r^{-1} - r^{-1}mr^{-1}$. Strictly this is only a _two_-sided inverse when the left and right actions associate.

invertibleFstOfInvertible (x : tsze R M) [Invertible x] : Invertible x.fst where invOf := (⅟x).fst invOf_mul_self := by rw [← fst_mul, invOf_mul_self, fst_one] mul_invOf_self := by rw [← fst_mul, mul_invOf_self, fst_one]

abbrev

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

invertibleFstOfInvertible

`x.fst : R` is invertible when `x : tzre R M` is.

fst_invOf (x : tsze R M) [Invertible x] [Invertible x.fst] : (⅟x).fst = ⅟(x.fst) := by letI := invertibleFstOfInvertible x convert (rfl : _ = ⅟x.fst)

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

fst_invOf

null

mul_left_eq_one (r : R) (x : tsze R M) (h : r * x.fst = 1) : (inl r + inr (-((r •> x.snd) <• r))) * x = 1 := by ext <;> dsimp · rw [add_zero, h] · rw [add_zero, zero_add, smul_neg, op_smul_op_smul, h, op_one, one_smul, add_neg_cancel]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

mul_left_eq_one

null

mul_right_eq_one (x : tsze R M) (r : R) (h : x.fst * r = 1) : x * (inl r + inr (-(r •> (x.snd <• r)))) = 1 := by ext <;> dsimp · rw [add_zero, h] · rw [add_zero, zero_add, smul_neg, smul_smul, h, one_smul, neg_add_cancel] variable [SMulCommClass R Rᵐᵒᵖ M]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

mul_right_eq_one

null

invertibleOfInvertibleFst (x : tsze R M) [Invertible x.fst] : Invertible x where invOf := (⅟x.fst, -(⅟x.fst •> x.snd <• ⅟x.fst)) invOf_mul_self := by convert mul_left_eq_one _ _ (invOf_mul_self x.fst) ext <;> simp mul_invOf_self := by convert mul_right_eq_one _ _ (mul_invOf_self x.fst) ext <;> simp [smul_comm]

abbrev

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

invertibleOfInvertibleFst

`x : tzre R M` is invertible when `x.fst : R` is.

snd_invOf (x : tsze R M) [Invertible x] [Invertible x.fst] : (⅟x).snd = -(⅟x.fst •> x.snd <• ⅟x.fst) := by letI := invertibleOfInvertibleFst x convert congr_arg (TrivSqZeroExt.snd (R := R) (M := M)) (_ : _ = ⅟x) convert rfl

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

snd_invOf

null

@[simps] invertibleEquivInvertibleFst (x : tsze R M) : Invertible x ≃ Invertible x.fst where toFun _ := invertibleFstOfInvertible x invFun _ := invertibleOfInvertibleFst x left_inv _ := Subsingleton.elim _ _ right_inv _ := Subsingleton.elim _ _

def

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

invertibleEquivInvertibleFst

Together `TrivSqZeroExt.detInvertibleOfInvertible` and `TrivSqZeroExt.invertibleOfDetInvertible` form an equivalence, although both sides of the equiv are subsingleton anyway.

isUnit_iff_isUnit_fst {x : tsze R M} : IsUnit x ↔ IsUnit x.fst := by simp only [← nonempty_invertible_iff_isUnit, (invertibleEquivInvertibleFst x).nonempty_congr] @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

isUnit_iff_isUnit_fst

When lowered to a prop, `Matrix.invertibleEquivInvertibleFst` forms an `iff`.

isUnit_inl_iff {r : R} : IsUnit (inl r : tsze R M) ↔ IsUnit r := by rw [isUnit_iff_isUnit_fst, fst_inl] @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

isUnit_inl_iff

null

isUnit_inr_iff {m : M} : IsUnit (inr m : tsze R M) ↔ Subsingleton R := by simp_rw [isUnit_iff_isUnit_fst, fst_inr, isUnit_zero_iff, subsingleton_iff_zero_eq_one]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

isUnit_inr_iff

null

protected inv_inl (r : R) : (inl r)⁻¹ = (inl (r⁻¹ : R) : tsze R M) := by ext · rw [fst_inv, fst_inl, fst_inl] · rw [snd_inv, fst_inl, snd_inl, snd_inl, smul_zero, smul_zero, neg_zero] @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inv_inl

null

inv_inr (m : M) : (inr m)⁻¹ = (0 : tsze R M) := by ext · rw [fst_inv, fst_inr, fst_zero, inv_zero] · rw [snd_inv, snd_inr, fst_inr, inv_zero, op_zero, zero_smul, snd_zero, neg_zero] @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inv_inr

null

protected inv_zero : (0 : tsze R M)⁻¹ = (0 : tsze R M) := by rw [← inl_zero, TrivSqZeroExt.inv_inl, inv_zero] @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inv_zero

null

protected inv_one : (1 : tsze R M)⁻¹ = (1 : tsze R M) := by rw [← inl_one, TrivSqZeroExt.inv_inl, inv_one]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inv_one

null

protected inv_mul_cancel {x : tsze R M} (hx : fst x ≠ 0) : x⁻¹ * x = 1 := by convert mul_left_eq_one _ _ (_root_.inv_mul_cancel₀ hx) using 2 ext <;> simp variable [SMulCommClass R Rᵐᵒᵖ M] @[simp] theorem invOf_eq_inv (x : tsze R M) [Invertible x] : ⅟x = x⁻¹ := by letI := invertibleFstOfInvertible x ext <;> simp [fst_invOf, snd_invOf]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inv_mul_cancel

null

protected mul_inv_cancel {x : tsze R M} (hx : fst x ≠ 0) : x * x⁻¹ = 1 := by have : Invertible x.fst := Units.invertible (.mk0 _ hx) have := invertibleOfInvertibleFst x rw [← invOf_eq_inv, mul_invOf_self]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

mul_inv_cancel

null

protected mul_inv_rev (a b : tsze R M) : (a * b)⁻¹ = b⁻¹ * a⁻¹ := by ext · rw [fst_inv, fst_mul, fst_mul, mul_inv_rev, fst_inv, fst_inv] · simp only [snd_inv, snd_mul, fst_mul, fst_inv] simp only [smul_neg, smul_add] simp_rw [mul_inv_rev, smul_comm (_ : R), op_smul_op_smul, smul_smul, add_comm, neg_add] obtain ha0 | ha := eq_or_ne (fst a) 0 · simp [ha0] obtain hb0 | hb := eq_or_ne (fst b) 0 · simp [hb0] rw [inv_mul_cancel_right₀ ha, mul_inv_cancel_left₀ hb]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

mul_inv_rev

null

protected inv_inv {x : tsze R M} (hx : fst x ≠ 0) : x⁻¹⁻¹ = x := calc x⁻¹⁻¹ = 1 * x⁻¹⁻¹ := by rw [one_mul] _ = x * x⁻¹ * x⁻¹⁻¹ := by rw [TrivSqZeroExt.mul_inv_cancel hx] _ = x := by rw [mul_assoc, TrivSqZeroExt.mul_inv_cancel, mul_one] rw [fst_inv] apply inv_ne_zero hx @[simp]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inv_inv

null

isUnit_inv_iff {x : tsze R M} : IsUnit x⁻¹ ↔ IsUnit x := by simp_rw [isUnit_iff_isUnit_fst, fst_inv, isUnit_iff_ne_zero, ne_eq, inv_eq_zero]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

isUnit_inv_iff

null

protected inv_neg {x : tsze R M} : (-x)⁻¹ = -(x⁻¹) := by ext <;> simp [inv_neg]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inv_neg

null

algebra' : Algebra S (tsze R M) where algebraMap := (TrivSqZeroExt.inlHom R M).comp (algebraMap S R) smul := (· • ·) commutes' := fun s x => ext (Algebra.commutes _ _) <| 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, op_one, smul_assoc, one_smul, smul_assoc, one_smul] smul_def' := fun s x => ext (Algebra.smul_def _ _) <| show s • x.snd = algebraMap S R s •> x.snd + (0 : M) <• x.fst by rw [smul_zero, add_zero, algebraMap_smul]

instance

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

algebra'

null

algebraMap_eq_inl : ⇑(algebraMap R' (tsze R' M)) = inl := rfl

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

algebraMap_eq_inl

null

algebraMap_eq_inlHom : algebraMap R' (tsze R' M) = inlHom R' M := rfl

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

algebraMap_eq_inlHom

null

algebraMap_eq_inl' (s : S) : algebraMap S (tsze R M) s = inl (algebraMap S R s) := rfl

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

algebraMap_eq_inl'

null

@[simps] fstHom : tsze R M →ₐ[S] R where toFun := fst map_one' := fst_one map_mul' := fst_mul map_zero' := fst_zero (M := M) map_add' := fst_add commutes' _r := fst_inl M _

def

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

fstHom

The canonical `S`-algebra projection `TrivSqZeroExt R M → R`.

@[simps] inlAlgHom : R →ₐ[S] tsze R M where toFun := inl map_one' := inl_one _ map_mul' := inl_mul _ map_zero' := inl_zero (M := M) map_add' := inl_add _ commutes' _r := (algebraMap_eq_inl' _ _ _ _).symm variable {R R' S M}

def

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

inlAlgHom

The canonical `S`-algebra inclusion `R → TrivSqZeroExt R M`.

algHom_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 := AlgHom.toLinearMap_injective <| linearMap_ext (fun _r => (f.commutes _).trans (g.commutes _).symm) h @[ext]

theorem

Algebra

[ "Mathlib.Algebra.BigOperators.GroupWithZero.Action", "Mathlib.Algebra.GroupWithZero.Invertible", "Mathlib.LinearAlgebra.Prod", "Mathlib.Algebra.Algebra.Subalgebra.Lattice" ]

Mathlib/Algebra/TrivSqZeroExt.lean

algHom_ext

null