declaration stringlengths 27 11.3k | file stringlengths 52 114 | context dict | tactic_states listlengths 1 1.24k |
|---|---|---|---|
theorem subgroupOf_range_eq_of_le {G₁ G₂ : Type*} [Group G₁] [Group G₂] {K : Subgroup G₂}
(f : G₁ →* G₂) (h : f.range ≤ K) :
f.range.subgroupOf K = (f.codRestrict K fun x => h ⟨x, rfl⟩).range := by
ext k
refine exists_congr ?_
simp [Subtype.ext_iff]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)"
]
} | [
{
"line": "ext k",
"before_state": "G₁ : Type u_7\nG₂ : Type u_8\ninst✝¹ : Group G₁\ninst✝ : Group G₂\nK : Subgroup G₂\nf : G₁ →* G₂\nh : f.range ≤ K\n⊢ f.range.subgroupOf K = (f.codRestrict K ⋯).range",
"after_state": "case h\nG₁ : Type u_7\nG₂ : Type u_8\ninst✝¹ : Group G₁\ninst✝ : Group G₂\nK : Subgr... |
theorem ker_toHomUnits {M} [Monoid M] (f : G →* M) : f.toHomUnits.ker = f.ker := by
ext x
simp [mem_ker, Units.ext_iff]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)",
"{M : Type*} [MulOneClass M]"
]
} | [
{
"line": "ext x",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nM : Type u_8\ninst✝ : Monoid M\nf : G →* M\n⊢ f.toHomUnits.ker = f.ker",
"after_state": "case h\nG : Type u_1\ninst✝¹ : Group G\nM : Type u_8\ninst✝ : Monoid M\nf : G →* M\nx : G\n⊢ x ∈ f.toHomUnits.ker ↔ x ∈ f.ker"
},
{
"line":... |
theorem range_le_ker_iff (f : G →* G') (g : G' →* G'') : f.range ≤ g.ker ↔ g.comp f = 1 :=
⟨fun h => ext fun x => h ⟨x, rfl⟩, by rintro h _ ⟨y, rfl⟩; exact DFunLike.congr_fun h y⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)",
"{M : Type*} [MulOneClass M]"
]
} | [
{
"line": "rintro h _ ⟨y, rfl⟩",
"before_state": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : Group G''\nf : G →* G'\ng : G' →* G''\n⊢ g.comp f = 1 → f.range ≤ g.ker",
"after_state": "case intro\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝² : Group G\... |
theorem map_comap_eq (H : Subgroup N) : map f (comap f H) = f.range ⊓ H :=
SetLike.ext' <| by
rw [coe_map]
rw [coe_comap]
rw [Set.image_preimage_eq_inter_range]
rw [coe_inf]
rw [coe_range]
rw [Set.inter_comm]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)",
"{M : Type*} [MulOneClass M]",
"{M : Type*} [Monoid... | [
{
"line": "rw [coe_map]",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf : G →* N\nH : Subgroup N\n⊢ ↑(map f (comap f H)) = ↑(f.range ⊓ H)",
"after_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf : G →* N\nH : Subgroup N\n⊢ ⇑f '' ↑(comap f H) ... |
theorem comap_map_eq (H : Subgroup G) : comap f (map f H) = H ⊔ f.ker := by
refine le_antisymm ?_ (sup_le (le_comap_map _ _) (ker_le_comap _ _))
intro x hx; simp only [exists_prop, mem_map, mem_comap] at hx
rcases hx with ⟨y, hy, hy'⟩
rw [← mul_inv_cancel_left y x]
exact mul_mem_sup hy (by simp [mem_ker, hy']... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)",
"{M : Type*} [MulOneClass M]",
"{M : Type*} [Monoid... | [
{
"line": "refine le_antisymm ?_ (sup_le (le_comap_map _ _) (ker_le_comap _ _))",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf : G →* N\nH : Subgroup G\n⊢ comap f (map f H) = H ⊔ f.ker",
"after_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf... |
theorem map_comap_eq_self {f : G →* N} {H : Subgroup N} (h : H ≤ f.range) :
map f (comap f H) = H := by
rwa [map_comap_eq, inf_eq_right]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)",
"{M : Type*} [MulOneClass M]",
"{M : Type*} [Monoid... | [
{
"line": "rwa [map_comap_eq, inf_eq_right]",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf : G →* N\nH : Subgroup N\nh : H ≤ f.range\n⊢ map f (comap f H) = H",
"after_state": "No Goals!"
},
{
"line": "rw [map_comap_eq, inf_eq_right]",
"before_state": "G :... |
theorem comap_lt_comap_of_surjective {f : G →* N} {K L : Subgroup N} (hf : Function.Surjective f) :
K.comap f < L.comap f ↔ K < L := by simp_rw [lt_iff_le_not_le, comap_le_comap_of_surjective hf]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)",
"{M : Type*} [MulOneClass M]",
"{M : Type*} [Monoid... | [
{
"line": "simp_rw [lt_iff_le_not_le, comap_le_comap_of_surjective hf]",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf : G →* N\nK L : Subgroup N\nhf : Surjective ⇑f\n⊢ comap f K < comap f L ↔ K < L",
"after_state": "No Goals!"
},
{
"line": "simp (failIfUnchan... |
theorem comap_injective {f : G →* N} (h : Function.Surjective f) : Function.Injective (comap f) :=
fun K L => by simp only [le_antisymm_iff, comap_le_comap_of_surjective h, imp_self]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)",
"{M : Type*} [MulOneClass M]",
"{M : Type*} [Monoid... | [
{
"line": "simp only [le_antisymm_iff, comap_le_comap_of_surjective h, imp_self]",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf : G →* N\nh : Surjective ⇑f\nK L : Subgroup N\n⊢ comap f K = comap f L → K = L",
"after_state": "No Goals!"
}
] |
theorem comap_map_eq_self {f : G →* N} {H : Subgroup G} (h : f.ker ≤ H) :
comap f (map f H) = H := by
rwa [comap_map_eq, sup_eq_left]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)",
"{M : Type*} [MulOneClass M]",
"{M : Type*} [Monoid... | [
{
"line": "rwa [comap_map_eq, sup_eq_left]",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf : G →* N\nH : Subgroup G\nh : f.ker ≤ H\n⊢ comap f (map f H) = H",
"after_state": "No Goals!"
},
{
"line": "rw [comap_map_eq, sup_eq_left]",
"before_state": "G : Typ... |
theorem map_le_map_iff {f : G →* N} {H K : Subgroup G} : H.map f ≤ K.map f ↔ H ≤ K ⊔ f.ker := by
rw [map_le_iff_le_comap]
rw [comap_map_eq]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)",
"{M : Type*} [MulOneClass M]",
"{M : Type*} [Monoid... | [
{
"line": "rw [map_le_iff_le_comap]",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf : G →* N\nH K : Subgroup G\n⊢ map f H ≤ map f K ↔ H ≤ K ⊔ f.ker",
"after_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf : G →* N\nH K : Subgroup G\n⊢ H ≤ com... |
theorem map_eq_map_iff {f : G →* N} {H K : Subgroup G} :
H.map f = K.map f ↔ H ⊔ f.ker = K ⊔ f.ker := by simp only [le_antisymm_iff, map_le_map_iff']
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)",
"{M : Type*} [MulOneClass M]",
"{M : Type*} [Monoid... | [
{
"line": "simp only [le_antisymm_iff, map_le_map_iff']",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf : G →* N\nH K : Subgroup G\n⊢ map f H = map f K ↔ H ⊔ f.ker = K ⊔ f.ker",
"after_state": "No Goals!"
}
] |
theorem map_eq_range_iff {f : G →* N} {H : Subgroup G} :
H.map f = f.range ↔ Codisjoint H f.ker := by
rw [f.range_eq_map]
rw [map_eq_map_iff]
rw [codisjoint_iff]
rw [top_sup_eq]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)",
"{M : Type*} [MulOneClass M]",
"{M : Type*} [Monoid... | [
{
"line": "rw [f.range_eq_map]",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf : G →* N\nH : Subgroup G\n⊢ map f H = f.range ↔ Codisjoint H f.ker",
"after_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf : G →* N\nH : Subgroup G\n⊢ map f H = m... |
theorem map_le_map_iff_of_injective {f : G →* N} (hf : Function.Injective f) {H K : Subgroup G} :
H.map f ≤ K.map f ↔ H ≤ K := by rw [map_le_iff_le_comap, comap_map_eq_self_of_injective hf]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)",
"{M : Type*} [MulOneClass M]",
"{M : Type*} [Monoid... | [
{
"line": "rw [map_le_iff_le_comap, comap_map_eq_self_of_injective hf]",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf : G →* N\nhf : Injective ⇑f\nH K : Subgroup G\n⊢ map f H ≤ map f K ↔ H ≤ K",
"after_state": "No Goals!"
},
{
"line": "rewrite [map_le_iff_le_... |
theorem map_injective_of_ker_le {H K : Subgroup G} (hH : f.ker ≤ H) (hK : f.ker ≤ K)
(hf : map f H = map f K) : H = K := by
apply_fun comap f at hf
rwa [comap_map_eq, comap_map_eq, sup_of_le_left hH, sup_of_le_left hK] at hf
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)",
"{M : Type*} [MulOneClass M]",
"{M : Type*} [Monoid... | [
{
"line": "apply_fun comap f at hf",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf : G →* N\nH K : Subgroup G\nhH : f.ker ≤ H\nhK : f.ker ≤ K\nhf : map f H = map f K\n⊢ H = K",
"after_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf : G →* N\n... |
theorem closure_preimage_eq_top (s : Set G) : closure ((closure s).subtype ⁻¹' s) = ⊤ := by
apply map_injective (closure s).subtype_injective
rw [MonoidHom.map_closure]
rw [← MonoidHom.range_eq_map]
rw [range_subtype]
rw [Set.image_preimage_eq_of_subset]
rw [coe_subtype]
rw [Subtype.range_coe_subtype]
e... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)",
"{M : Type*} [MulOneClass M]",
"{M : Type*} [Monoid... | [
{
"line": "apply map_injective (closure s).subtype_injective",
"before_state": "G : Type u_1\ninst✝ : Group G\ns : Set G\n⊢ Subgroup.closure (⇑(Subgroup.closure s).subtype ⁻¹' s) = ⊤",
"after_state": "case a\nG : Type u_1\ninst✝ : Group G\ns : Set G\n⊢ map (Subgroup.closure s).subtype (Subgroup.closure ... |
theorem codisjoint_subgroupOf_sup (H K : Subgroup G) :
Codisjoint (H.subgroupOf (H ⊔ K)) (K.subgroupOf (H ⊔ K)) := by
rw [codisjoint_iff]
rw [sup_subgroupOf_eq]
rw [subgroupOf_self]
exacts [le_sup_left, le_sup_right]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Ker.lean | {
"open": [
"Function",
"scoped Int",
"Subgroup",
"MonoidHom"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"{N : Type*} {P : Type*} [Group N] [Group P] (K : Subgroup G)",
"{M : Type*} [MulOneClass M]",
"{M : Type*} [Monoid... | [
{
"line": "rw [codisjoint_iff]",
"before_state": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\n⊢ Codisjoint (H.subgroupOf (H ⊔ K)) (K.subgroupOf (H ⊔ K))",
"after_state": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\n⊢ H.subgroupOf (H ⊔ K) ⊔ K.subgroupOf (H ⊔ K) = ⊤"
},
{
"line": "rewrit... |
theorem coe_eq_univ {H : Subgroup G} : (H : Set G) = Set.univ ↔ H = ⊤ :=
(SetLike.ext'_iff.trans (by rfl)).symm
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Lattice.lean | {
"open": [
"Function",
"scoped Int"
],
"variables": [
"{G : Type*} [Group G]",
"{A : Type*} [AddGroup A]",
"(H K : Subgroup G)"
]
} | [
{
"line": "rfl",
"before_state": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ ↑H = ↑⊤ ↔ ↑H = Set.univ",
"after_state": "No Goals!"
},
{
"line": "exact Iff.rfl✝",
"before_state": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ ↑H = ↑⊤ ↔ ↑H = Set.univ",
"after_state": "No Goals!"
... |
theorem coe_eq_singleton {H : Subgroup G} : (∃ g : G, (H : Set G) = {g}) ↔ H = ⊥ :=
⟨fun ⟨g, hg⟩ =>
haveI : Subsingleton (H : Set G) := by
rw [hg]
infer_instance
H.eq_bot_of_subsingleton,
fun h => ⟨1, SetLike.ext'_iff.mp h⟩⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Lattice.lean | {
"open": [
"Function",
"scoped Int"
],
"variables": [
"{G : Type*} [Group G]",
"{A : Type*} [AddGroup A]",
"(H K : Subgroup G)"
]
} | [
{
"line": "rw [hg]",
"before_state": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nx✝ : ∃ g, ↑H = {g}\ng : G\nhg : ↑H = {g}\n⊢ Subsingleton ↑↑H",
"after_state": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nx✝ : ∃ g, ↑H = {g}\ng : G\nhg : ↑H = {g}\n⊢ Subsingleton ↑{g}"
},
{
"line": "rewrite ... |
theorem nontrivial_iff_exists_ne_one (H : Subgroup G) : Nontrivial H ↔ ∃ x ∈ H, x ≠ (1 : G) := by
rw [Subtype.nontrivial_iff_exists_ne (fun x => x ∈ H) (1 : H)]
simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Lattice.lean | {
"open": [
"Function",
"scoped Int"
],
"variables": [
"{G : Type*} [Group G]",
"{A : Type*} [AddGroup A]",
"(H K : Subgroup G)"
]
} | [
{
"line": "rw [Subtype.nontrivial_iff_exists_ne (fun x => x ∈ H) (1 : H)]",
"before_state": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ Nontrivial ↥H ↔ ∃ x ∈ H, x ≠ 1",
"after_state": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ (∃ y, ∃ (_ : y ∈ H), y ≠ ↑1) ↔ ∃ x ∈ H, x ≠ 1"
},
{
"lin... |
theorem exists_ne_one_of_nontrivial (H : Subgroup G) [Nontrivial H] :
∃ x ∈ H, x ≠ 1 := by
rwa [← Subgroup.nontrivial_iff_exists_ne_one]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Lattice.lean | {
"open": [
"Function",
"scoped Int"
],
"variables": [
"{G : Type*} [Group G]",
"{A : Type*} [AddGroup A]",
"(H K : Subgroup G)"
]
} | [
{
"line": "rwa [← Subgroup.nontrivial_iff_exists_ne_one]",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Nontrivial ↥H\n⊢ ∃ x ∈ H, x ≠ 1",
"after_state": "No Goals!"
},
{
"line": "rw [← Subgroup.nontrivial_iff_exists_ne_one]",
"before_state": "G : Type u_1\ninst✝¹ ... |
theorem comap_id (K : Subgroup N) : K.comap (MonoidHom.id _) = K := by
ext
rfl
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Map.lean | {
"open": [
"Function",
"scoped Int",
"Set"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"(H K : Subgroup G) {k : Set G}",
"{N : Type*} [Group N] {P : Type*} [Group P]"
]
} | [
{
"line": "ext",
"before_state": "N : Type u_5\ninst✝ : Group N\nK : Subgroup N\n⊢ Subgroup.comap (MonoidHom.id N) K = K",
"after_state": "case h\nN : Type u_5\ninst✝ : Group N\nK : Subgroup N\nx✝ : N\n⊢ x✝ ∈ Subgroup.comap (MonoidHom.id N) K ↔ x✝ ∈ K"
},
{
"line": "rfl",
"before_state": "ca... |
theorem map_one_eq_bot : K.map (1 : G →* N) = ⊥ :=
eq_bot_iff.mpr <| by
rintro x ⟨y, _, rfl⟩
simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Map.lean | {
"open": [
"Function",
"scoped Int",
"Set"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"(H K : Subgroup G) {k : Set G}",
"{N : Type*} [Group N] {P : Type*} [Group P]"
]
} | [
{
"line": "rintro x ⟨y, _, rfl⟩",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nK : Subgroup G\nN : Type u_5\ninst✝ : Group N\n⊢ Subgroup.map 1 K ≤ ⊥",
"after_state": "case intro.intro\nG : Type u_1\ninst✝¹ : Group G\nK : Subgroup G\nN : Type u_5\ninst✝ : Group N\ny : G\nleft✝ : y ∈ ↑K\n⊢ 1 y ∈ ⊥"
... |
theorem map_top_of_surjective (f : G →* N) (h : Function.Surjective f) : Subgroup.map f ⊤ = ⊤ := by
rw [eq_top_iff]
intro x _
obtain ⟨y, hy⟩ := h x
exact ⟨y, trivial, hy⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Map.lean | {
"open": [
"Function",
"scoped Int",
"Set"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"(H K : Subgroup G) {k : Set G}",
"{N : Type*} [Group N] {P : Type*} [Group P]"
]
} | [
{
"line": "rw [eq_top_iff]",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_5\ninst✝ : Group N\nf : G →* N\nh : Surjective ⇑f\n⊢ Subgroup.map f ⊤ = ⊤",
"after_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_5\ninst✝ : Group N\nf : G →* N\nh : Surjective ⇑f\n⊢ ⊤ ≤ Subgroup.map f ⊤"
},
... |
theorem subgroupOf_map_subtype (H K : Subgroup G) : (H.subgroupOf K).map K.subtype = H ⊓ K :=
SetLike.ext' <| by refine Subtype.image_preimage_coe _ _ |>.trans ?_; apply Set.inter_comm
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Map.lean | {
"open": [
"Function",
"scoped Int",
"Set"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"(H K : Subgroup G) {k : Set G}",
"{N : Type*} [Group N] {P : Type*} [Group P]"
]
} | [
{
"line": "refine Subtype.image_preimage_coe _ _ |>.trans ?_",
"before_state": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\n⊢ ↑(Subgroup.map K.subtype (H.subgroupOf K)) = ↑(H ⊓ K)",
"after_state": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\n⊢ ↑K ∩ H.toSubsemigroup.1 = ↑(H ⊓ K)"
},
{
"... |
theorem closure_preimage_le (f : G →* N) (s : Set N) : closure (f ⁻¹' s) ≤ (closure s).comap f :=
(closure_le _).2 fun x hx => by rw [SetLike.mem_coe, mem_comap]; exact subset_closure hx
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Map.lean | {
"open": [
"Function",
"scoped Int",
"Set",
"MonoidHom",
"Subgroup"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"(H K : Subgroup G) {k : Set G}",
"{N : Type*} [Group N] {P : Type*} [Group P]",
"(H : Subgroup G)",
... | [
{
"line": "rw [SetLike.mem_coe, mem_comap]",
"before_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf : G →* N\ns : Set N\nx : G\nhx : x ∈ ⇑f ⁻¹' s\n⊢ x ∈ ↑(comap f (Subgroup.closure s))",
"after_state": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_9\ninst✝ : Group N\nf : G →* N\... |
theorem equivMapOfInjective_coe_mulEquiv (H : Subgroup G) (e : G ≃* G') :
H.equivMapOfInjective (e : G →* G') (EquivLike.injective e) = e.subgroupMap H := by
ext
rfl
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Map.lean | {
"open": [
"Function",
"scoped Int",
"Set",
"MonoidHom",
"Subgroup"
],
"variables": [
"{G G' G'' : Type*} [Group G] [Group G'] [Group G'']",
"{A : Type*} [AddGroup A]",
"(H K : Subgroup G) {k : Set G}",
"{N : Type*} [Group N] {P : Type*} [Group P]",
"(H : Subgroup G)",
... | [
{
"line": "ext",
"before_state": "G : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nH : Subgroup G\ne : G ≃* G'\n⊢ H.equivMapOfInjective ↑e ⋯ = e.subgroupMap H",
"after_state": "case h.a\nG : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nH : Subgroup G\ne : G ≃* G'\nx✝ : ↥H... |
lemma mul_subgroupClosure (hs : s.Nonempty) : s * closure s = closure s := by
rw [← smul_eq_mul]
rw [← Set.iUnion_smul_set]
have h a (ha : a ∈ s) : a • (closure s : Set G) = closure s :=
smul_coe_set <| subset_closure ha
simp +contextual [h, hs]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "rw [← smul_eq_mul]",
"before_state": "G : Type u_2\ninst✝ : Group G\ns : Set G\nhs : s.Nonempty\n⊢ s * ↑(Subgroup.closure s) = ↑(Subgroup.closure s)",
"after_state": "G : Type u_2\ninst✝ : Group G\ns : Set G\nhs : s.Nonempty\n⊢ s • ↑(Subgroup.closure s) = ↑(Subgroup.closure s)"
},
{
"... |
lemma pow_mul_subgroupClosure (hs : s.Nonempty) : ∀ n, s ^ n * closure s = closure s
| 0 => by simp
| n + 1 => by rw [pow_succ, mul_assoc, mul_subgroupClosure hs, pow_mul_subgroupClosure hs]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "simp",
"before_state": "G : Type u_1\ninst✝ : Group G\ns : Set G\nhs : s.Nonempty\n⊢ s ^ 0 * ↑(Subgroup.closure s) = ↑(Subgroup.closure s)",
"after_state": "No Goals!"
},
{
"line": "rw [pow_succ, mul_assoc, mul_subgroupClosure hs, pow_mul_subgroupClosure hs]",
"before_state": "G :... |
lemma subgroupClosure_mul_pow (hs : s.Nonempty) : ∀ n, closure s * s ^ n = closure s
| 0 => by simp
| n + 1 => by rw [pow_succ', ← mul_assoc, subgroupClosure_mul hs, subgroupClosure_mul_pow hs]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "simp",
"before_state": "G : Type u_1\ninst✝ : Group G\ns : Set G\nhs : s.Nonempty\n⊢ ↑(Subgroup.closure s) * s ^ 0 = ↑(Subgroup.closure s)",
"after_state": "No Goals!"
},
{
"line": "rw [pow_succ', ← mul_assoc, subgroupClosure_mul hs, subgroupClosure_mul_pow hs]",
"before_state": "... |
theorem inv_subset_closure (S : Set G) : S⁻¹ ⊆ closure S := fun s hs => by
rw [SetLike.mem_coe]
rw [← Subgroup.inv_mem_iff]
exact subset_closure (mem_inv.mp hs)
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "rw [SetLike.mem_coe]",
"before_state": "G : Type u_1\ninst✝ : Group G\nS : Set G\ns : G\nhs : s ∈ S⁻¹\n⊢ s ∈ ↑(Subgroup.closure S)",
"after_state": "G : Type u_1\ninst✝ : Group G\nS : Set G\ns : G\nhs : s ∈ S⁻¹\n⊢ s ∈ Subgroup.closure S"
},
{
"line": "rewrite [SetLike.mem_coe]",
"... |
theorem closure_induction_left {p : (x : G) → x ∈ closure s → Prop} (one : p 1 (one_mem _))
(mul_left : ∀ x (hx : x ∈ s), ∀ (y) hy, p y hy → p (x * y) (mul_mem (subset_closure hx) hy))
(inv_mul_cancel : ∀ x (hx : x ∈ s), ∀ (y) hy, p y hy →
p (x⁻¹ * y) (mul_mem (inv_mem (subset_closure hx)) hy))
{x : G... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "revert h",
"before_state": "G : Type u_1\ninst✝ : Group G\ns : Set G\np : (x : G) → x ∈ Subgroup.closure s → Prop\none : p 1 ⋯\nmul_left : ∀ (x : G) (hx : x ∈ s) (y : G) (hy : y ∈ Subgroup.closure s), p y hy → p (x * y) ⋯\ninv_mul_cancel : ∀ (x : G) (hx : x ∈ s) (y : G) (hy : y ∈ Subgroup.closure... |
theorem closure_induction_right {p : (x : G) → x ∈ closure s → Prop} (one : p 1 (one_mem _))
(mul_right : ∀ (x) hx, ∀ y (hy : y ∈ s), p x hx → p (x * y) (mul_mem hx (subset_closure hy)))
(mul_inv_cancel : ∀ (x) hx, ∀ y (hy : y ∈ s), p x hx →
p (x * y⁻¹) (mul_mem hx (inv_mem (subset_closure hy))))
{x :... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "rwa [← op_closure] at hm",
"before_state": "G : Type u_1\ninst✝ : Group G\ns : Set G\np : (x : G) → x ∈ Subgroup.closure s → Prop\none : p 1 ⋯\nmul_right : ∀ (x : G) (hx : x ∈ Subgroup.closure s) (y : G) (hy : y ∈ s), p x hx → p (x * y) ⋯\nmul_inv_cancel : ∀ (x : G) (hx : x ∈ Subgroup.closure s) ... |
theorem closure_inv (s : Set G) : closure s⁻¹ = closure s := by
simp only [← toSubmonoid_inj]
simp only [closure_toSubmonoid]
simp only [inv_inv]
simp only [union_comm]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "simp only [← toSubmonoid_inj]",
"before_state": "G : Type u_1\ninst✝ : Group G\ns : Set G\n⊢ Subgroup.closure s⁻¹ = Subgroup.closure s",
"after_state": "G : Type u_1\ninst✝ : Group G\ns : Set G\n⊢ (Subgroup.closure s⁻¹).toSubmonoid = (Subgroup.closure s).toSubmonoid"
},
{
"line": "sim... |
lemma closure_singleton_inv (x : G) : closure {x⁻¹} = closure {x} := by
rw [← Set.inv_singleton]
rw [closure_inv]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "rw [← Set.inv_singleton]",
"before_state": "G : Type u_1\ninst✝ : Group G\nx : G\n⊢ Subgroup.closure {x⁻¹} = Subgroup.closure {x}",
"after_state": "G : Type u_1\ninst✝ : Group G\nx : G\n⊢ Subgroup.closure {x}⁻¹ = Subgroup.closure {x}"
},
{
"line": "rewrite [← Set.inv_singleton]",
... |
theorem iSup_induction {ι : Sort*} (S : ι → Subgroup G) {C : G → Prop} {x : G} (hx : x ∈ ⨆ i, S i)
(mem : ∀ (i), ∀ x ∈ S i, C x) (one : C 1) (mul : ∀ x y, C x → C y → C (x * y)) : C x := by
rw [iSup_eq_closure] at hx
induction hx using closure_induction'' with
| one => exact one
| mem x hx =>
obtain ⟨i,... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "rw [iSup_eq_closure] at hx",
"before_state": "G : Type u_2\ninst✝ : Group G\nι : Sort u_1\nS : ι → Subgroup G\nC : G → Prop\nx : G\nhx : x ∈ ⨆ i, S i\nmem : ∀ (i : ι), ∀ x ∈ S i, C x\none : C 1\nmul : ∀ (x y : G), C x → C y → C (x * y)\n⊢ C x",
"after_state": "G : Type u_2\ninst✝ : Group G\nι... |
theorem iSup_induction' {ι : Sort*} (S : ι → Subgroup G) {C : ∀ x, (x ∈ ⨆ i, S i) → Prop}
(hp : ∀ (i), ∀ x (hx : x ∈ S i), C x (mem_iSup_of_mem i hx)) (h1 : C 1 (one_mem _))
(hmul : ∀ x y hx hy, C x hx → C y hy → C (x * y) (mul_mem ‹_› ‹_›)) {x : G}
(hx : x ∈ ⨆ i, S i) : C x hx := by
suffices ∃ h, C x h f... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "assumption",
"before_state": "G : Type ?u.49\nA : Type ?u.52\ninst✝¹ : Group G\ninst✝ : AddGroup A\ns : Set G\nι : Sort u_1\nS : ι → Subgroup G\nC : (x : G) → x ∈ ⨆ i, S i → Prop\nhp : ∀ (i : ι) (x : G) (hx : x ∈ S i), C x ⋯\nh1 : C 1 ⋯\nx y : G\nhx : x ∈ ⨆ i, S i\nhy : y ∈ ⨆ i, S i\n⊢ x ∈ ⨆ i, S... |
lemma closure_pow {n : ℕ} (hs : 1 ∈ s) (hn : n ≠ 0) : closure (s ^ n) = closure s :=
(closure_pow_le hn).antisymm <| by gcongr; exact subset_pow hs hn
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "gcongr",
"before_state": "G : Type u_1\ninst✝ : Group G\ns : Set G\nn : ℕ\nhs : 1 ∈ s\nhn : n ≠ 0\n⊢ Subgroup.closure s ≤ Subgroup.closure (s ^ n)",
"after_state": "case h'\nG : Type u_1\ninst✝ : Group G\ns : Set G\nn : ℕ\nhs : 1 ∈ s\nhn : n ≠ 0\n⊢ s ⊆ s ^ n"
},
{
"line": "exact subse... |
theorem sup_eq_closure_mul (H K : Subgroup G) : H ⊔ K = closure ((H : Set G) * (K : Set G)) :=
le_antisymm
(sup_le (fun h hh => subset_closure ⟨h, hh, 1, K.one_mem, mul_one h⟩) fun k hk =>
subset_closure ⟨1, H.one_mem, k, hk, one_mul k⟩)
((closure_mul_le _ _).trans <| by rw [closure_eq, closure_eq])
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "rw [closure_eq, closure_eq]",
"before_state": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\n⊢ Subgroup.closure ↑H ⊔ Subgroup.closure ↑K ≤ H ⊔ K",
"after_state": "No Goals!"
},
{
"line": "rewrite [closure_eq, closure_eq]",
"before_state": "G : Type u_1\ninst✝ : Group G\nH K : S... |
theorem set_mul_normalizer_comm (S : Set G) (N : Subgroup G) (hLE : S ⊆ N.normalizer) :
S * N = N * S := by
rw [← iUnion_mul_left_image]
rw [← iUnion_mul_right_image]
simp only [image_mul_left]
simp only [image_mul_right]
simp only [Set.preimage]
congr! 5 with s hs x
exact (mem_normalizer_iff'.mp (inv... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "rw [← iUnion_mul_left_image]",
"before_state": "G : Type u_1\ninst✝ : Group G\nS : Set G\nN : Subgroup G\nhLE : S ⊆ ↑N.normalizer\n⊢ S * ↑N = ↑N * S",
"after_state": "G : Type u_1\ninst✝ : Group G\nS : Set G\nN : Subgroup G\nhLE : S ⊆ ↑N.normalizer\n⊢ ⋃ a ∈ S, (fun x => a * x) '' ↑N = ↑N * S"... |
theorem coe_mul_of_left_le_normalizer_right (H N : Subgroup G) (hLE : H ≤ N.normalizer) :
(↑(H ⊔ N) : Set G) = H * N := by
rw [sup_eq_closure_mul]
refine Set.Subset.antisymm (fun x hx => ?_) subset_closure
induction hx using closure_induction'' with
| one => exact ⟨1, one_mem _, 1, one_mem _, mul_one 1⟩
|... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "rw [sup_eq_closure_mul]",
"before_state": "G : Type u_1\ninst✝ : Group G\nH N : Subgroup G\nhLE : H ≤ N.normalizer\n⊢ ↑(H ⊔ N) = ↑H * ↑N",
"after_state": "G : Type u_1\ninst✝ : Group G\nH N : Subgroup G\nhLE : H ≤ N.normalizer\n⊢ ↑(Subgroup.closure (↑H * ↑N)) = ↑H * ↑N"
},
{
"line": "... |
theorem coe_mul_of_right_le_normalizer_left (N H : Subgroup G) (hLE : H ≤ N.normalizer) :
(↑(N ⊔ H) : Set G) = N * H := by
rw [← set_mul_normalizer_comm _ _ hLE]
rw [sup_comm]
rw [coe_mul_of_left_le_normalizer_right _ _ hLE]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "rw [← set_mul_normalizer_comm _ _ hLE]",
"before_state": "G : Type u_1\ninst✝ : Group G\nN H : Subgroup G\nhLE : H ≤ N.normalizer\n⊢ ↑(N ⊔ H) = ↑N * ↑H",
"after_state": "G : Type u_1\ninst✝ : Group G\nN H : Subgroup G\nhLE : H ≤ N.normalizer\n⊢ ↑(N ⊔ H) = ↑H * ↑N"
},
{
"line": "rewrit... |
theorem mul_inf_assoc (A B C : Subgroup G) (h : A ≤ C) :
(A : Set G) * ↑(B ⊓ C) = (A : Set G) * (B : Set G) ∩ C := by
ext
simp only [coe_inf]
simp only [Set.mem_mul]
simp only [Set.mem_inter_iff]
constructor
· rintro ⟨y, hy, z, ⟨hzB, hzC⟩, rfl⟩
refine ⟨?_, mul_mem (h hy) hzC⟩
exact ⟨y, hy, z, hz... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "ext",
"before_state": "G : Type u_1\ninst✝ : Group G\nA B C : Subgroup G\nh : A ≤ C\n⊢ ↑A * ↑(B ⊓ C) = ↑A * ↑B ∩ ↑C",
"after_state": "case h\nG : Type u_1\ninst✝ : Group G\nA B C : Subgroup G\nh : A ≤ C\nx✝ : G\n⊢ x✝ ∈ ↑A * ↑(B ⊓ C) ↔ x✝ ∈ ↑A * ↑B ∩ ↑C"
},
{
"line": "simp only [coe_in... |
theorem inf_mul_assoc (A B C : Subgroup G) (h : C ≤ A) :
((A ⊓ B : Subgroup G) : Set G) * C = (A : Set G) ∩ (↑B * ↑C) := by
ext
simp only [coe_inf]
simp only [Set.mem_mul]
simp only [Set.mem_inter_iff]
constructor
· rintro ⟨y, ⟨hyA, hyB⟩, z, hz, rfl⟩
refine ⟨A.mul_mem hyA (h hz), ?_⟩
exact ⟨y, h... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "ext",
"before_state": "G : Type u_1\ninst✝ : Group G\nA B C : Subgroup G\nh : C ≤ A\n⊢ ↑(A ⊓ B) * ↑C = ↑A ∩ (↑B * ↑C)",
"after_state": "case h\nG : Type u_1\ninst✝ : Group G\nA B C : Subgroup G\nh : C ≤ A\nx✝ : G\n⊢ x✝ ∈ ↑(A ⊓ B) * ↑C ↔ x✝ ∈ ↑A ∩ (↑B * ↑C)"
},
{
"line": "simp only [co... |
theorem smul_mem_of_mem_closure_of_mem {X : Type*} [MulAction G X] {s : Set G} {t : Set X}
(hs : ∀ g ∈ s, g⁻¹ ∈ s) (hst : ∀ᵉ (g ∈ s) (x ∈ t), g • x ∈ t) {g : G}
(hg : g ∈ Subgroup.closure s) {x : X} (hx : x ∈ t) : g • x ∈ t := by
induction hg using Subgroup.closure_induction'' generalizing x with
| one => s... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "induction hg using Subgroup.closure_induction'' generalizing x with\n| one => simpa\n| mem g' hg' => exact hst g' hg' x hx\n| inv_mem g' hg' => exact hst g'⁻¹ (hs g' hg') x hx\n| mul _ _ _ _ h₁ h₂ => rw [mul_smul]; exact h₁ (h₂ hx)",
"before_state": "G : Type u_2\ninst✝¹ : Group G\nX : Type u_1\n... |
theorem smul_opposite_image_mul_preimage' (g : G) (h : Gᵐᵒᵖ) (s : Set G) :
(fun y => h • y) '' ((g * ·) ⁻¹' s) = (g * ·) ⁻¹' ((fun y => h • y) '' s) := by
simp [preimage_preimage, mul_assoc]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}"
]
} | [
{
"line": "simp [preimage_preimage, mul_assoc]",
"before_state": "G : Type u_1\ninst✝ : Group G\ng : G\nh : Gᵐᵒᵖ\ns : Set G\n⊢ (fun y => h • y) '' ((fun x => g * x) ⁻¹' s) = (fun x => g * x) ⁻¹' ((fun y => h • y) '' s)",
"after_state": "No Goals!"
}
] |
theorem conj_smul_le_of_le {P H : Subgroup G} (hP : P ≤ H) (h : H) :
MulAut.conj (h : G) • P ≤ H := by
rintro - ⟨g, hg, rfl⟩
exact H.mul_mem (H.mul_mem h.2 (hP hg)) (H.inv_mem h.2)
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}",
"[Monoid α] [MulDistribMulAction α G]"
]
} | [
{
"line": "rintro - ⟨g, hg, rfl⟩",
"before_state": "G : Type u_1\ninst✝ : Group G\nP H : Subgroup G\nhP : P ≤ H\nh : ↥H\n⊢ MulAut.conj ↑h • P ≤ H",
"after_state": "case intro.intro\nG : Type u_1\ninst✝ : Group G\nP H : Subgroup G\nhP : P ≤ H\nh : ↥H\ng : G\nhg : g ∈ ↑P\n⊢ ((MulDistribMulAction.toMonoidE... |
theorem conj_smul_subgroupOf {P H : Subgroup G} (hP : P ≤ H) (h : H) :
MulAut.conj h • P.subgroupOf H = (MulAut.conj (h : G) • P).subgroupOf H := by
refine le_antisymm ?_ ?_
· rintro - ⟨g, hg, rfl⟩
exact ⟨g, hg, rfl⟩
· rintro p ⟨g, hg, hp⟩
exact ⟨⟨g, hP hg⟩, hg, Subtype.ext hp⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}",
"[Monoid α] [MulDistribMulAction α G]"
]
} | [
{
"line": "refine le_antisymm ?_ ?_",
"before_state": "G : Type u_1\ninst✝ : Group G\nP H : Subgroup G\nhP : P ≤ H\nh : ↥H\n⊢ MulAut.conj h • P.subgroupOf H = (MulAut.conj ↑h • P).subgroupOf H",
"after_state": "case refine_1\nG : Type u_1\ninst✝ : Group G\nP H : Subgroup G\nhP : P ≤ H\nh : ↥H\n⊢ MulAut.... |
theorem subgroup_mul_singleton {H : Subgroup G} {h : G} (hh : h ∈ H) : (H : Set G) * {h} = H := by
simp [preimage, mul_mem_cancel_right (inv_mem hh)]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}",
"[Monoid α] [MulDistribMulAction α G]",
"[Group α] [MulDistribMulAction α G]"
]
} | [
{
"line": "simp [preimage, mul_mem_cancel_right (inv_mem hh)]",
"before_state": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : G\nhh : h ∈ H\n⊢ ↑H * {h} = ↑H",
"after_state": "No Goals!"
}
] |
theorem singleton_mul_subgroup {H : Subgroup G} {h : G} (hh : h ∈ H) : {h} * (H : Set G) = H := by
simp [preimage, mul_mem_cancel_left (inv_mem hh)]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}",
"[Monoid α] [MulDistribMulAction α G]",
"[Group α] [MulDistribMulAction α G]"
]
} | [
{
"line": "simp [preimage, mul_mem_cancel_left (inv_mem hh)]",
"before_state": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : G\nhh : h ∈ H\n⊢ {h} * ↑H = ↑H",
"after_state": "No Goals!"
}
] |
theorem Normal.of_conjugate_fixed {H : Subgroup G} (h : ∀ g : G, (MulAut.conj g) • H = H) :
H.Normal := by
constructor
intro n hn g
rw [← h g]
rw [Subgroup.mem_pointwise_smul_iff_inv_smul_mem]
rw [← map_inv]
rw [MulAut.smul_def]
rw [MulAut.conj_apply]
rw [inv_inv]
rw [mul_assoc]
rw [mul_assoc]
... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}",
"[Monoid α] [MulDistribMulAction α G]",
"[Group α] [MulDistribMulAction α G]"
]
} | [
{
"line": "constructor",
"before_state": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : ∀ (g : G), MulAut.conj g • H = H\n⊢ H.Normal",
"after_state": "case conj_mem\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : ∀ (g : G), MulAut.conj g • H = H\n⊢ ∀ n ∈ H, ∀ (g : G), g * n * g⁻¹ ∈ H"
},
{
... |
theorem normalCore_eq_iInf_conjAct (H : Subgroup G) :
H.normalCore = ⨅ (g : ConjAct G), g • H := by
ext g
simp only [Subgroup.normalCore]
simp only [Subgroup.mem_iInf]
simp only [Subgroup.mem_pointwise_smul_iff_inv_smul_mem]
refine ⟨fun h x ↦ h x⁻¹, fun h x ↦ ?_⟩
simpa only [ConjAct.toConjAct_inv,inv_in... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subgroup/Pointwise.lean | {
"open": [
"Set",
"Pointwise",
"Subgroup",
"scoped RightActions in"
],
"variables": [
"{α G A S : Type*}",
"[Group G] [AddGroup A] {s : Set G}",
"[Monoid α] [MulDistribMulAction α G]",
"[Group α] [MulDistribMulAction α G]"
]
} | [
{
"line": "ext g",
"before_state": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ H.normalCore = ⨅ g, g • H",
"after_state": "case h\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\ng : G\n⊢ g ∈ H.normalCore ↔ g ∈ ⨅ g, g • H"
},
{
"line": "simp only [Subgroup.normalCore]",
"before_state": ... |
theorem iSup_eq_closure {ι : Sort*} (p : ι → Submonoid M) :
⨆ i, p i = Submonoid.closure (⋃ i, (p i : Set M)) := by
simp_rw [Submonoid.closure_iUnion, Submonoid.closure_eq]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Basic.lean | {
"open": [
"Set"
],
"variables": [
"{M : Type*} {N : Type*}",
"{A : Type*}",
"[MulOneClass M] {s : Set M}",
"[AddZeroClass A] {t : Set A}",
"(S : Submonoid M)",
"{S}",
"(S)",
"(M)",
"{M}"
]
} | [
{
"line": "simp_rw [Submonoid.closure_iUnion, Submonoid.closure_eq]",
"before_state": "M : Type u_1\ninst✝ : MulOneClass M\nι : Sort u_4\np : ι → Submonoid M\n⊢ ⨆ i, p i = Submonoid.closure (⋃ i, ↑(p i))",
"after_state": "No Goals!"
},
{
"line": "simp (failIfUnchanged✝ := false✝) only",
"bef... |
theorem multiset_noncommProd_mem (S : Submonoid M) (m : Multiset M) (comm) (h : ∀ x ∈ m, x ∈ S) :
m.noncommProd comm ∈ S := by
induction m using Quotient.inductionOn with | h l => ?_
simp only [Multiset.quot_mk_to_coe]
simp only [Multiset.noncommProd_coe]
exact Submonoid.list_prod_mem _ h
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/BigOperators.lean | {
"open": [
"SubmonoidClass"
],
"variables": [
"{M A B : Type*}",
"[Monoid M] [SetLike B M] [SubmonoidClass B M] {x : M} {S : B}",
"[Monoid M] {x : M} (s : Submonoid M)"
]
} | [
{
"line": "induction m using Quotient.inductionOn with\n| h l => ?_",
"before_state": "M : Type u_1\ninst✝¹ inst✝ : Monoid M\nS : Submonoid M\nm : Multiset M\ncomm : {x | x ∈ m}.Pairwise Commute\nh : ∀ x ∈ m, x ∈ S\n⊢ m.noncommProd comm ∈ S",
"after_state": "case h\nM : Type u_1\ninst✝¹ inst✝ : Monoid M... |
theorem noncommProd_mem (S : Submonoid M) {ι : Type*} (t : Finset ι) (f : ι → M) (comm)
(h : ∀ c ∈ t, f c ∈ S) : t.noncommProd f comm ∈ S := by
apply multiset_noncommProd_mem
intro y
rw [Multiset.mem_map]
rintro ⟨x, ⟨hx, rfl⟩⟩
exact h x hx
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/BigOperators.lean | {
"open": [
"SubmonoidClass"
],
"variables": [
"{M A B : Type*}",
"[Monoid M] [SetLike B M] [SubmonoidClass B M] {x : M} {S : B}",
"[Monoid M] {x : M} (s : Submonoid M)"
]
} | [
{
"line": "apply multiset_noncommProd_mem",
"before_state": "M : Type u_1\ninst✝¹ inst✝ : Monoid M\nS : Submonoid M\nι : Type u_4\nt : Finset ι\nf : ι → M\ncomm : (↑t).Pairwise (Function.onFun Commute f)\nh : ∀ c ∈ t, f c ∈ S\n⊢ t.noncommProd f comm ∈ S",
"after_state": "No Goals!"
}
] |
lemma mem_closure_finset {s : Finset M} :
x ∈ closure s ↔ ∃ f : M → ℕ, f.support ⊆ s ∧ ∏ a ∈ s, a ^ f a = x where
mp := by
rw [mem_closure_iff_exists_finset_subset]
rintro ⟨f, t, hts, hf, rfl⟩
refine ⟨f, hf.trans hts, .symm <| Finset.prod_subset hts ?_⟩
simp +contextual [Function.support_subset_if... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/BigOperators.lean | {
"open": [
"SubmonoidClass"
],
"variables": [
"{M A B : Type*}",
"[Monoid M] [SetLike B M] [SubmonoidClass B M] {x : M} {S : B}",
"[Monoid M] {x : M} (s : Submonoid M)",
"[CommMonoid M] {x : M}"
]
} | [
{
"line": "rw [mem_closure_iff_exists_finset_subset]",
"before_state": "M : Type u_1\ninst✝² inst✝¹ : Monoid M\ninst✝ : CommMonoid M\nx : M\ns : Finset M\n⊢ ?m.3514 → ?m.3515",
"after_state": "M : Type u_1\ninst✝² inst✝¹ : Monoid M\ninst✝ : CommMonoid M\nx : M\ns : Finset M\n⊢ ?m.3514 → ?m.3515"
},
... |
theorem mem_sup_left {S T : Submonoid M} : ∀ {x : M}, x ∈ S → x ∈ S ⊔ T := by
rw [← SetLike.le_def]
exact le_sup_left
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Membership.lean | {
"open": [
"Set"
],
"variables": [
"{M A B : Type*}",
"[Monoid M] [SetLike B M] [SubmonoidClass B M] {S : B}",
"[MulOneClass M]"
]
} | [
{
"line": "rw [← SetLike.le_def]",
"before_state": "M : Type u_1\ninst✝¹ : Monoid M\ninst✝ : MulOneClass M\nS T : Submonoid M\n⊢ ∀ {x : M}, x ∈ S → x ∈ S ⊔ T",
"after_state": "M : Type u_1\ninst✝¹ : Monoid M\ninst✝ : MulOneClass M\nS T : Submonoid M\n⊢ S ≤ S ⊔ T"
},
{
"line": "rewrite [← SetLike... |
theorem mem_sup_right {S T : Submonoid M} : ∀ {x : M}, x ∈ T → x ∈ S ⊔ T := by
rw [← SetLike.le_def]
exact le_sup_right
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Membership.lean | {
"open": [
"Set"
],
"variables": [
"{M A B : Type*}",
"[Monoid M] [SetLike B M] [SubmonoidClass B M] {S : B}",
"[MulOneClass M]"
]
} | [
{
"line": "rw [← SetLike.le_def]",
"before_state": "M : Type u_1\ninst✝¹ : Monoid M\ninst✝ : MulOneClass M\nS T : Submonoid M\n⊢ ∀ {x : M}, x ∈ T → x ∈ S ⊔ T",
"after_state": "M : Type u_1\ninst✝¹ : Monoid M\ninst✝ : MulOneClass M\nS T : Submonoid M\n⊢ T ≤ S ⊔ T"
},
{
"line": "rewrite [← SetLike... |
theorem mem_iSup_of_mem {ι : Sort*} {S : ι → Submonoid M} (i : ι) :
∀ {x : M}, x ∈ S i → x ∈ iSup S := by
rw [← SetLike.le_def]
exact le_iSup _ _
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Membership.lean | {
"open": [
"Set"
],
"variables": [
"{M A B : Type*}",
"[Monoid M] [SetLike B M] [SubmonoidClass B M] {S : B}",
"[MulOneClass M]"
]
} | [
{
"line": "rw [← SetLike.le_def]",
"before_state": "M : Type u_1\ninst✝¹ : Monoid M\ninst✝ : MulOneClass M\nι : Sort u_4\nS : ι → Submonoid M\ni : ι\n⊢ ∀ {x : M}, x ∈ S i → x ∈ iSup S",
"after_state": "M : Type u_1\ninst✝¹ : Monoid M\ninst✝ : MulOneClass M\nι : Sort u_4\nS : ι → Submonoid M\ni : ι\n⊢ S ... |
theorem mem_sSup_of_mem {S : Set (Submonoid M)} {s : Submonoid M} (hs : s ∈ S) :
∀ {x : M}, x ∈ s → x ∈ sSup S := by
rw [← SetLike.le_def]
exact le_sSup hs
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Membership.lean | {
"open": [
"Set"
],
"variables": [
"{M A B : Type*}",
"[Monoid M] [SetLike B M] [SubmonoidClass B M] {S : B}",
"[MulOneClass M]"
]
} | [
{
"line": "rw [← SetLike.le_def]",
"before_state": "M : Type u_1\ninst✝¹ : Monoid M\ninst✝ : MulOneClass M\nS : Set (Submonoid M)\ns : Submonoid M\nhs : s ∈ S\n⊢ ∀ {x : M}, x ∈ s → x ∈ sSup S",
"after_state": "M : Type u_1\ninst✝¹ : Monoid M\ninst✝ : MulOneClass M\nS : Set (Submonoid M)\ns : Submonoid M... |
theorem iSup_induction' {ι : Sort*} (S : ι → Submonoid M) {motive : ∀ x, (x ∈ ⨆ i, S i) → Prop}
(mem : ∀ (i), ∀ (x) (hxS : x ∈ S i), motive x (mem_iSup_of_mem i hxS))
(one : motive 1 (one_mem _))
(mul : ∀ x y hx hy, motive x hx → motive y hy → motive (x * y) (mul_mem ‹_› ‹_›)) {x : M}
(hx : x ∈ ⨆ i, S i... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Membership.lean | {
"open": [
"Set"
],
"variables": [
"{M A B : Type*}",
"[Monoid M] [SetLike B M] [SubmonoidClass B M] {S : B}",
"[MulOneClass M]"
]
} | [
{
"line": "assumption",
"before_state": "M : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝³ : Monoid M\ninst✝² : SetLike B M\ninst✝¹ : SubmonoidClass B M\nS✝ : B\ninst✝ : MulOneClass M\nmem_iSup_of_mem : ?m.1390\nι : Sort u_4\nS : ι → Submonoid M\nmotive : (x : M) → x ∈ ⨆ i, S i → Prop\nmem : ∀ (i : ι), ∀ x ∈... |
theorem card_le_one_iff_eq_bot : card S ≤ 1 ↔ S = ⊥ :=
⟨fun h =>
(eq_bot_iff_forall _).2 fun x hx => by
simpa [Subtype.ext_iff] using card_le_one_iff.1 h ⟨x, hx⟩ 1,
fun h => by simp [h]⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Membership.lean | {
"open": [
"Set",
"Submonoid",
"MonoidHom",
"Fintype"
],
"variables": [
"{M A B : Type*}",
"[Monoid M] [SetLike B M] [SubmonoidClass B M] {S : B}",
"[MulOneClass M]",
"{α : Type*}",
"[Monoid M] {a : M}",
"{S : Submonoid M} [Fintype S]"
]
} | [
{
"line": "simpa [Subtype.ext_iff] using card_le_one_iff.1 h ⟨x, hx⟩ 1",
"before_state": "M : Type u_1\ninst✝³ : Monoid M\ninst✝² : MulOneClass M\ninst✝¹ : Monoid M\nS : Submonoid M\ninst✝ : Fintype ↥S\nh : card ↥S ≤ 1\nx : M\nhx : x ∈ S\n⊢ x = 1",
"after_state": "No Goals!"
},
{
"line": "simp [... |
theorem exists_multiset_of_mem_closure {M : Type*} [CommMonoid M] {s : Set M} {x : M}
(hx : x ∈ closure s) : ∃ l : Multiset M, (∀ y ∈ l, y ∈ s) ∧ l.prod = x := by
obtain ⟨l, h1, h2⟩ := exists_list_of_mem_closure hx
exact ⟨l, h1, (Multiset.prod_coe l).trans h2⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Membership.lean | {
"open": [
"Set",
"Submonoid",
"MonoidHom",
"Fintype"
],
"variables": [
"{M A B : Type*}",
"[Monoid M] [SetLike B M] [SubmonoidClass B M] {S : B}",
"[MulOneClass M]",
"{α : Type*}",
"[Monoid M] {a : M}",
"{S : Submonoid M} [Fintype S]"
]
} | [
{
"line": "obtain ⟨l, h1, h2⟩ := exists_list_of_mem_closure hx",
"before_state": "M : Type u_5\ninst✝ : CommMonoid M\ns : Set M\nx : M\nhx : x ∈ Submonoid.closure s\n⊢ ∃ l, (∀ y ∈ l, y ∈ s) ∧ l.prod = x",
"after_state": "case intro.intro\nM : Type u_5\ninst✝ : CommMonoid M\ns : Set M\nx : M\nhx : x ∈ Su... |
theorem mem_sup {s t : Submonoid N} {x : N} : x ∈ s ⊔ t ↔ ∃ y ∈ s, ∃ z ∈ t, y * z = x := by
simp only [sup_eq_range]
simp only [mem_mrange]
simp only [coprod_apply]
simp only [coe_subtype]
simp only [Prod.exists]
simp only [Subtype.exists]
simp only [exists_prop]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Membership.lean | {
"open": [
"Set",
"Submonoid",
"MonoidHom",
"Fintype",
"MonoidHom"
],
"variables": [
"{M A B : Type*}",
"[Monoid M] [SetLike B M] [SubmonoidClass B M] {S : B}",
"[MulOneClass M]",
"{α : Type*}",
"[Monoid M] {a : M}",
"{S : Submonoid M} [Fintype S]",
"{N : Type*} [C... | [
{
"line": "simp only [sup_eq_range]",
"before_state": "N : Type u_5\ninst✝ : CommMonoid N\ns t : Submonoid N\nx : N\n⊢ x ∈ s ⊔ t ↔ ∃ y ∈ s, ∃ z ∈ t, y * z = x",
"after_state": "N : Type u_5\ninst✝ : CommMonoid N\ns t : Submonoid N\nx : N\n⊢ x ∈ mrange (s.subtype.coprod t.subtype) ↔ ∃ y ∈ s, ∃ z ∈ t, y *... |
theorem mem_closure_pair {A : Type*} [CommMonoid A] (a b c : A) :
c ∈ Submonoid.closure ({a, b} : Set A) ↔ ∃ m n : ℕ, a ^ m * b ^ n = c := by
rw [← Set.singleton_union]
rw [Submonoid.closure_union]
rw [mem_sup]
simp_rw [mem_closure_singleton, exists_exists_eq_and]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Membership.lean | {
"open": [
"Set",
"Submonoid",
"MonoidHom",
"Fintype",
"MonoidHom",
"Set"
],
"variables": [
"{M A B : Type*}",
"[Monoid M] [SetLike B M] [SubmonoidClass B M] {S : B}",
"[MulOneClass M]",
"{α : Type*}",
"[Monoid M] {a : M}",
"{S : Submonoid M} [Fintype S]",
"{N ... | [
{
"line": "rw [← Set.singleton_union]",
"before_state": "A : Type u_6\ninst✝ : CommMonoid A\na b c : A\n⊢ c ∈ Submonoid.closure {a, b} ↔ ∃ m n, a ^ m * b ^ n = c",
"after_state": "A : Type u_6\ninst✝ : CommMonoid A\na b c : A\n⊢ c ∈ Submonoid.closure ({a} ∪ {b}) ↔ ∃ m n, a ^ m * b ^ n = c"
},
{
... |
example {S : Submonoid M'} : IsScalarTower S M' M' := by infer_instance
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/MulAction.lean | {
"open": [],
"variables": [
"{M' : Type*} {α β : Type*}",
"{S' : Type*} [SetLike S' M'] (s : S')",
"[MulOneClass M']",
"[Monoid M'] [SubmonoidClass S' M']",
"[MulOneClass M']",
"[SMul M' α] {S : Submonoid M'}",
"[Monoid M']"
]
} | [
{
"line": "infer_instance",
"before_state": "M' : Type u_1\nα : Type u_2\nβ : Type u_3\nS' : Type u_4\ninst✝⁶ : SetLike S' M'\ns : S'\ninst✝⁵ : MulOneClass M'\ninst✝⁴ : Monoid M'\ninst✝³ : SubmonoidClass S' M'\ninst✝² : MulOneClass M'\ninst✝¹ : SMul M' α\nS✝ : Submonoid M'\ninst✝ : Monoid M'\nS : Submonoid ... |
theorem le_prod_iff {s : Submonoid M} {t : Submonoid N} {u : Submonoid (M × N)} :
u ≤ s.prod t ↔ u.map (fst M N) ≤ s ∧ u.map (snd M N) ≤ t := by
constructor
· intro h
constructor
· rintro x ⟨⟨y1, y2⟩, ⟨hy1, rfl⟩⟩
exact (h hy1).1
· rintro x ⟨⟨y1, y2⟩, ⟨hy1, rfl⟩⟩
exact (h hy1).2
· rintr... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M N]",
"{ι : Type*} {f : F}",
"(hf : Function.Inject... | [
{
"line": "constructor",
"before_state": "N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\n⊢ u ≤ s.prod t ↔ Submonoid.map (fst M N) u ≤ s ∧ Submonoid.map (snd M N) u ≤ t",
"after_state": "case mp\nN : Type u_2\ninst✝¹ : M... |
theorem prod_le_iff {s : Submonoid M} {t : Submonoid N} {u : Submonoid (M × N)} :
s.prod t ≤ u ↔ s.map (inl M N) ≤ u ∧ t.map (inr M N) ≤ u := by
constructor
· intro h
constructor
· rintro _ ⟨x, hx, rfl⟩
apply h
exact ⟨hx, Submonoid.one_mem _⟩
· rintro _ ⟨x, hx, rfl⟩
apply h
e... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M N]",
"{ι : Type*} {f : F}",
"(hf : Function.Inject... | [
{
"line": "constructor",
"before_state": "N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\n⊢ s.prod t ≤ u ↔ Submonoid.map (inl M N) s ≤ u ∧ Submonoid.map (inr M N) t ≤ u",
"after_state": "case mp\nN : Type u_2\ninst✝¹ : M... |
theorem mrange_id : mrange (MonoidHom.id M) = ⊤ := by
simp [mrange_eq_map]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom",
"Submonoid"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M N]",
"{ι : Type*} {f : F}",
"(hf ... | [
{
"line": "simp [mrange_eq_map]",
"before_state": "M : Type u_8\ninst✝ : MulOneClass M\n⊢ mrange (MonoidHom.id M) = ⊤",
"after_state": "No Goals!"
}
] |
theorem mrange_eq_top {f : F} : mrange f = (⊤ : Submonoid N) ↔ Surjective f :=
SetLike.ext'_iff.trans <| Iff.trans (by rw [coe_mrange, coe_top]) Set.range_eq_univ
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom",
"Submonoid"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M N]",
"{ι : Type*} {f : F}",
"(hf ... | [
{
"line": "rw [coe_mrange, coe_top]",
"before_state": "N : Type u_2\ninst✝² : MulOneClass N\nM : Type u_8\ninst✝¹ : MulOneClass M\nF : Type u_9\ninst✝ : FunLike F M N\nmc : MonoidHomClass F M N\nf : F\n⊢ ↑(mrange f) = ↑⊤ ↔ Set.range ⇑f = univ",
"after_state": "No Goals!"
},
{
"line": "rewrite [c... |
theorem restrict_mrange (f : M →* N) : mrange (f.restrict S) = S.map f := by
simp [SetLike.ext_iff]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom",
"Submonoid"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M N]",
"{ι : Type*} {f : F}",
"(hf ... | [
{
"line": "simp [SetLike.ext_iff]",
"before_state": "N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M\nS : Submonoid M\nf : M →* N\n⊢ mrange (f.restrict S) = map f S",
"after_state": "No Goals!"
}
] |
theorem mrangeRestrict_mker (f : M →* N) : mker (mrangeRestrict f) = mker f := by
ext x
change (⟨f x, _⟩ : mrange f) = ⟨1, _⟩ ↔ f x = 1
simp
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom",
"Submonoid"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M N]",
"{ι : Type*} {f : F}",
"(hf ... | [
{
"line": "ext x",
"before_state": "N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M\nf : M →* N\n⊢ mker f.mrangeRestrict = mker f",
"after_state": "case h\nN : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M\nf : M →* N\nx : M\n⊢ x ∈ mker f.mrangeRestrict ... |
theorem mker_one : mker (1 : M →* N) = ⊤ := by
ext
simp [mem_mker]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom",
"Submonoid"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M N]",
"{ι : Type*} {f : F}",
"(hf ... | [
{
"line": "ext",
"before_state": "N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M\n⊢ mker 1 = ⊤",
"after_state": "case h\nN : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M\nx✝ : M\n⊢ x✝ ∈ mker 1 ↔ x✝ ∈ ⊤"
},
{
"line": "simp [mem_mker]",
"befo... |
theorem mker_prod_map {M' : Type*} {N' : Type*} [MulOneClass M'] [MulOneClass N'] (f : M →* N)
(g : M' →* N') : mker (prodMap f g) = (mker f).prod (mker g) := by
rw [← comap_bot']
rw [← comap_bot']
rw [← comap_bot']
rw [← prod_map_comap_prod']
rw [bot_prod_bot]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom",
"Submonoid"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M N]",
"{ι : Type*} {f : F}",
"(hf ... | [
{
"line": "rw [← comap_bot']",
"before_state": "N : Type u_2\ninst✝³ : MulOneClass N\nM : Type u_8\ninst✝² : MulOneClass M\nM' : Type u_10\nN' : Type u_11\ninst✝¹ : MulOneClass M'\ninst✝ : MulOneClass N'\nf : M →* N\ng : M' →* N'\n⊢ mker (f.prodMap g) = (mker f).prod (mker g)",
"after_state": "N : Type ... |
theorem mker_inl : mker (inl M N) = ⊥ := by
ext x
simp [mem_mker]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom",
"Submonoid"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M N]",
"{ι : Type*} {f : F}",
"(hf ... | [
{
"line": "ext x",
"before_state": "N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M\n⊢ mker (inl M N) = ⊥",
"after_state": "case h\nN : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M\nx : M\n⊢ x ∈ mker (inl M N) ↔ x ∈ ⊥"
},
{
"line": "simp [mem_mk... |
theorem mker_inr : mker (inr M N) = ⊥ := by
ext x
simp [mem_mker]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom",
"Submonoid"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M N]",
"{ι : Type*} {f : F}",
"(hf ... | [
{
"line": "ext x",
"before_state": "N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M\n⊢ mker (inr M N) = ⊥",
"after_state": "case h\nN : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M\nx : N\n⊢ x ∈ mker (inr M N) ↔ x ∈ ⊥"
},
{
"line": "simp [mem_mk... |
lemma submonoidComap_surjective_of_surjective (f : M →* N) (N' : Submonoid N) (hf : Surjective f) :
Surjective (f.submonoidComap N') := fun y ↦ by
obtain ⟨x, hx⟩ := hf y
use ⟨x, mem_comap.mpr (hx ▸ y.2)⟩
apply Subtype.val_injective
simp [hx]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom",
"Submonoid"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M N]",
"{ι : Type*} {f : F}",
"(hf ... | [
{
"line": "obtain ⟨x, hx⟩ := hf y",
"before_state": "N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M\nf : M →* N\nN' : Submonoid N\nhf : Surjective ⇑f\ny : ↥N'\n⊢ ∃ a, (f.submonoidComap N') a = y",
"after_state": "case intro\nN : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\n... |
theorem submonoidMap_surjective (f : M →* N) (M' : Submonoid M) :
Function.Surjective (f.submonoidMap M') := by
rintro ⟨_, x, hx, rfl⟩
exact ⟨⟨x, hx⟩, rfl⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom",
"Submonoid"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M N]",
"{ι : Type*} {f : F}",
"(hf ... | [
{
"line": "rintro ⟨_, x, hx, rfl⟩",
"before_state": "N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M\nf : M →* N\nM' : Submonoid M\n⊢ Surjective ⇑(f.submonoidMap M')",
"after_state": "case mk.intro.intro\nN : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M... |
theorem prod_eq_bot_iff {s : Submonoid M} {t : Submonoid N} : s.prod t = ⊥ ↔ s = ⊥ ∧ t = ⊥ := by
simp only [eq_bot_iff]
simp only [prod_le_iff]
simp only [(gc_map_comap _).le_iff_le]
simp only [comap_bot']
simp only [mker_inl]
simp only [mker_inr]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom",
"Submonoid",
"MonoidHom"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M N]",
"{ι : Type*} {f... | [
{
"line": "simp only [eq_bot_iff]",
"before_state": "N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M\ns : Submonoid M\nt : Submonoid N\n⊢ s.prod t = ⊥ ↔ s = ⊥ ∧ t = ⊥",
"after_state": "N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M\ns : Submonoid M\nt... |
theorem mrange_inl_sup_mrange_inr : mrange (inl M N) ⊔ mrange (inr M N) = ⊤ := by
simp only [mrange_inl]
simp only [mrange_inr]
simp only [prod_bot_sup_bot_prod]
simp only [top_prod_top]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom",
"Submonoid",
"MonoidHom"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M N]",
"{ι : Type*} {f... | [
{
"line": "simp only [mrange_inl]",
"before_state": "N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M\n⊢ mrange (inl M N) ⊔ mrange (inr M N) = ⊤",
"after_state": "N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_8\ninst✝ : MulOneClass M\n⊢ ⊤.prod ⊥ ⊔ mrange (inr M N) = ⊤"
},
... |
theorem eq_bot_iff_forall : S = ⊥ ↔ ∀ x ∈ S, x = (1 : M) :=
SetLike.ext_iff.trans <| by simp +contextual [iff_def, S.one_mem]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom",
"Submonoid",
"MonoidHom"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M N]",
"{ι : Type*} {f... | [
{
"line": "simp +contextual [iff_def, S.one_mem]",
"before_state": "M : Type u_8\ninst✝ : MulOneClass M\nS : Submonoid M\n⊢ (∀ (x : M), x ∈ S ↔ x ∈ ⊥) ↔ ∀ x ∈ S, x = 1",
"after_state": "No Goals!"
}
] |
theorem eq_bot_of_subsingleton [Subsingleton S] : S = ⊥ := by
rw [eq_bot_iff_forall]
intro y hy
simpa using congr_arg ((↑) : S → M) <| Subsingleton.elim (⟨y, hy⟩ : S) 1
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom",
"Submonoid",
"MonoidHom"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M N]",
"{ι : Type*} {f... | [
{
"line": "rw [eq_bot_iff_forall]",
"before_state": "M : Type u_8\ninst✝¹ : MulOneClass M\nS : Submonoid M\ninst✝ : Subsingleton ↥S\n⊢ S = ⊥",
"after_state": "M : Type u_8\ninst✝¹ : MulOneClass M\nS : Submonoid M\ninst✝ : Subsingleton ↥S\n⊢ ∀ x ∈ S, x = 1"
},
{
"line": "rewrite [eq_bot_iff_foral... |
theorem nontrivial_iff_exists_ne_one (S : Submonoid M) : Nontrivial S ↔ ∃ x ∈ S, x ≠ (1 : M) :=
calc
Nontrivial S ↔ ∃ x : S, x ≠ 1 := nontrivial_iff_exists_ne 1
_ ↔ ∃ (x : _) (hx : x ∈ S), (⟨x, hx⟩ : S) ≠ ⟨1, S.one_mem⟩ := Subtype.exists
_ ↔ ∃ x ∈ S, x ≠ (1 : M) := by simp [Ne]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom",
"Submonoid",
"MonoidHom"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M N]",
"{ι : Type*} {f... | [
{
"line": "simp [Ne]",
"before_state": "M : Type u_8\ninst✝ : MulOneClass M\nS : Submonoid M\n⊢ (∃ x, ∃ (hx : x ∈ S), ⟨x, hx⟩ ≠ ⟨1, ⋯⟩) ↔ ∃ x ∈ S, x ≠ 1",
"after_state": "No Goals!"
}
] |
theorem map_comap_eq_self {f : F} {S : Submonoid N} (h : S ≤ MonoidHom.mrange f) :
(S.comap f).map f = S := by
simpa only [inf_of_le_left h] using map_comap_eq f S
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Operations.lean | {
"open": [
"Function",
"Set",
"MonoidHom",
"Submonoid",
"MonoidHom",
"AddSubmonoid Set"
],
"variables": [
"{M N P : Type*} [MulOneClass M] [MulOneClass N] [MulOneClass P] (S : Submonoid M)",
"{A : Type*} [AddZeroClass A]",
"{F : Type*} [FunLike F M N] [mc : MonoidHomClass F M ... | [
{
"line": "simpa only [inf_of_le_left h] using map_comap_eq f S",
"before_state": "N : Type u_2\ninst✝² : MulOneClass N\nM : Type u_8\ninst✝¹ : MulOneClass M\nF : Type u_10\ninst✝ : FunLike F M N\nmc : MonoidHomClass F M N\nf : F\nS : Submonoid N\nh : S ≤ mrange f\n⊢ Submonoid.map f (Submonoid.comap f S) = ... |
theorem coe_mul_self_eq (s : Submonoid M) : (s : Set M) * s = s := by
ext x
refine ⟨?_, fun h => ⟨x, h, 1, s.one_mem, mul_one x⟩⟩
rintro ⟨a, ha, b, hb, rfl⟩
exact s.mul_mem ha hb
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Pointwise.lean | {
"open": [
"Set Pointwise"
],
"variables": [
"{α G M R A S : Type*}",
"[Monoid M] [AddMonoid A]",
"{s t u : Set M}"
]
} | [
{
"line": "ext x",
"before_state": "M : Type u_3\ninst✝ : Monoid M\ns : Submonoid M\n⊢ ↑s * ↑s = ↑s",
"after_state": "case h\nM : Type u_3\ninst✝ : Monoid M\ns : Submonoid M\nx : M\n⊢ x ∈ ↑s * ↑s ↔ x ∈ ↑s"
},
{
"line": "refine ⟨?_, fun h => ⟨x, h, 1, s.one_mem, mul_one x⟩⟩",
"before_state": ... |
theorem submonoid_closure (hpos : ∀ x : α, x ∈ s → 1 ≤ x) (h : s.IsPWO) :
IsPWO (Submonoid.closure s : Set α) := by
rw [Submonoid.closure_eq_image_prod]
refine (h.partiallyWellOrderedOn_sublistForall₂ (· ≤ ·)).image_of_monotone_on ?_
exact fun l1 _ l2 hl2 h12 => h12.prod_le_prod' fun x hx => hpos x <| hl2 x h... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Submonoid/Pointwise.lean | {
"open": [
"Set Pointwise"
],
"variables": [
"{α G M R A S : Type*}",
"[Monoid M] [AddMonoid A]",
"{s t u : Set M}",
"[Group G]",
"[Monoid α] [MulDistribMulAction α M]",
"[Group α] [MulDistribMulAction α M]",
"[CommMonoid α] [PartialOrder α] [IsOrderedCancelMonoid α] {s : Set α}"
... | [
{
"line": "rw [Submonoid.closure_eq_image_prod]",
"before_state": "α : Type u_1\ninst✝⁴ : Monoid α\ninst✝³ : Group α\ninst✝² : CommMonoid α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedCancelMonoid α\ns : Set α\nhpos : ∀ x ∈ s, 1 ≤ x\nh : s.IsPWO\n⊢ (↑(Submonoid.closure s)).IsPWO",
"after_state": "α : Typ... |
theorem iSup_eq_closure {ι : Sort*} (p : ι → Subsemigroup M) :
⨆ i, p i = Subsemigroup.closure (⋃ i, (p i : Set M)) := by
simp_rw [Subsemigroup.closure_iUnion, Subsemigroup.closure_eq]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subsemigroup/Basic.lean | {
"open": [
"Set"
],
"variables": [
"{M : Type*} {N : Type*}",
"[Mul M] {s : Set M}",
"(S : Subsemigroup M)",
"{S}",
"(S)",
"(M)",
"{M}"
]
} | [
{
"line": "simp_rw [Subsemigroup.closure_iUnion, Subsemigroup.closure_eq]",
"before_state": "M : Type u_1\ninst✝ : Mul M\nι : Sort u_3\np : ι → Subsemigroup M\n⊢ ⨆ i, p i = Subsemigroup.closure (⋃ i, ↑(p i))",
"after_state": "No Goals!"
},
{
"line": "simp (failIfUnchanged✝ := false✝) only",
... |
theorem subsingleton_of_subsingleton [Subsingleton (Subsemigroup M)] : Subsingleton M := by
constructor; intro x y
have : ∀ a : M, a ∈ (⊥ : Subsemigroup M) := by simp [Subsingleton.elim (⊥ : Subsemigroup M) ⊤]
exact absurd (this x) not_mem_bot
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subsemigroup/Defs.lean | {
"open": [],
"variables": [
"{M : Type*} {N : Type*}",
"[Mul M] {s : Set M}",
"{S : Subsemigroup M}",
"(S)"
]
} | [
{
"line": "constructor",
"before_state": "M : Type u_1\ninst✝¹ : Mul M\ninst✝ : Subsingleton (Subsemigroup M)\n⊢ Subsingleton M",
"after_state": "case allEq\nM : Type u_1\ninst✝¹ : Mul M\ninst✝ : Subsingleton (Subsemigroup M)\n⊢ ∀ (a b : M), a = b"
},
{
"line": "intro x y",
"before_state": "... |
theorem mem_sup_left {S T : Subsemigroup M} : ∀ {x : M}, x ∈ S → x ∈ S ⊔ T := by
have : S ≤ S ⊔ T := le_sup_left
tauto
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subsemigroup/Membership.lean | {
"open": [
"Set"
],
"variables": [
"{ι : Sort*} {M : Type*}",
"[Mul M]"
]
} | [
{
"line": "have : S ≤ S ⊔ T := le_sup_left",
"before_state": "M : Type u_2\ninst✝ : Mul M\nS T : Subsemigroup M\n⊢ ∀ {x : M}, x ∈ S → x ∈ S ⊔ T",
"after_state": "M : Type u_2\ninst✝ : Mul M\nS T : Subsemigroup M\nthis : S ≤ S ⊔ T\n⊢ ∀ {x : M}, x ∈ S → x ∈ S ⊔ T"
},
{
"line": "refine_lift\n have... |
theorem mem_sup_right {S T : Subsemigroup M} : ∀ {x : M}, x ∈ T → x ∈ S ⊔ T := by
have : T ≤ S ⊔ T := le_sup_right
tauto
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subsemigroup/Membership.lean | {
"open": [
"Set"
],
"variables": [
"{ι : Sort*} {M : Type*}",
"[Mul M]"
]
} | [
{
"line": "have : T ≤ S ⊔ T := le_sup_right",
"before_state": "M : Type u_2\ninst✝ : Mul M\nS T : Subsemigroup M\n⊢ ∀ {x : M}, x ∈ T → x ∈ S ⊔ T",
"after_state": "M : Type u_2\ninst✝ : Mul M\nS T : Subsemigroup M\nthis : T ≤ S ⊔ T\n⊢ ∀ {x : M}, x ∈ T → x ∈ S ⊔ T"
},
{
"line": "refine_lift\n hav... |
theorem mem_iSup_of_mem {S : ι → Subsemigroup M} (i : ι) : ∀ {x : M}, x ∈ S i → x ∈ iSup S := by
have : S i ≤ iSup S := le_iSup _ _
tauto
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subsemigroup/Membership.lean | {
"open": [
"Set"
],
"variables": [
"{ι : Sort*} {M : Type*}",
"[Mul M]"
]
} | [
{
"line": "have : S i ≤ iSup S := le_iSup _ _",
"before_state": "ι : Sort u_1\nM : Type u_2\ninst✝ : Mul M\nS : ι → Subsemigroup M\ni : ι\n⊢ ∀ {x : M}, x ∈ S i → x ∈ iSup S",
"after_state": "ι : Sort u_1\nM : Type u_2\ninst✝ : Mul M\nS : ι → Subsemigroup M\ni : ι\nthis : S i ≤ iSup S\n⊢ ∀ {x : M}, x ∈ S... |
theorem mem_sSup_of_mem {S : Set (Subsemigroup M)} {s : Subsemigroup M} (hs : s ∈ S) :
∀ {x : M}, x ∈ s → x ∈ sSup S := by
have : s ≤ sSup S := le_sSup hs
tauto
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subsemigroup/Membership.lean | {
"open": [
"Set"
],
"variables": [
"{ι : Sort*} {M : Type*}",
"[Mul M]"
]
} | [
{
"line": "have : s ≤ sSup S := le_sSup hs",
"before_state": "M : Type u_2\ninst✝ : Mul M\nS : Set (Subsemigroup M)\ns : Subsemigroup M\nhs : s ∈ S\n⊢ ∀ {x : M}, x ∈ s → x ∈ sSup S",
"after_state": "M : Type u_2\ninst✝ : Mul M\nS : Set (Subsemigroup M)\ns : Subsemigroup M\nhs : s ∈ S\nthis : s ≤ sSup S\... |
theorem iSup_induction' (S : ι → Subsemigroup M) {C : ∀ x, (x ∈ ⨆ i, S i) → Prop}
(mem : ∀ (i) (x) (hxS : x ∈ S i), C x (mem_iSup_of_mem i ‹_›))
(mul : ∀ x y hx hy, C x hx → C y hy → C (x * y) (mul_mem ‹_› ‹_›)) {x₁ : M}
(hx₁ : x₁ ∈ ⨆ i, S i) : C x₁ hx₁ := by
refine Exists.elim ?_ fun (hx₁' : x₁ ∈ ⨆ i, S ... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subsemigroup/Membership.lean | {
"open": [
"Set"
],
"variables": [
"{ι : Sort*} {M : Type*}",
"[Mul M]"
]
} | [
{
"line": "assumption",
"before_state": "ι : Sort u_1\nM : Type u_2\ninst✝ : Mul M\nmem_iSup_of_mem : ?m.376\nS : ι → Subsemigroup M\nC : (x : M) → x ∈ ⨆ i, S i → Prop\nmem : ∀ (i : ι), ∀ x ∈ S i, C x ⋯\nx y : M\nhx : x ∈ ⨆ i, S i\nhy : y ∈ ⨆ i, S i\n⊢ x ∈ ⨆ i, S i",
"after_state": "No Goals!"
},
{
... |
theorem le_prod_iff {s : Subsemigroup M} {t : Subsemigroup N} {u : Subsemigroup (M × N)} :
u ≤ s.prod t ↔ u.map (fst M N) ≤ s ∧ u.map (snd M N) ≤ t := by
constructor
· intro h
constructor
· rintro x ⟨⟨y1, y2⟩, ⟨hy1, rfl⟩⟩
exact (h hy1).1
· rintro x ⟨⟨y1, y2⟩, ⟨hy1, rfl⟩⟩
exact (h hy1).2
... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subsemigroup/Operations.lean | {
"open": [
"Set",
"MulHom"
],
"variables": [
"{M N P σ : Type*}",
"[Mul M]",
"{A : Type*} [Add A]",
"[Mul M] [Mul N] [Mul P] (S : Subsemigroup M)",
"{ι : Type*} {f : M →ₙ* N}",
"(hf : Function.Injective f)",
"{ι : Type*} {f : M →ₙ* N} (hf : Function.Surjective f)",
"[Mul M... | [
{
"line": "constructor",
"before_state": "M : Type u_1\nN : Type u_2\ninst✝⁴ inst✝³ : Mul M\ninst✝² : Mul N\ninst✝¹ : Mul M\ninst✝ : Mul N\ns : Subsemigroup M\nt : Subsemigroup N\nu : Subsemigroup (M × N)\n⊢ u ≤ s.prod t ↔ Subsemigroup.map (fst M N) u ≤ s ∧ Subsemigroup.map (snd M N) u ≤ t",
"after_stat... |
theorem map_srange (g : N →ₙ* P) (f : M →ₙ* N) : f.srange.map g = (g.comp f).srange := by
simpa only [srange_eq_map] using (⊤ : Subsemigroup M).map_map g f
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subsemigroup/Operations.lean | {
"open": [
"Set",
"MulHom",
"Subsemigroup"
],
"variables": [
"{M N P σ : Type*}",
"[Mul M]",
"{A : Type*} [Add A]",
"[Mul M] [Mul N] [Mul P] (S : Subsemigroup M)",
"{ι : Type*} {f : M →ₙ* N}",
"(hf : Function.Injective f)",
"{ι : Type*} {f : M →ₙ* N} (hf : Function.Surject... | [
{
"line": "simpa only [srange_eq_map] using (⊤ : Subsemigroup M).map_map g f",
"before_state": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝⁹ inst✝⁸ : Mul M\ninst✝⁷ : Mul N\ninst✝⁶ : Mul P\ninst✝⁵ : Mul M\ninst✝⁴ : Mul N\ninst✝³ : Mul P\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\ng : N →ₙ* P\nf : M →... |
theorem srange_eq_top_iff_surjective {N} [Mul N] {f : M →ₙ* N} :
f.srange = (⊤ : Subsemigroup N) ↔ Function.Surjective f :=
SetLike.ext'_iff.trans <| Iff.trans (by rw [coe_srange, coe_top]) Set.range_eq_univ
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subsemigroup/Operations.lean | {
"open": [
"Set",
"MulHom",
"Subsemigroup"
],
"variables": [
"{M N P σ : Type*}",
"[Mul M]",
"{A : Type*} [Add A]",
"[Mul M] [Mul N] [Mul P] (S : Subsemigroup M)",
"{ι : Type*} {f : M →ₙ* N}",
"(hf : Function.Injective f)",
"{ι : Type*} {f : M →ₙ* N} (hf : Function.Surject... | [
{
"line": "rw [coe_srange, coe_top]",
"before_state": "M : Type u_1\ninst✝⁴ inst✝³ inst✝² inst✝¹ : Mul M\nN : Type u_8\ninst✝ : Mul N\nf : M →ₙ* N\n⊢ ↑f.srange = ↑⊤ ↔ range ⇑f = univ",
"after_state": "No Goals!"
},
{
"line": "rewrite [coe_srange, coe_top]",
"before_state": "M : Type u_1\nins... |
theorem subsemigroupMap_surjective (f : M →ₙ* N) (M' : Subsemigroup M) :
Function.Surjective (f.subsemigroupMap M') := by
rintro ⟨_, x, hx, rfl⟩
exact ⟨⟨x, hx⟩, rfl⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subsemigroup/Operations.lean | {
"open": [
"Set",
"MulHom",
"Subsemigroup"
],
"variables": [
"{M N P σ : Type*}",
"[Mul M]",
"{A : Type*} [Add A]",
"[Mul M] [Mul N] [Mul P] (S : Subsemigroup M)",
"{ι : Type*} {f : M →ₙ* N}",
"(hf : Function.Injective f)",
"{ι : Type*} {f : M →ₙ* N} (hf : Function.Surject... | [
{
"line": "rintro ⟨_, x, hx, rfl⟩",
"before_state": "M : Type u_1\nN : Type u_2\ninst✝⁶ inst✝⁵ : Mul M\ninst✝⁴ : Mul N\ninst✝³ : Mul M\ninst✝² : Mul N\ninst✝¹ : Mul M\ninst✝ : Mul N\nf : M →ₙ* N\nM' : Subsemigroup M\n⊢ Function.Surjective ⇑(f.subsemigroupMap M')",
"after_state": "case mk.intro.intro\nM ... |
theorem map_comap_eq_self {f : M →ₙ* N} {S : Subsemigroup N} (h : S ≤ f.srange) :
(S.comap f).map f = S := by
simpa only [inf_of_le_left h] using map_comap_eq f S
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/Subsemigroup/Operations.lean | {
"open": [
"Set",
"MulHom",
"Subsemigroup",
"MulHom"
],
"variables": [
"{M N P σ : Type*}",
"[Mul M]",
"{A : Type*} [Add A]",
"[Mul M] [Mul N] [Mul P] (S : Subsemigroup M)",
"{ι : Type*} {f : M →ₙ* N}",
"(hf : Function.Injective f)",
"{ι : Type*} {f : M →ₙ* N} (hf : Fu... | [
{
"line": "simpa only [inf_of_le_left h] using map_comap_eq f S",
"before_state": "M : Type u_1\nN : Type u_2\ninst✝¹² inst✝¹¹ : Mul M\ninst✝¹⁰ : Mul N\ninst✝⁹ : Mul M\ninst✝⁸ : Mul N\ninst✝⁷ : Mul M\ninst✝⁶ : Mul N\ninst✝⁵ : Mul M\ninst✝⁴ : Mul N\ninst✝³ : Mul M\ninst✝² : Mul N\ninst✝¹ : Mul M\ninst✝ : Mul... |
theorem of_subsingleton [Subsingleton G] : UniqueMul A B a0 b0 := by
simp [UniqueMul, eq_iff_true_of_subsingleton]
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/UniqueProds/Basic.lean | {
"open": [
"Finset"
],
"variables": [
"{G H : Type*} [Mul G] [Mul H] {A B : Finset G} {a0 b0 : G}"
]
} | [
{
"line": "simp [UniqueMul, eq_iff_true_of_subsingleton]",
"before_state": "G : Type u_1\ninst✝¹ : Mul G\nA B : Finset G\na0 b0 : G\ninst✝ : Subsingleton G\n⊢ UniqueMul A B a0 b0",
"after_state": "No Goals!"
}
] |
theorem of_card_le_one (hA : A.Nonempty) (hB : B.Nonempty) (hA1 : #A ≤ 1) (hB1 : #B ≤ 1) :
∃ a ∈ A, ∃ b ∈ B, UniqueMul A B a b := by
rw [Finset.card_le_one_iff] at hA1 hB1
obtain ⟨a, ha⟩ := hA; obtain ⟨b, hb⟩ := hB
exact ⟨a, ha, b, hb, fun _ _ ha' hb' _ ↦ ⟨hA1 ha' ha, hB1 hb' hb⟩⟩
| /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/UniqueProds/Basic.lean | {
"open": [
"Finset"
],
"variables": [
"{G H : Type*} [Mul G] [Mul H] {A B : Finset G} {a0 b0 : G}"
]
} | [
{
"line": "rw [Finset.card_le_one_iff] at hA1 hB1",
"before_state": "G : Type u_1\ninst✝ : Mul G\nA B : Finset G\nhA : A.Nonempty\nhB : B.Nonempty\nhA1 : #A ≤ 1\nhB1 : #B ≤ 1\n⊢ ∃ a ∈ A, ∃ b ∈ B, UniqueMul A B a b",
"after_state": "G : Type u_1\ninst✝ : Mul G\nA B : Finset G\nhA : A.Nonempty\nhB : B.Non... |
theorem set_subsingleton (h : UniqueMul A B a0 b0) :
Set.Subsingleton { ab : G × G | ab.1 ∈ A ∧ ab.2 ∈ B ∧ ab.1 * ab.2 = a0 * b0 } := by
rintro ⟨x1, y1⟩ (hx : x1 ∈ A ∧ y1 ∈ B ∧ x1 * y1 = a0 * b0) ⟨x2, y2⟩
(hy : x2 ∈ A ∧ y2 ∈ B ∧ x2 * y2 = a0 * b0)
rcases h hx.1 hx.2.1 hx.2.2 with ⟨rfl, rfl⟩
rcases h hy.1 ... | /root/DuelModelResearch/mathlib4/Mathlib/Algebra/Group/UniqueProds/Basic.lean | {
"open": [
"Finset"
],
"variables": [
"{G H : Type*} [Mul G] [Mul H] {A B : Finset G} {a0 b0 : G}"
]
} | [
{
"line": "rintro ⟨x1, y1⟩ (hx : x1 ∈ A ∧ y1 ∈ B ∧ x1 * y1 = a0 * b0) ⟨x2, y2⟩ (hy : x2 ∈ A ∧ y2 ∈ B ∧ x2 * y2 = a0 * b0)",
"before_state": "G : Type u_1\ninst✝ : Mul G\nA B : Finset G\na0 b0 : G\nh : UniqueMul A B a0 b0\n⊢ {ab | ab.1 ∈ A ∧ ab.2 ∈ B ∧ ab.1 * ab.2 = a0 * b0}.Subsingleton",
"after_state":... |
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