Theorem eqgen 19061

Description: Each coset is equipotent to the subgroup itself (which is also the coset containing the identity). (Contributed by Mario Carneiro, 20-Sep-2015.)

Hypotheses

Ref	Expression
eqger.x	⊢ 𝑋 = (Base‘𝐺)
eqger.r	⊢ ∼ = (𝐺 ~QG 𝑌)

Assertion

Ref	Expression
eqgen	⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (𝑋 / ∼ )) → 𝑌 ≈ 𝐴)

Proof of Theorem eqgen

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2733	. 2 ⊢ (𝑋 / ∼ ) = (𝑋 / ∼ )
2		breq2 5153	. 2 ⊢ ([𝑥] ∼ = 𝐴 → (𝑌 ≈ [𝑥] ∼ ↔ 𝑌 ≈ 𝐴))
3		simpl 484	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ (SubGrp‘𝐺))
4		subgrcl 19011	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5		eqger.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
6	5	subgss 19007	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
7	4, 6	jca 513	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋))
8		eqger.r	. . . . . . . 8 ⊢ ∼ = (𝐺 ~QG 𝑌)
9		eqid 2733	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
10	5, 8, 9	eqglact 19059	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑋) → [𝑥] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
11	10	3expa 1119	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → [𝑥] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
12	7, 11	sylan 581	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → [𝑥] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
13	8	ovexi 7443	. . . . . 6 ⊢ ∼ ∈ V
14		ecexg 8707	. . . . . 6 ⊢ ( ∼ ∈ V → [𝑥] ∼ ∈ V)
15	13, 14	ax-mp 5	. . . . 5 ⊢ [𝑥] ∼ ∈ V
16	12, 15	eqeltrrdi 2843	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌) ∈ V)
17		eqid 2733	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧))) = (𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))
18	17, 5, 9	grplactf1o 18927	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))‘𝑥):𝑋–1-1-onto→𝑋)
19	17, 5	grplactfval 18924	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 → ((𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))‘𝑥) = (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)))
20	19	adantl 483	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))‘𝑥) = (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)))
21	20	f1oeq1d 6829	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (((𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))‘𝑥):𝑋–1-1-onto→𝑋 ↔ (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1-onto→𝑋))
22	18, 21	mpbid 231	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1-onto→𝑋)
23	4, 22	sylan 581	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1-onto→𝑋)
24		f1of1 6833	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1-onto→𝑋 → (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1→𝑋)
25	23, 24	syl 17	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1→𝑋)
26	6	adantr 482	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑌 ⊆ 𝑋)
27		f1ores 6848	. . . . 5 ⊢ (((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1→𝑋 ∧ 𝑌 ⊆ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) ↾ 𝑌):𝑌–1-1-onto→((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
28	25, 26, 27	syl2anc 585	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) ↾ 𝑌):𝑌–1-1-onto→((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
29		f1oen2g 8964	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌) ∈ V ∧ ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) ↾ 𝑌):𝑌–1-1-onto→((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌)) → 𝑌 ≈ ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
30	3, 16, 28, 29	syl3anc 1372	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑌 ≈ ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
31	30, 12	breqtrrd 5177	. 2 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑌 ≈ [𝑥] ∼ )
32	1, 2, 31	ectocld 8778	1 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (𝑋 / ∼ )) → 𝑌 ≈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ⊆ wss 3949   class class class wbr 5149   ↦ cmpt 5232   ↾ cres 5679   “ cima 5680  –1-1→wf1 6541  –1-1-onto→wf1o 6543  ‘cfv 6544  (class class class)co 7409  [cec 8701   / cqs 8702   ≈ cen 8936  Basecbs 17144  +_gcplusg 17197  Grpcgrp 18819  SubGrpcsubg 19000   ~_QG cqg 19002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-ec 8705  df-qs 8709  df-en 8940  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-subg 19003  df-eqg 19005

This theorem is referenced by:  lagsubg2  19071  sylow2blem1  19488