Theorem eqgen 17647

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 2622	. 2 ⊢ (𝑋 / ∼ ) = (𝑋 / ∼ )
2		breq2 4657	. 2 ⊢ ([𝑥] ∼ = 𝐴 → (𝑌 ≈ [𝑥] ∼ ↔ 𝑌 ≈ 𝐴))
3		simpl 473	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ (SubGrp‘𝐺))
4		subgrcl 17599	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5		eqger.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
6	5	subgss 17595	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
7	4, 6	jca 554	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋))
8		eqger.r	. . . . . . . 8 ⊢ ∼ = (𝐺 ~QG 𝑌)
9		eqid 2622	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
10	5, 8, 9	eqglact 17645	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑋) → [𝑥] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
11	10	3expa 1265	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → [𝑥] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
12	7, 11	sylan 488	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → [𝑥] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
13		ovex 6678	. . . . . . 7 ⊢ (𝐺 ~QG 𝑌) ∈ V
14	8, 13	eqeltri 2697	. . . . . 6 ⊢ ∼ ∈ V
15		ecexg 7746	. . . . . 6 ⊢ ( ∼ ∈ V → [𝑥] ∼ ∈ V)
16	14, 15	ax-mp 5	. . . . 5 ⊢ [𝑥] ∼ ∈ V
17	12, 16	syl6eqelr 2710	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌) ∈ V)
18		eqid 2622	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧))) = (𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))
19	18, 5, 9	grplactf1o 17519	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))‘𝑥):𝑋–1-1-onto→𝑋)
20	18, 5	grplactfval 17516	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 → ((𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))‘𝑥) = (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)))
21	20	adantl 482	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))‘𝑥) = (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)))
22		f1oeq1 6127	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))‘𝑥) = (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) → (((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))
23	21, 22	syl 17	. . . . . . . 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))
24	19, 23	mpbid 222	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1-onto→𝑋)
25	4, 24	sylan 488	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1-onto→𝑋)
26		f1of1 6136	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1-onto→𝑋 → (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1→𝑋)
27	25, 26	syl 17	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1→𝑋)
28	6	adantr 481	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑌 ⊆ 𝑋)
29		f1ores 6151	. . . . 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))
30	27, 28, 29	syl2anc 693	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) ↾ 𝑌):𝑌–1-1-onto→((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
31		f1oen2g 7972	. . . 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))
32	3, 17, 30, 31	syl3anc 1326	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑌 ≈ ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
33	32, 12	breqtrrd 4681	. 2 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑌 ≈ [𝑥] ∼ )
34	1, 2, 33	ectocld 7814	1 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (𝑋 / ∼ )) → 𝑌 ≈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574   class class class wbr 4653   ↦ cmpt 4729   ↾ cres 5116   “ cima 5117  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  [cec 7740   / cqs 7741   ≈ cen 7952  Basecbs 15857  +_gcplusg 15941  Grpcgrp 17422  SubGrpcsubg 17588   ~_QG cqg 17590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-ec 7744  df-qs 7748  df-en 7956  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-eqg 17593

This theorem is referenced by:  lagsubg2  17655  sylow2blem1  18035