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 ⌀ |
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noncommCoprod_unique (f : M × N →* P) :
(f.comp (inl M N)).noncommCoprod (f.comp (inr M N)) (fun _ _ => (commute_inl_inr _ _).map f)
= f :=
ext fun x => by simp [inl_apply, inr_apply, ← map_mul]
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Algebra.Group.Subgroup.Ker",
"Mathlib.Algebra.Group.Subgroup.Lattice",
"Mathlib.Order.Disjoint"
] | Mathlib/GroupTheory/NoncommCoprod.lean | noncommCoprod_unique | null |
noncommCoprod_inl_inr {M N : Type*} [Monoid M] [Monoid N] :
(inl M N).noncommCoprod (inr M N) commute_inl_inr = id (M × N) :=
noncommCoprod_unique <| .id (M × N)
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Algebra.Group.Subgroup.Ker",
"Mathlib.Algebra.Group.Subgroup.Lattice",
"Mathlib.Order.Disjoint"
] | Mathlib/GroupTheory/NoncommCoprod.lean | noncommCoprod_inl_inr | null |
comp_noncommCoprod {Q : Type*} [Monoid Q] (h : P →* Q) :
h.comp (f.noncommCoprod g comm) =
(h.comp f).noncommCoprod (h.comp g) (fun m n ↦ (comm m n).map h) :=
ext fun x => by simp | theorem | GroupTheory | [
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Algebra.Group.Subgroup.Ker",
"Mathlib.Algebra.Group.Subgroup.Lattice",
"Mathlib.Order.Disjoint"
] | Mathlib/GroupTheory/NoncommCoprod.lean | comp_noncommCoprod | null |
noncommCoprod_injective {M N P : Type*} [Group M] [Group N] [Group P]
(f : M →* P) (g : N →* P) (comm : ∀ (m : M) (n : N), Commute (f m) (g n)) :
Function.Injective (noncommCoprod f g comm) ↔
(Function.Injective f ∧ Function.Injective g ∧ _root_.Disjoint f.range g.range) := by
simp only [injective_iff_map_eq_one, disjoint_iff_inf_le,
noncommCoprod_apply, Prod.forall, Prod.mk_eq_one]
refine ⟨fun h ↦ ⟨fun x ↦ ?_, fun x ↦ ?_, ?_⟩, ?_⟩
· simpa using h x 1
· simpa using h 1 x
· intro x ⟨⟨y, hy⟩, z, hz⟩
rwa [(h y z⁻¹ (by rw [map_inv, hy, hz, mul_inv_cancel])).1, map_one, eq_comm] at hy
· intro ⟨hf, hg, hp⟩ a b h
have key := hp ⟨⟨a⁻¹, by rwa [map_inv, inv_eq_iff_mul_eq_one]⟩, b, rfl⟩
exact ⟨hf a (by rwa [key, mul_one] at h), hg b key⟩ | lemma | GroupTheory | [
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Algebra.Group.Subgroup.Ker",
"Mathlib.Algebra.Group.Subgroup.Lattice",
"Mathlib.Order.Disjoint"
] | Mathlib/GroupTheory/NoncommCoprod.lean | noncommCoprod_injective | null |
noncommCoprod_range {M N P : Type*} [Group M] [Group N] [Group P]
(f : M →* P) (g : N →* P) (comm : ∀ (m : M) (n : N), Commute (f m) (g n)) :
(noncommCoprod f g comm).range = f.range ⊔ g.range := by
apply le_antisymm
· rintro - ⟨a, rfl⟩
exact mul_mem (mem_sup_left ⟨a.1, rfl⟩) (mem_sup_right ⟨a.2, rfl⟩)
· rw [sup_le_iff]
constructor
· rintro - ⟨a, rfl⟩
exact ⟨(a, 1), by rw [noncommCoprod_apply, map_one, mul_one]⟩
· rintro - ⟨a, rfl⟩
exact ⟨(1, a), by rw [noncommCoprod_apply, map_one, one_mul]⟩ | lemma | GroupTheory | [
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Algebra.Group.Subgroup.Ker",
"Mathlib.Algebra.Group.Subgroup.Lattice",
"Mathlib.Order.Disjoint"
] | Mathlib/GroupTheory/NoncommCoprod.lean | noncommCoprod_range | null |
@[to_additive /-- `Finset.noncommSum` is “injective” in `f` if `f` maps into independent subgroups.
This generalizes (one direction of) `AddSubgroup.disjoint_iff_add_eq_zero`. -/]
eq_one_of_noncommProd_eq_one_of_iSupIndep {ι : Type*} (s : Finset ι) (f : ι → G) (comm)
(K : ι → Subgroup G) (hind : iSupIndep K) (hmem : ∀ x ∈ s, f x ∈ K x)
(heq1 : s.noncommProd f comm = 1) : ∀ i ∈ s, f i = 1 := by
classical
revert heq1
induction s using Finset.induction_on with
| empty => simp
| insert i s hnotMem ih =>
have hcomm := comm.mono (Finset.coe_subset.2 <| Finset.subset_insert _ _)
simp only [Finset.forall_mem_insert] at hmem
have hmem_bsupr : s.noncommProd f hcomm ∈ ⨆ i ∈ (s : Set ι), K i := by
refine Subgroup.noncommProd_mem _ _ ?_
intro x hx
have : K x ≤ ⨆ i ∈ (s : Set ι), K i := le_iSup₂ (f := fun i _ => K i) x hx
exact this (hmem.2 x hx)
intro heq1
rw [Finset.noncommProd_insert_of_notMem _ _ _ _ hnotMem] at heq1
have hnotMem' : i ∉ (s : Set ι) := by simpa
obtain ⟨heq1i : f i = 1, heq1S : s.noncommProd f _ = 1⟩ :=
Subgroup.disjoint_iff_mul_eq_one.mp (hind.disjoint_biSup hnotMem') hmem.1 hmem_bsupr heq1
intro i h
simp only [Finset.mem_insert] at h
rcases h with (rfl | h)
· exact heq1i
· refine ih hcomm hmem.2 heq1S _ h | theorem | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | eq_one_of_noncommProd_eq_one_of_iSupIndep | `Finset.noncommProd` is “injective” in `f` if `f` maps into independent subgroups. This
generalizes (one direction of) `Subgroup.disjoint_iff_mul_eq_one`. |
@[to_additive /-- The canonical homomorphism from a family of additive monoids. See also
`LinearMap.lsum` for a linear version without the commutativity assumption. -/]
noncommPiCoprod : (∀ i : ι, N i) →* M where
toFun f := Finset.univ.noncommProd (fun i => ϕ i (f i)) fun _ _ _ _ h => hcomm h _ _
map_one' := by
apply (Finset.noncommProd_eq_pow_card _ _ _ _ _).trans (one_pow _)
simp
map_mul' f g := by
classical
convert @Finset.noncommProd_mul_distrib _ _ _ _ (fun i => ϕ i (f i)) (fun i => ϕ i (g i)) _ _ _
· exact map_mul _ _ _
· rintro i - j - h
exact hcomm h _ _
variable {hcomm}
@[to_additive (attr := simp)] | def | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | noncommPiCoprod | The canonical homomorphism from a family of monoids. |
noncommPiCoprod_mulSingle [DecidableEq ι] (i : ι) (y : N i) :
noncommPiCoprod ϕ hcomm (Pi.mulSingle i y) = ϕ i y := by
change Finset.univ.noncommProd (fun j => ϕ j (Pi.mulSingle i y j)) (fun _ _ _ _ h => hcomm h _ _)
= ϕ i y
rw [← Finset.insert_erase (Finset.mem_univ i)]
rw [Finset.noncommProd_insert_of_notMem _ _ _ _ (Finset.notMem_erase i _)]
rw [Pi.mulSingle_eq_same]
rw [Finset.noncommProd_eq_pow_card]
· rw [one_pow]
exact mul_one _
· intro j hj
simp only [Finset.mem_erase] at hj
simp [hj] | theorem | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | noncommPiCoprod_mulSingle | null |
@[to_additive /-- The universal property of `MonoidHom.noncommPiCoprod`
Given monoid morphisms `φᵢ : Nᵢ → M` whose images pairwise commute,
there exists a unique monoid morphism `φ : Πᵢ Nᵢ → M` that induces the `φᵢ`,
and it is given by `AddMonoidHom.noncommPiCoprod`. -/]
noncommPiCoprodEquiv [DecidableEq ι] :
{ ϕ : ∀ i, N i →* M // Pairwise fun i j => ∀ x y, Commute (ϕ i x) (ϕ j y) } ≃
((∀ i, N i) →* M) where
toFun ϕ := noncommPiCoprod ϕ.1 ϕ.2
invFun f :=
⟨fun i => f.comp (MonoidHom.mulSingle N i), fun _ _ hij x y =>
Commute.map (Pi.mulSingle_commute hij x y) f⟩
left_inv ϕ := by
ext
simp only [coe_comp, Function.comp_apply, mulSingle_apply, noncommPiCoprod_mulSingle]
right_inv f := pi_ext fun i x => by
simp only [noncommPiCoprod_mulSingle, coe_comp, Function.comp_apply, mulSingle_apply]
@[to_additive] | def | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | noncommPiCoprodEquiv | The universal property of `MonoidHom.noncommPiCoprod`
Given monoid morphisms `φᵢ : Nᵢ → M` whose images pairwise commute,
there exists a unique monoid morphism `φ : Πᵢ Nᵢ → M` that induces the `φᵢ`,
and it is given by `MonoidHom.noncommPiCoprod`. |
noncommPiCoprod_mrange :
MonoidHom.mrange (noncommPiCoprod ϕ hcomm) = ⨆ i : ι, MonoidHom.mrange (ϕ i) := by
letI := Classical.decEq ι
apply le_antisymm
· rintro x ⟨f, rfl⟩
refine Submonoid.noncommProd_mem _ _ _ (fun _ _ _ _ h => hcomm h _ _) (fun i _ => ?_)
apply Submonoid.mem_sSup_of_mem
· use i
simp
· refine iSup_le ?_
rintro i x ⟨y, rfl⟩
exact ⟨Pi.mulSingle i y, noncommPiCoprod_mulSingle _ _ _⟩
@[to_additive] | theorem | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | noncommPiCoprod_mrange | null |
commute_noncommPiCoprod {m : M}
(comm : ∀ i (x : N i), Commute m ((ϕ i x))) (h : (i : ι) → N i) :
Commute m (MonoidHom.noncommPiCoprod ϕ hcomm h) := by
dsimp only [MonoidHom.noncommPiCoprod, MonoidHom.coe_mk, OneHom.coe_mk]
apply Finset.noncommProd_induction
· exact fun x y ↦ Commute.mul_right
· exact Commute.one_right _
· exact fun x _ ↦ comm x (h x)
@[to_additive] | lemma | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | commute_noncommPiCoprod | null |
noncommPiCoprod_apply (h : (i : ι) → N i) :
MonoidHom.noncommPiCoprod ϕ hcomm h = Finset.noncommProd Finset.univ (fun i ↦ ϕ i (h i))
(Pairwise.set_pairwise (fun ⦃i j⦄ a ↦ hcomm a (h i) (h j)) _) := by
dsimp only [MonoidHom.noncommPiCoprod, MonoidHom.coe_mk, OneHom.coe_mk] | lemma | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | noncommPiCoprod_apply | null |
@[to_additive]
comp_noncommPiCoprod {P : Type*} [Monoid P] {f : M →* P}
(hcomm' : Pairwise fun i j => ∀ x y, Commute (f.comp (ϕ i) x) (f.comp (ϕ j) y) :=
Pairwise.mono hcomm (fun i j ↦ forall_imp (fun x h y ↦ by
simp only [MonoidHom.coe_comp, Function.comp_apply, Commute.map (h y) f]))) :
f.comp (MonoidHom.noncommPiCoprod ϕ hcomm) =
MonoidHom.noncommPiCoprod (fun i ↦ f.comp (ϕ i)) hcomm' :=
MonoidHom.ext fun _ ↦ by
simp only [MonoidHom.noncommPiCoprod, MonoidHom.coe_comp, MonoidHom.coe_mk, OneHom.coe_mk,
Function.comp_apply, Finset.map_noncommProd] | theorem | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | comp_noncommPiCoprod | Given monoid morphisms `φᵢ : Nᵢ → M` and `f : M → P`, if we have sufficient commutativity, then
`f ∘ (∐ᵢ φᵢ) = ∐ᵢ (f ∘ φᵢ)` |
@[to_additive]
noncommPiCoprod_range [Fintype ι]
{hcomm : Pairwise fun i j : ι => ∀ (x : H i) (y : H j), Commute (ϕ i x) (ϕ j y)} :
(noncommPiCoprod ϕ hcomm).range = ⨆ i : ι, (ϕ i).range := by
letI := Classical.decEq ι
apply le_antisymm
· rintro x ⟨f, rfl⟩
refine Subgroup.noncommProd_mem _ (fun _ _ _ _ h => hcomm h _ _) ?_
intro i _hi
apply Subgroup.mem_sSup_of_mem
· use i
simp
· refine iSup_le ?_
rintro i x ⟨y, rfl⟩
exact ⟨Pi.mulSingle i y, noncommPiCoprod_mulSingle _ _ _⟩
@[to_additive] | theorem | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | noncommPiCoprod_range | null |
injective_noncommPiCoprod_of_iSupIndep [Fintype ι]
{hcomm : Pairwise fun i j : ι => ∀ (x : H i) (y : H j), Commute (ϕ i x) (ϕ j y)}
(hind : iSupIndep fun i => (ϕ i).range)
(hinj : ∀ i, Function.Injective (ϕ i)) : Function.Injective (noncommPiCoprod ϕ hcomm) := by
classical
apply (MonoidHom.ker_eq_bot_iff _).mp
rw [eq_bot_iff]
intro f heq1
have : ∀ i, i ∈ Finset.univ → ϕ i (f i) = 1 :=
Subgroup.eq_one_of_noncommProd_eq_one_of_iSupIndep _ _ (fun _ _ _ _ h => hcomm h _ _)
_ hind (by simp) heq1
ext i
apply hinj
simp [this i (Finset.mem_univ i)]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | injective_noncommPiCoprod_of_iSupIndep | null |
independent_range_of_coprime_order
(hcomm : Pairwise fun i j : ι => ∀ (x : H i) (y : H j), Commute (ϕ i x) (ϕ j y))
[Finite ι] [∀ i, Fintype (H i)]
(hcoprime : Pairwise fun i j => Nat.Coprime (Fintype.card (H i)) (Fintype.card (H j))) :
iSupIndep fun i => (ϕ i).range := by
cases nonempty_fintype ι
letI := Classical.decEq ι
rintro i
rw [disjoint_iff_inf_le]
rintro f ⟨hxi, hxp⟩
dsimp at hxi hxp
rw [iSup_subtype', ← noncommPiCoprod_range] at hxp
rotate_left
· intro _ _ hj
apply hcomm
exact hj ∘ Subtype.ext
obtain ⟨g, hgf⟩ := hxp
obtain ⟨g', hg'f⟩ := hxi
have hxi : orderOf f ∣ Fintype.card (H i) := by
rw [← hg'f]
exact (orderOf_map_dvd _ _).trans orderOf_dvd_card
have hxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H j) := by
rw [← hgf, ← Fintype.card_pi]
exact (orderOf_map_dvd _ _).trans orderOf_dvd_card
change f = 1
rw [← pow_one f, ← orderOf_dvd_iff_pow_eq_one]
obtain ⟨c, hc⟩ := Nat.dvd_gcd hxp hxi
use c
rw [← hc]
symm
rw [← Nat.coprime_iff_gcd_eq_one, Nat.coprime_fintype_prod_left_iff, Subtype.forall]
intro j h
exact hcoprime h | theorem | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | independent_range_of_coprime_order | null |
@[to_additive]
commute_subtype_of_commute
(hcomm : Pairwise fun i j : ι => ∀ x y : G, x ∈ H i → y ∈ H j → Commute x y) (i j : ι)
(hne : i ≠ j) :
∀ (x : H i) (y : H j), Commute ((H i).subtype x) ((H j).subtype y) := by
rintro ⟨x, hx⟩ ⟨y, hy⟩
exact hcomm hne x y hx hy
@[to_additive] | theorem | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | commute_subtype_of_commute | null |
independent_of_coprime_order
(hcomm : Pairwise fun i j : ι => ∀ x y : G, x ∈ H i → y ∈ H j → Commute x y)
[Finite ι] [∀ i, Fintype (H i)]
(hcoprime : Pairwise fun i j => Nat.Coprime (Fintype.card (H i)) (Fintype.card (H j))) :
iSupIndep H := by
simpa using
MonoidHom.independent_range_of_coprime_order (fun i => (H i).subtype)
(commute_subtype_of_commute hcomm) hcoprime
variable [Fintype ι] | theorem | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | independent_of_coprime_order | null |
@[to_additive /-- The canonical homomorphism from a family of additive subgroups where elements from
different subgroups commute -/]
noncommPiCoprod (hcomm : Pairwise fun i j : ι => ∀ x y : G, x ∈ H i → y ∈ H j → Commute x y) :
(∀ i : ι, H i) →* G :=
MonoidHom.noncommPiCoprod (fun i => (H i).subtype) (commute_subtype_of_commute hcomm)
@[to_additive (attr := simp)] | def | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | noncommPiCoprod | The canonical homomorphism from a family of subgroups where elements from different subgroups
commute |
noncommPiCoprod_mulSingle [DecidableEq ι]
{hcomm : Pairwise fun i j : ι => ∀ x y : G, x ∈ H i → y ∈ H j → Commute x y}(i : ι) (y : H i) :
noncommPiCoprod hcomm (Pi.mulSingle i y) = y := by apply MonoidHom.noncommPiCoprod_mulSingle
@[to_additive] | theorem | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | noncommPiCoprod_mulSingle | null |
noncommPiCoprod_range
{hcomm : Pairwise fun i j : ι => ∀ x y : G, x ∈ H i → y ∈ H j → Commute x y} :
(noncommPiCoprod hcomm).range = ⨆ i : ι, H i := by
simp [noncommPiCoprod, MonoidHom.noncommPiCoprod_range]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | noncommPiCoprod_range | null |
injective_noncommPiCoprod_of_iSupIndep
{hcomm : Pairwise fun i j : ι => ∀ x y : G, x ∈ H i → y ∈ H j → Commute x y}
(hind : iSupIndep H) :
Function.Injective (noncommPiCoprod hcomm) := by
apply MonoidHom.injective_noncommPiCoprod_of_iSupIndep
· simpa using hind
· intro i
exact Subtype.coe_injective
@[to_additive] | theorem | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | injective_noncommPiCoprod_of_iSupIndep | null |
noncommPiCoprod_apply (comm) (u : (i : ι) → H i) :
Subgroup.noncommPiCoprod comm u = Finset.noncommProd Finset.univ (fun i ↦ u i)
(fun i _ j _ h ↦ comm h _ _ (u i).prop (u j).prop) := by
simp only [Subgroup.noncommPiCoprod, MonoidHom.noncommPiCoprod,
coe_subtype, MonoidHom.coe_mk, OneHom.coe_mk] | theorem | GroupTheory | [
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.Order.SupIndep"
] | Mathlib/GroupTheory/NoncommPiCoprod.lean | noncommPiCoprod_apply | null |
@[to_additive]
isPeriodicPt_mul_iff_pow_eq_one (x : G) : IsPeriodicPt (x * ·) n 1 ↔ x ^ n = 1 := by
rw [IsPeriodicPt, IsFixedPt, mul_left_iterate_apply_one] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | isPeriodicPt_mul_iff_pow_eq_one | null |
@[to_additive /-- `IsOfFinAddOrder` is a predicate on an element `a` of an
additive monoid to be of finite order, i.e. there exists `n ≥ 1` such that `n • a = 0`. -/]
IsOfFinOrder (x : G) : Prop :=
(1 : G) ∈ periodicPts (x * ·) | def | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder | `IsOfFinOrder` is a predicate on an element `x` of a monoid to be of finite order, i.e. there
exists `n ≥ 1` such that `x ^ n = 1`. |
isOfFinAddOrder_ofMul_iff : IsOfFinAddOrder (Additive.ofMul x) ↔ IsOfFinOrder x :=
Iff.rfl | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | isOfFinAddOrder_ofMul_iff | null |
isOfFinOrder_ofAdd_iff {α : Type*} [AddMonoid α] {x : α} :
IsOfFinOrder (Multiplicative.ofAdd x) ↔ IsOfFinAddOrder x := Iff.rfl
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | isOfFinOrder_ofAdd_iff | null |
isOfFinOrder_iff_pow_eq_one : IsOfFinOrder x ↔ ∃ n, 0 < n ∧ x ^ n = 1 := by
simp [IsOfFinOrder, mem_periodicPts, isPeriodicPt_mul_iff_pow_eq_one]
@[to_additive] alias ⟨IsOfFinOrder.exists_pow_eq_one, _⟩ := isOfFinOrder_iff_pow_eq_one
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | isOfFinOrder_iff_pow_eq_one | null |
isOfFinOrder_iff_zpow_eq_one {G} [DivisionMonoid G] {x : G} :
IsOfFinOrder x ↔ ∃ (n : ℤ), n ≠ 0 ∧ x ^ n = 1 := by
rw [isOfFinOrder_iff_pow_eq_one]
refine ⟨fun ⟨n, hn, hn'⟩ ↦ ⟨n, Int.natCast_ne_zero_iff_pos.mpr hn, zpow_natCast x n ▸ hn'⟩,
fun ⟨n, hn, hn'⟩ ↦ ⟨n.natAbs, Int.natAbs_pos.mpr hn, ?_⟩⟩
rcases (Int.natAbs_eq_iff (a := n)).mp rfl with h | h
· rwa [h, zpow_natCast] at hn'
· rwa [h, zpow_neg, inv_eq_one, zpow_natCast] at hn' | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | isOfFinOrder_iff_zpow_eq_one | null |
@[to_additive /-- See also `injective_nsmul_iff_not_isOfFinAddOrder`. -/]
not_isOfFinOrder_of_injective_pow {x : G} (h : Injective fun n : ℕ => x ^ n) :
¬IsOfFinOrder x := by
simp_rw [isOfFinOrder_iff_pow_eq_one, not_exists, not_and]
intro n hn_pos hnx
rw [← pow_zero x] at hnx
rw [h hnx] at hn_pos
exact irrefl 0 hn_pos | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | not_isOfFinOrder_of_injective_pow | See also `injective_pow_iff_not_isOfFinOrder`. |
@[to_additive (attr := simp) /-- 0 is of finite order in any additive monoid. -/]
IsOfFinOrder.one : IsOfFinOrder (1 : G) :=
isOfFinOrder_iff_pow_eq_one.mpr ⟨1, Nat.one_pos, one_pow 1⟩
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.one | 1 is of finite order in any monoid. |
IsOfFinOrder.pow {n : ℕ} : IsOfFinOrder a → IsOfFinOrder (a ^ n) := by
simp_rw [isOfFinOrder_iff_pow_eq_one]
rintro ⟨m, hm, ha⟩
exact ⟨m, hm, by simp [pow_right_comm _ n, ha]⟩
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.pow | null |
IsOfFinOrder.of_pow {n : ℕ} (h : IsOfFinOrder (a ^ n)) (hn : n ≠ 0) : IsOfFinOrder a := by
rw [isOfFinOrder_iff_pow_eq_one] at *
rcases h with ⟨m, hm, ha⟩
exact ⟨n * m, mul_pos hn.bot_lt hm, by rwa [pow_mul]⟩
@[to_additive (attr := simp)] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.of_pow | null |
isOfFinOrder_pow {n : ℕ} : IsOfFinOrder (a ^ n) ↔ IsOfFinOrder a ∨ n = 0 := by
rcases Decidable.eq_or_ne n 0 with rfl | hn
· simp
· exact ⟨fun h ↦ .inl <| h.of_pow hn, fun h ↦ (h.resolve_right hn).pow⟩
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | isOfFinOrder_pow | null |
not_isOfFinOrder_of_isMulTorsionFree [IsMulTorsionFree G] (ha : a ≠ 1) :
¬ IsOfFinOrder a := by
rw [isOfFinOrder_iff_pow_eq_one]
rintro ⟨n, hn, han⟩
exact ha <| pow_left_injective hn.ne' <| by simpa using han | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | not_isOfFinOrder_of_isMulTorsionFree | null |
@[to_additive /-- Elements of finite order are of finite order in submonoids. -/]
Submonoid.isOfFinOrder_coe {H : Submonoid G} {x : H} :
IsOfFinOrder (x : G) ↔ IsOfFinOrder x := by
rw [isOfFinOrder_iff_pow_eq_one, isOfFinOrder_iff_pow_eq_one]
norm_cast | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | Submonoid.isOfFinOrder_coe | Elements of finite order are of finite order in submonoids. |
IsConj.isOfFinOrder (h : IsConj x y) : IsOfFinOrder x → IsOfFinOrder y := by
simp_rw [isOfFinOrder_iff_pow_eq_one]
rintro ⟨n, n_gt_0, eq'⟩
exact ⟨n, n_gt_0, by rw [← isConj_one_right, ← eq']; exact h.pow n⟩ | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsConj.isOfFinOrder | null |
@[to_additive /-- The image of an element of finite additive order has finite additive order. -/]
MonoidHom.isOfFinOrder [Monoid H] (f : G →* H) {x : G} (h : IsOfFinOrder x) :
IsOfFinOrder <| f x :=
isOfFinOrder_iff_pow_eq_one.mpr <| by
obtain ⟨n, npos, hn⟩ := h.exists_pow_eq_one
exact ⟨n, npos, by rw [← f.map_pow, hn, f.map_one]⟩ | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | MonoidHom.isOfFinOrder | The image of an element of finite order has finite order. |
@[to_additive /-- If a direct product has finite additive order then so does each component. -/]
IsOfFinOrder.apply {η : Type*} {Gs : η → Type*} [∀ i, Monoid (Gs i)] {x : ∀ i, Gs i}
(h : IsOfFinOrder x) : ∀ i, IsOfFinOrder (x i) := by
obtain ⟨n, npos, hn⟩ := h.exists_pow_eq_one
exact fun _ => isOfFinOrder_iff_pow_eq_one.mpr ⟨n, npos, (congr_fun hn.symm _).symm⟩ | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.apply | If a direct product has finite order then so does each component. |
@[to_additive /-- The additive submonoid generated by an element is
an additive group if that element has finite order. -/]
noncomputable IsOfFinOrder.groupPowers (hx : IsOfFinOrder x) :
Group (Submonoid.powers x) := by
obtain ⟨hpos, hx⟩ := hx.exists_pow_eq_one.choose_spec
exact Submonoid.groupPowers hpos hx | abbrev | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.groupPowers | The submonoid generated by an element is a group if that element has finite order. |
@[to_additive
/-- `addOrderOf a` is the order of the element `a`, i.e. the `n ≥ 1`, s.t. `n • a = 0` if it
exists. Otherwise, i.e. if `a` is of infinite order, then `addOrderOf a` is `0` by convention. -/]
noncomputable orderOf (x : G) : ℕ :=
minimalPeriod (x * ·) 1
@[to_additive (attr := nontriviality)] | def | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf | `orderOf x` is the order of the element `x`, i.e. the `n ≥ 1`, s.t. `x ^ n = 1` if it exists.
Otherwise, i.e. if `x` is of infinite order, then `orderOf x` is `0` by convention. |
Subsingleton.orderOf_eq [Subsingleton G] (x : G) : orderOf x = 1 := by
simp [orderOf, nontriviality]
@[simp] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | Subsingleton.orderOf_eq | null |
addOrderOf_ofMul_eq_orderOf (x : G) : addOrderOf (Additive.ofMul x) = orderOf x :=
rfl
@[simp] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | addOrderOf_ofMul_eq_orderOf | null |
orderOf_ofAdd_eq_addOrderOf {α : Type*} [AddMonoid α] (a : α) :
orderOf (Multiplicative.ofAdd a) = addOrderOf a := rfl
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_ofAdd_eq_addOrderOf | null |
protected IsOfFinOrder.orderOf_pos (h : IsOfFinOrder x) : 0 < orderOf x :=
minimalPeriod_pos_of_mem_periodicPts h
@[to_additive (attr := simp) addOrderOf_nsmul_eq_zero] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.orderOf_pos | null |
pow_orderOf_eq_one (x : G) : x ^ orderOf x = 1 := by
convert Eq.trans _ (isPeriodicPt_minimalPeriod (x * ·) 1)
rw [orderOf, mul_left_iterate_apply_one]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | pow_orderOf_eq_one | null |
orderOf_eq_zero (h : ¬IsOfFinOrder x) : orderOf x = 0 := by
rwa [orderOf, minimalPeriod, dif_neg]
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_eq_zero | null |
orderOf_eq_zero_iff : orderOf x = 0 ↔ ¬IsOfFinOrder x :=
⟨fun h H ↦ H.orderOf_pos.ne' h, orderOf_eq_zero⟩
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_eq_zero_iff | null |
orderOf_eq_zero_iff' : orderOf x = 0 ↔ ∀ n : ℕ, 0 < n → x ^ n ≠ 1 := by
simp_rw [orderOf_eq_zero_iff, isOfFinOrder_iff_pow_eq_one, not_exists, not_and]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_eq_zero_iff' | null |
orderOf_ne_zero_iff : orderOf x ≠ 0 ↔ IsOfFinOrder x := orderOf_eq_zero_iff.not_left | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_ne_zero_iff | null |
@[simp]
orderOf_zero (M₀ : Type*) [MonoidWithZero M₀] [Nontrivial M₀] : orderOf (0 : M₀) = 0 := by
rw [orderOf_eq_zero_iff, isOfFinOrder_iff_pow_eq_one]
simp +contextual [ne_of_gt]
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_zero | In a nontrivial monoid with zero, the order of the zero element is zero. |
orderOf_eq_iff {n} (h : 0 < n) :
orderOf x = n ↔ x ^ n = 1 ∧ ∀ m, m < n → 0 < m → x ^ m ≠ 1 := by
simp_rw [Ne, ← isPeriodicPt_mul_iff_pow_eq_one, orderOf, minimalPeriod]
split_ifs with h1
· classical
rw [find_eq_iff]
simp only [h, true_and]
push_neg
rfl
· rw [iff_false_left h.ne]
rintro ⟨h', -⟩
exact h1 ⟨n, h, h'⟩ | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_eq_iff | null |
@[to_additive (attr := simp)
/-- A group element has finite additive order iff its order is positive. -/]
orderOf_pos_iff : 0 < orderOf x ↔ IsOfFinOrder x := by
rw [iff_not_comm.mp orderOf_eq_zero_iff, pos_iff_ne_zero]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_pos_iff | A group element has finite order iff its order is positive. |
IsOfFinOrder.mono [Monoid β] {y : β} (hx : IsOfFinOrder x) (h : orderOf y ∣ orderOf x) :
IsOfFinOrder y := by rw [← orderOf_pos_iff] at hx ⊢; exact Nat.pos_of_dvd_of_pos h hx
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.mono | null |
pow_ne_one_of_lt_orderOf (n0 : n ≠ 0) (h : n < orderOf x) : x ^ n ≠ 1 := fun j =>
not_isPeriodicPt_of_pos_of_lt_minimalPeriod n0 h ((isPeriodicPt_mul_iff_pow_eq_one x).mpr j)
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | pow_ne_one_of_lt_orderOf | null |
orderOf_le_of_pow_eq_one (hn : 0 < n) (h : x ^ n = 1) : orderOf x ≤ n :=
IsPeriodicPt.minimalPeriod_le hn (by rwa [isPeriodicPt_mul_iff_pow_eq_one])
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_le_of_pow_eq_one | null |
orderOf_one : orderOf (1 : G) = 1 := by
rw [orderOf, ← minimalPeriod_id (x := (1 : G)), ← one_mul_eq_id]
@[to_additive (attr := simp) AddMonoid.addOrderOf_eq_one_iff] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_one | null |
orderOf_eq_one_iff : orderOf x = 1 ↔ x = 1 := by
rw [orderOf, minimalPeriod_eq_one_iff_isFixedPt, IsFixedPt, mul_one]
@[to_additive (attr := simp) mod_addOrderOf_nsmul] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_eq_one_iff | null |
pow_mod_orderOf (x : G) (n : ℕ) : x ^ (n % orderOf x) = x ^ n :=
calc
x ^ (n % orderOf x) = x ^ (n % orderOf x + orderOf x * (n / orderOf x)) := by
simp [pow_add, pow_mul, pow_orderOf_eq_one]
_ = x ^ n := by rw [Nat.mod_add_div]
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | pow_mod_orderOf | null |
orderOf_dvd_of_pow_eq_one (h : x ^ n = 1) : orderOf x ∣ n :=
IsPeriodicPt.minimalPeriod_dvd ((isPeriodicPt_mul_iff_pow_eq_one _).mpr h)
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_dvd_of_pow_eq_one | null |
orderOf_dvd_iff_pow_eq_one {n : ℕ} : orderOf x ∣ n ↔ x ^ n = 1 :=
⟨fun h => by rw [← pow_mod_orderOf, Nat.mod_eq_zero_of_dvd h, _root_.pow_zero],
orderOf_dvd_of_pow_eq_one⟩
@[to_additive addOrderOf_smul_dvd] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_dvd_iff_pow_eq_one | null |
orderOf_pow_dvd (n : ℕ) : orderOf (x ^ n) ∣ orderOf x := by
rw [orderOf_dvd_iff_pow_eq_one, pow_right_comm, pow_orderOf_eq_one, one_pow]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_pow_dvd | null |
pow_injOn_Iio_orderOf : (Set.Iio <| orderOf x).InjOn (x ^ ·) := by
simpa only [mul_left_iterate_apply_one]
using iterate_injOn_Iio_minimalPeriod (f := (x * ·)) (x := 1)
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | pow_injOn_Iio_orderOf | null |
protected IsOfFinOrder.mem_powers_iff_mem_range_orderOf [DecidableEq G]
(hx : IsOfFinOrder x) :
y ∈ Submonoid.powers x ↔ y ∈ (Finset.range (orderOf x)).image (x ^ ·) :=
Finset.mem_range_iff_mem_finset_range_of_mod_eq' hx.orderOf_pos <| pow_mod_orderOf _
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.mem_powers_iff_mem_range_orderOf | null |
protected IsOfFinOrder.powers_eq_image_range_orderOf [DecidableEq G] (hx : IsOfFinOrder x) :
(Submonoid.powers x : Set G) = (Finset.range (orderOf x)).image (x ^ ·) :=
Set.ext fun _ ↦ hx.mem_powers_iff_mem_range_orderOf
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.powers_eq_image_range_orderOf | null |
pow_eq_one_iff_modEq : x ^ n = 1 ↔ n ≡ 0 [MOD orderOf x] := by
rw [modEq_zero_iff_dvd, orderOf_dvd_iff_pow_eq_one]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | pow_eq_one_iff_modEq | null |
orderOf_map_dvd {H : Type*} [Monoid H] (ψ : G →* H) (x : G) :
orderOf (ψ x) ∣ orderOf x := by
apply orderOf_dvd_of_pow_eq_one
rw [← map_pow, pow_orderOf_eq_one]
apply map_one
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_map_dvd | null |
exists_pow_eq_self_of_coprime (h : n.Coprime (orderOf x)) : ∃ m : ℕ, (x ^ n) ^ m = x := by
by_cases h0 : orderOf x = 0
· rw [h0, coprime_zero_right] at h
exact ⟨1, by rw [h, pow_one, pow_one]⟩
by_cases h1 : orderOf x = 1
· exact ⟨0, by rw [orderOf_eq_one_iff.mp h1, one_pow, one_pow]⟩
obtain ⟨m, h⟩ := exists_mul_emod_eq_one_of_coprime h (one_lt_iff_ne_zero_and_ne_one.mpr ⟨h0, h1⟩)
exact ⟨m, by rw [← pow_mul, ← pow_mod_orderOf, h, pow_one]⟩ | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | exists_pow_eq_self_of_coprime | null |
@[to_additive addOrderOf_eq_of_nsmul_and_div_prime_nsmul /-- If `n * x = 0`, but `n/p * x ≠ 0` for
all prime factors `p` of `n`, then `x` has order `n` in `G`. -/]
orderOf_eq_of_pow_and_pow_div_prime (hn : 0 < n) (hx : x ^ n = 1)
(hd : ∀ p : ℕ, p.Prime → p ∣ n → x ^ (n / p) ≠ 1) : orderOf x = n := by
obtain ⟨a, ha⟩ := exists_eq_mul_right_of_dvd (orderOf_dvd_of_pow_eq_one hx)
suffices a = 1 by simp [this, ha]
by_contra h
have a_min_fac_dvd_p_sub_one : a.minFac ∣ n := by
obtain ⟨b, hb⟩ : ∃ b : ℕ, a = b * a.minFac := exists_eq_mul_left_of_dvd a.minFac_dvd
rw [hb, ← mul_assoc] at ha
exact Dvd.intro_left (orderOf x * b) ha.symm
refine hd a.minFac (Nat.minFac_prime h) a_min_fac_dvd_p_sub_one ?_
rw [← orderOf_dvd_iff_pow_eq_one, Nat.dvd_div_iff_mul_dvd a_min_fac_dvd_p_sub_one, ha, mul_comm,
Nat.mul_dvd_mul_iff_left (IsOfFinOrder.orderOf_pos _)]
· exact Nat.minFac_dvd a
· rw [isOfFinOrder_iff_pow_eq_one]
exact Exists.intro n (id ⟨hn, hx⟩)
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_eq_of_pow_and_pow_div_prime | If `x^n = 1`, but `x^(n/p) ≠ 1` for all prime factors `p` of `n`,
then `x` has order `n` in `G`. |
orderOf_eq_orderOf_iff {H : Type*} [Monoid H] {y : H} :
orderOf x = orderOf y ↔ ∀ n : ℕ, x ^ n = 1 ↔ y ^ n = 1 := by
simp_rw [← isPeriodicPt_mul_iff_pow_eq_one, ← minimalPeriod_eq_minimalPeriod_iff, orderOf] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_eq_orderOf_iff | null |
@[to_additive /-- An injective homomorphism of additive monoids preserves orders of elements. -/]
orderOf_injective {H : Type*} [Monoid H] (f : G →* H) (hf : Function.Injective f) (x : G) :
orderOf (f x) = orderOf x := by
simp_rw [orderOf_eq_orderOf_iff, ← f.map_pow, ← f.map_one, hf.eq_iff, forall_const] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_injective | An injective homomorphism of monoids preserves orders of elements. |
@[to_additive (attr := simp) /-- An additive equivalence preserves orders of elements. -/]
MulEquiv.orderOf_eq {H : Type*} [Monoid H] (e : G ≃* H) (x : G) :
orderOf (e x) = orderOf x :=
orderOf_injective e.toMonoidHom e.injective x
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | MulEquiv.orderOf_eq | A multiplicative equivalence preserves orders of elements. |
Function.Injective.isOfFinOrder_iff [Monoid H] {f : G →* H} (hf : Injective f) :
IsOfFinOrder (f x) ↔ IsOfFinOrder x := by
rw [← orderOf_pos_iff, orderOf_injective f hf x, ← orderOf_pos_iff]
@[to_additive (attr := norm_cast, simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | Function.Injective.isOfFinOrder_iff | null |
orderOf_submonoid {H : Submonoid G} (y : H) : orderOf (y : G) = orderOf y :=
orderOf_injective H.subtype Subtype.coe_injective y
@[to_additive (attr := norm_cast)] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_submonoid | null |
orderOf_units {y : Gˣ} : orderOf (y : G) = orderOf y :=
orderOf_injective (Units.coeHom G) Units.val_injective y
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_units | null |
IsUnit.orderOf_eq_one [Subsingleton Gˣ] {x : G} (h : IsUnit x) :
orderOf x = 1 := by
simp [isUnit_iff_eq_one.mp h]
@[to_additive (attr := norm_cast)] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsUnit.orderOf_eq_one | null |
Units.isOfFinOrder_val {u : Gˣ} : IsOfFinOrder (u : G) ↔ IsOfFinOrder u :=
Units.coeHom_injective.isOfFinOrder_iff | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | Units.isOfFinOrder_val | null |
@[to_additive (attr := simps) /-- If the additive order of `x` is finite, then `x` is an additive
unit with inverse `(addOrderOf x - 1) • x`. -/]
noncomputable IsOfFinOrder.unit {M} [Monoid M] {x : M} (hx : IsOfFinOrder x) : Mˣ :=
⟨x, x ^ (orderOf x - 1),
by rw [← _root_.pow_succ', tsub_add_cancel_of_le (by exact hx.orderOf_pos), pow_orderOf_eq_one],
by rw [← _root_.pow_succ, tsub_add_cancel_of_le (by exact hx.orderOf_pos), pow_orderOf_eq_one]⟩
@[to_additive] | def | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.unit | If the order of `x` is finite, then `x` is a unit with inverse `x ^ (orderOf x - 1)`. |
IsOfFinOrder.isUnit {M} [Monoid M] {x : M} (hx : IsOfFinOrder x) : IsUnit x := ⟨hx.unit, rfl⟩
variable (x)
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.isUnit | null |
orderOf_pow' (h : n ≠ 0) : orderOf (x ^ n) = orderOf x / gcd (orderOf x) n := by
unfold orderOf
rw [← minimalPeriod_iterate_eq_div_gcd h, mul_left_iterate]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_pow' | null |
orderOf_pow_of_dvd {x : G} {n : ℕ} (hn : n ≠ 0) (dvd : n ∣ orderOf x) :
orderOf (x ^ n) = orderOf x / n := by rw [orderOf_pow' _ hn, Nat.gcd_eq_right dvd]
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_pow_of_dvd | null |
orderOf_pow_orderOf_div {x : G} {n : ℕ} (hx : orderOf x ≠ 0) (hn : n ∣ orderOf x) :
orderOf (x ^ (orderOf x / n)) = n := by
rw [orderOf_pow_of_dvd _ (Nat.div_dvd_of_dvd hn), Nat.div_div_self hn hx]
rw [← Nat.div_mul_cancel hn] at hx; exact left_ne_zero_of_mul hx
variable (n)
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_pow_orderOf_div | null |
protected IsOfFinOrder.orderOf_pow (h : IsOfFinOrder x) :
orderOf (x ^ n) = orderOf x / gcd (orderOf x) n := by
unfold orderOf
rw [← minimalPeriod_iterate_eq_div_gcd' h, mul_left_iterate]
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.orderOf_pow | null |
Nat.Coprime.orderOf_pow (h : (orderOf y).Coprime m) : orderOf (y ^ m) = orderOf y := by
by_cases hg : IsOfFinOrder y
· rw [hg.orderOf_pow y m, h.gcd_eq_one, Nat.div_one]
· rw [m.coprime_zero_left.1 (orderOf_eq_zero hg ▸ h), pow_one]
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | Nat.Coprime.orderOf_pow | null |
IsOfFinOrder.natCard_powers_le_orderOf (ha : IsOfFinOrder a) :
Nat.card (powers a : Set G) ≤ orderOf a := by
classical
simpa [ha.powers_eq_image_range_orderOf, Finset.card_range, Nat.Iio_eq_range]
using Finset.card_image_le (s := Finset.range (orderOf a))
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.natCard_powers_le_orderOf | null |
IsOfFinOrder.finite_powers (ha : IsOfFinOrder a) : (powers a : Set G).Finite := by
classical rw [ha.powers_eq_image_range_orderOf]; exact Finset.finite_toSet _ | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.finite_powers | null |
@[to_additive]
orderOf_mul_dvd_lcm (h : Commute x y) :
orderOf (x * y) ∣ Nat.lcm (orderOf x) (orderOf y) := by
rw [orderOf, ← comp_mul_left]
exact Function.Commute.minimalPeriod_of_comp_dvd_lcm h.function_commute_mul_left
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_mul_dvd_lcm | null |
orderOf_dvd_lcm_mul (h : Commute x y) :
orderOf y ∣ Nat.lcm (orderOf x) (orderOf (x * y)) := by
by_cases h0 : orderOf x = 0
· rw [h0, lcm_zero_left]
apply dvd_zero
conv_lhs =>
rw [← one_mul y, ← pow_orderOf_eq_one x, ← succ_pred_eq_of_pos (Nat.pos_of_ne_zero h0),
_root_.pow_succ, mul_assoc]
exact
(((Commute.refl x).mul_right h).pow_left _).orderOf_mul_dvd_lcm.trans
(lcm_dvd_iff.2 ⟨(orderOf_pow_dvd _).trans (dvd_lcm_left _ _), dvd_lcm_right _ _⟩)
@[to_additive addOrderOf_add_dvd_mul_addOrderOf] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_dvd_lcm_mul | null |
orderOf_mul_dvd_mul_orderOf (h : Commute x y) :
orderOf (x * y) ∣ orderOf x * orderOf y :=
dvd_trans h.orderOf_mul_dvd_lcm (lcm_dvd_mul _ _)
@[to_additive addOrderOf_add_eq_mul_addOrderOf_of_coprime] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_mul_dvd_mul_orderOf | null |
orderOf_mul_eq_mul_orderOf_of_coprime (h : Commute x y)
(hco : (orderOf x).Coprime (orderOf y)) : orderOf (x * y) = orderOf x * orderOf y := by
rw [orderOf, ← comp_mul_left]
exact h.function_commute_mul_left.minimalPeriod_of_comp_eq_mul_of_coprime hco | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_mul_eq_mul_orderOf_of_coprime | null |
@[to_additive /-- Commuting elements of finite additive order are closed under addition. -/]
isOfFinOrder_mul (h : Commute x y) (hx : IsOfFinOrder x) (hy : IsOfFinOrder y) :
IsOfFinOrder (x * y) :=
orderOf_pos_iff.mp <|
pos_of_dvd_of_pos h.orderOf_mul_dvd_mul_orderOf <| mul_pos hx.orderOf_pos hy.orderOf_pos | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | isOfFinOrder_mul | Commuting elements of finite order are closed under multiplication. |
@[to_additive addOrderOf_add_eq_right_of_forall_prime_mul_dvd
/-- If each prime factor of
`addOrderOf x` has higher multiplicity in `addOrderOf y`, and `x` commutes with `y`,
then `x + y` has the same order as `y`. -/]
orderOf_mul_eq_right_of_forall_prime_mul_dvd (h : Commute x y) (hy : IsOfFinOrder y)
(hdvd : ∀ p : ℕ, p.Prime → p ∣ orderOf x → p * orderOf x ∣ orderOf y) :
orderOf (x * y) = orderOf y := by
have hoy := hy.orderOf_pos
have hxy := dvd_of_forall_prime_mul_dvd hdvd
apply orderOf_eq_of_pow_and_pow_div_prime hoy <;> simp only [Ne, ← orderOf_dvd_iff_pow_eq_one]
· exact h.orderOf_mul_dvd_lcm.trans (lcm_dvd hxy dvd_rfl)
refine fun p hp hpy hd => hp.ne_one ?_
rw [← Nat.dvd_one, ← mul_dvd_mul_iff_right hoy.ne', one_mul, ← dvd_div_iff_mul_dvd hpy]
refine (orderOf_dvd_lcm_mul h).trans (lcm_dvd ((dvd_div_iff_mul_dvd hpy).2 ?_) hd)
by_cases h : p ∣ orderOf x
exacts [hdvd p hp h, (hp.coprime_iff_not_dvd.2 h).mul_dvd_of_dvd_of_dvd hpy hxy] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_mul_eq_right_of_forall_prime_mul_dvd | If each prime factor of `orderOf x` has higher multiplicity in `orderOf y`, and `x` commutes
with `y`, then `x * y` has the same order as `y`. |
@[to_additive]
orderOf_eq_prime_iff : orderOf x = p ↔ x ^ p = 1 ∧ x ≠ 1 := by
rw [orderOf, minimalPeriod_eq_prime_iff, isPeriodicPt_mul_iff_pow_eq_one, IsFixedPt, mul_one] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_eq_prime_iff | null |
@[to_additive /-- The backward direction of `addOrderOf_eq_prime_iff`. -/]
orderOf_eq_prime (hg : x ^ p = 1) (hg1 : x ≠ 1) : orderOf x = p :=
orderOf_eq_prime_iff.mpr ⟨hg, hg1⟩
@[to_additive addOrderOf_eq_prime_pow] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_eq_prime | The backward direction of `orderOf_eq_prime_iff`. |
orderOf_eq_prime_pow (hnot : ¬x ^ p ^ n = 1) (hfin : x ^ p ^ (n + 1) = 1) :
orderOf x = p ^ (n + 1) := by
apply minimalPeriod_eq_prime_pow <;> rwa [isPeriodicPt_mul_iff_pow_eq_one]
@[to_additive exists_addOrderOf_eq_prime_pow_iff] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_eq_prime_pow | null |
exists_orderOf_eq_prime_pow_iff :
(∃ k : ℕ, orderOf x = p ^ k) ↔ ∃ m : ℕ, x ^ (p : ℕ) ^ m = 1 :=
⟨fun ⟨k, hk⟩ => ⟨k, by rw [← hk, pow_orderOf_eq_one]⟩, fun ⟨_, hm⟩ => by
obtain ⟨k, _, hk⟩ := (Nat.dvd_prime_pow hp.elim).mp (orderOf_dvd_of_pow_eq_one hm)
exact ⟨k, hk⟩⟩ | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | exists_orderOf_eq_prime_pow_iff | null |
@[to_additive /-- The equivalence between `Fin (addOrderOf a)` and
`AddSubmonoid.multiples a`, sending `i` to `i • a` -/]
noncomputable finEquivPowers {x : G} (hx : IsOfFinOrder x) : Fin (orderOf x) ≃ powers x :=
Equiv.ofBijective (fun n ↦ ⟨x ^ (n : ℕ), ⟨n, rfl⟩⟩) ⟨fun ⟨_, h₁⟩ ⟨_, h₂⟩ ij ↦
Fin.ext (pow_injOn_Iio_orderOf h₁ h₂ (Subtype.mk_eq_mk.1 ij)), fun ⟨_, i, rfl⟩ ↦
⟨⟨i % orderOf x, mod_lt _ hx.orderOf_pos⟩, Subtype.eq <| pow_mod_orderOf _ _⟩⟩
@[to_additive (attr := simp)] | def | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | finEquivPowers | The equivalence between `Fin (orderOf x)` and `Submonoid.powers x`, sending `i` to `x ^ i` |
finEquivPowers_apply {x : G} (hx : IsOfFinOrder x) {n : Fin (orderOf x)} :
finEquivPowers hx n = ⟨x ^ (n : ℕ), n, rfl⟩ := rfl
@[to_additive (attr := simp)] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | finEquivPowers_apply | null |
finEquivPowers_symm_apply {x : G} (hx : IsOfFinOrder x) (n : ℕ) :
(finEquivPowers hx).symm ⟨x ^ n, _, rfl⟩ = ⟨n % orderOf x, Nat.mod_lt _ hx.orderOf_pos⟩ := by
rw [Equiv.symm_apply_eq, finEquivPowers_apply, Subtype.mk_eq_mk, ← pow_mod_orderOf, Fin.val_mk]
variable {x n} (hx : IsOfFinOrder x)
include hx
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | finEquivPowers_symm_apply | null |
IsOfFinOrder.pow_eq_pow_iff_modEq : x ^ n = x ^ m ↔ n ≡ m [MOD orderOf x] := by
wlog hmn : m ≤ n generalizing m n
· rw [eq_comm, ModEq.comm, this (le_of_not_ge hmn)]
obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le hmn
rw [pow_add, (hx.isUnit.pow _).mul_eq_left, pow_eq_one_iff_modEq]
exact ⟨fun h ↦ Nat.ModEq.add_left _ h, fun h ↦ Nat.ModEq.add_left_cancel' _ h⟩
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.pow_eq_pow_iff_modEq | null |
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