Theorem eqgen 19147

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 2739	. 2 ⊢ (𝑋 / ∼ ) = (𝑋 / ∼ )
2		breq2 5076	. 2 ⊢ ([𝑥] ∼ = 𝐴 → (𝑌 ≈ [𝑥] ∼ ↔ 𝑌 ≈ 𝐴))
3		simpl 483	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ (SubGrp‘𝐺))
4		subgrcl 19098	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5		eqger.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
6	5	subgss 19094	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
7	4, 6	jca 516	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋))
8		eqger.r	. . . . . . . 8 ⊢ ∼ = (𝐺 ~QG 𝑌)
9		eqid 2739	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
10	5, 8, 9	eqglact 19145	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑋) → [𝑥] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
11	10	3expa 1124	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → [𝑥] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
12	7, 11	sylan 586	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → [𝑥] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
13	8	ovexi 7390	. . . . . 6 ⊢ ∼ ∈ V
14		ecexg 8637	. . . . . 6 ⊢ ( ∼ ∈ V → [𝑥] ∼ ∈ V)
15	13, 14	ax-mp 5	. . . . 5 ⊢ [𝑥] ∼ ∈ V
16	12, 15	eqeltrrdi 2848	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌) ∈ V)
17		eqid 2739	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧))) = (𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))
18	17, 5, 9	grplactf1o 19011	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))‘𝑥):𝑋–1-1-onto→𝑋)
19	17, 5	grplactfval 19008	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 → ((𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))‘𝑥) = (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)))
20	19	adantl 482	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))‘𝑥) = (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)))
21	20	f1oeq1d 6762	. . . . . . . 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 233	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1-onto→𝑋)
23	4, 22	sylan 586	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1-onto→𝑋)
24		f1of1 6766	. . . . . 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 481	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑌 ⊆ 𝑋)
27		f1ores 6781	. . . . 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 590	. . . 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 8905	. . . 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 1379	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑌 ≈ ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
31	30, 12	breqtrrd 5100	. 2 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑌 ≈ [𝑥] ∼ )
32	1, 2, 31	ectocld 8719	1 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (𝑋 / ∼ )) → 𝑌 ≈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ⊆ wss 3883   class class class wbr 5072   ↦ cmpt 5153   ↾ cres 5620   “ cima 5621  –1-1→wf1 6482  –1-1-onto→wf1o 6484  ‘cfv 6485  (class class class)co 7356  [cec 8631   / cqs 8632   ≈ cen 8880  Basecbs 17170  +_gcplusg 17211  Grpcgrp 18900  SubGrpcsubg 19087   ~_QG cqg 19089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-ec 8635  df-qs 8639  df-en 8884  df-0g 17395  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18903  df-minusg 18904  df-subg 19090  df-eqg 19092

This theorem is referenced by:  lagsubg2  19160  sylow2blem1  19586