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Theorem eqgen 17960
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 / )) → 𝑌𝐴)

Proof of Theorem eqgen
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2799 . 2 (𝑋 / ) = (𝑋 / )
2 breq2 4847 . 2 ([𝑥] = 𝐴 → (𝑌 ≈ [𝑥] 𝑌𝐴))
3 simpl 475 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝑌 ∈ (SubGrp‘𝐺))
4 subgrcl 17912 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5 eqger.x . . . . . . . 8 𝑋 = (Base‘𝐺)
65subgss 17908 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
74, 6jca 508 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → (𝐺 ∈ Grp ∧ 𝑌𝑋))
8 eqger.r . . . . . . . 8 = (𝐺 ~QG 𝑌)
9 eqid 2799 . . . . . . . 8 (+g𝐺) = (+g𝐺)
105, 8, 9eqglact 17958 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝑥𝑋) → [𝑥] = ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
11103expa 1148 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑌𝑋) ∧ 𝑥𝑋) → [𝑥] = ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
127, 11sylan 576 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → [𝑥] = ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
13 ovex 6910 . . . . . . 7 (𝐺 ~QG 𝑌) ∈ V
148, 13eqeltri 2874 . . . . . 6 ∈ V
15 ecexg 7986 . . . . . 6 ( ∈ V → [𝑥] ∈ V)
1614, 15ax-mp 5 . . . . 5 [𝑥] ∈ V
1712, 16syl6eqelr 2887 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌) ∈ V)
18 eqid 2799 . . . . . . . . 9 (𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧))) = (𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))
1918, 5, 9grplactf1o 17835 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ((𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))‘𝑥):𝑋1-1-onto𝑋)
2018, 5grplactfval 17832 . . . . . . . . . 10 (𝑥𝑋 → ((𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))‘𝑥) = (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)))
2120adantl 474 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ((𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))‘𝑥) = (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)))
22 f1oeq1 6345 . . . . . . . . 9 (((𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))‘𝑥) = (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) → (((𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))‘𝑥):𝑋1-1-onto𝑋 ↔ (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1-onto𝑋))
2321, 22syl 17 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → (((𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))‘𝑥):𝑋1-1-onto𝑋 ↔ (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1-onto𝑋))
2419, 23mpbid 224 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1-onto𝑋)
254, 24sylan 576 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1-onto𝑋)
26 f1of1 6355 . . . . . 6 ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1-onto𝑋 → (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1𝑋)
2725, 26syl 17 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1𝑋)
286adantr 473 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝑌𝑋)
29 f1ores 6370 . . . . 5 (((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1𝑋𝑌𝑋) → ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) ↾ 𝑌):𝑌1-1-onto→((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
3027, 28, 29syl2anc 580 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) ↾ 𝑌):𝑌1-1-onto→((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
31 f1oen2g 8212 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌) ∈ V ∧ ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) ↾ 𝑌):𝑌1-1-onto→((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌)) → 𝑌 ≈ ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
323, 17, 30, 31syl3anc 1491 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝑌 ≈ ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
3332, 12breqtrrd 4871 . 2 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝑌 ≈ [𝑥] )
341, 2, 33ectocld 8052 1 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (𝑋 / )) → 𝑌𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  wss 3769   class class class wbr 4843  cmpt 4922  cres 5314  cima 5315  1-1wf1 6098  1-1-ontowf1o 6100  cfv 6101  (class class class)co 6878  [cec 7980   / cqs 7981  cen 8192  Basecbs 16184  +gcplusg 16267  Grpcgrp 17738  SubGrpcsubg 17901   ~QG cqg 17903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-ec 7984  df-qs 7988  df-en 8196  df-0g 16417  df-mgm 17557  df-sgrp 17599  df-mnd 17610  df-grp 17741  df-minusg 17742  df-subg 17904  df-eqg 17906
This theorem is referenced by:  lagsubg2  17968  sylow2blem1  18348
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