Theorem eqgen 14032

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 2238	. 2 ⊢ (𝑋 / ∼ ) = (𝑋 / ∼ )
2		breq2 4134	. 2 ⊢ ([𝑥] ∼ = 𝐴 → (𝑌 ≈ [𝑥] ∼ ↔ 𝑌 ≈ 𝐴))
3		simpl 109	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑌 ∈ (SubGrp‘𝐺))
4		subgrcl 13984	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5		eqger.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
6	5	subgss 13979	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
7	4, 6	jca 306	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋))
8		eqger.r	. . . . . . . 8 ⊢ ∼ = (𝐺 ~QG 𝑌)
9		eqid 2238	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
10	5, 8, 9	eqglact 14030	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑋) → [𝑥] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
11	10	3expa 1234	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → [𝑥] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
12	7, 11	sylan 283	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → [𝑥] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
13	5, 8	eqger 14029	. . . . . . . 8 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)
14		basfn 13413	. . . . . . . . . 10 ⊢ Base Fn V
15	4	elexd 2835	. . . . . . . . . 10 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ V)
16		funfvex 5712	. . . . . . . . . . 11 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
17	16	funfni 5483	. . . . . . . . . 10 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
18	14, 15, 17	sylancr 418	. . . . . . . . 9 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (Base‘𝐺) ∈ V)
19	5, 18	eqeltrid 2325	. . . . . . . 8 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑋 ∈ V)
20		erex 6831	. . . . . . . 8 ⊢ ( ∼ Er 𝑋 → (𝑋 ∈ V → ∼ ∈ V))
21	13, 19, 20	sylc 62	. . . . . . 7 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ∼ ∈ V)
22		ecexg 6811	. . . . . . 7 ⊢ ( ∼ ∈ V → [𝑥] ∼ ∈ V)
23	21, 22	syl 14	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → [𝑥] ∼ ∈ V)
24	23	adantr 276	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → [𝑥] ∼ ∈ V)
25	12, 24	eqeltrrd 2316	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌) ∈ V)
26		eqid 2238	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧))) = (𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))
27	26, 5, 9	grplactf1o 13910	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))‘𝑥):𝑋–1-1-onto→𝑋)
28	26, 5	grplactfval 13908	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 → ((𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))‘𝑥) = (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)))
29	28	adantl 277	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑦(+g‘𝐺)𝑧)))‘𝑥) = (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)))
30	29	f1oeq1d 5634	. . . . . . . 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))
31	27, 30	mpbid 147	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1-onto→𝑋)
32	4, 31	sylan 283	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1-onto→𝑋)
33		f1of1 5638	. . . . . 6 ⊢ ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1-onto→𝑋 → (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1→𝑋)
34	32, 33	syl 14	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → (𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)):𝑋–1-1→𝑋)
35	6	adantr 276	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑌 ⊆ 𝑋)
36		f1ores 5654	. . . . 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))
37	34, 35, 36	syl2anc 415	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) ↾ 𝑌):𝑌–1-1-onto→((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
38		f1oen2g 7041	. . . 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))
39	3, 25, 37, 38	syl3anc 1278	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑌 ≈ ((𝑧 ∈ 𝑋 ↦ (𝑥(+g‘𝐺)𝑧)) “ 𝑌))
40	39, 12	breqtrrd 4158	. 2 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑌 ≈ [𝑥] ∼ )
41	1, 2, 40	ectocld 6875	1 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (𝑋 / ∼ )) → 𝑌 ≈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220   class class class wbr 4130   ↦ cmpt 4192   ↾ cres 4776   “ cima 4777   Fn wfn 5372  –1-1→wf1 5374  –1-1-onto→wf1o 5376  ‘cfv 5377  (class class class)co 6085   Er wer 6804  [cec 6805   / cqs 6806   ≈ cen 7020  Basecbs 13354  +_gcplusg 13433  Grpcgrp 13807  SubGrpcsubg 13972   ~_QG cqg 13974

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-er 6807  df-ec 6809  df-qs 6813  df-en 7023  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9306  df-2 9364  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-iress 13362  df-plusg 13446  df-0g 13614  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-grp 13810  df-minusg 13811  df-subg 13975  df-eqg 13977

This theorem is used by: (None)