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Theorem eqgen 19199
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 2737 . 2 (𝑋 / ) = (𝑋 / )
2 breq2 5147 . 2 ([𝑥] = 𝐴 → (𝑌 ≈ [𝑥] 𝑌𝐴))
3 simpl 482 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝑌 ∈ (SubGrp‘𝐺))
4 subgrcl 19149 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5 eqger.x . . . . . . . 8 𝑋 = (Base‘𝐺)
65subgss 19145 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
74, 6jca 511 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → (𝐺 ∈ Grp ∧ 𝑌𝑋))
8 eqger.r . . . . . . . 8 = (𝐺 ~QG 𝑌)
9 eqid 2737 . . . . . . . 8 (+g𝐺) = (+g𝐺)
105, 8, 9eqglact 19197 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝑥𝑋) → [𝑥] = ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
11103expa 1119 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑌𝑋) ∧ 𝑥𝑋) → [𝑥] = ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
127, 11sylan 580 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → [𝑥] = ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
138ovexi 7465 . . . . . 6 ∈ V
14 ecexg 8749 . . . . . 6 ( ∈ V → [𝑥] ∈ V)
1513, 14ax-mp 5 . . . . 5 [𝑥] ∈ V
1612, 15eqeltrrdi 2850 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌) ∈ V)
17 eqid 2737 . . . . . . . . 9 (𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧))) = (𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))
1817, 5, 9grplactf1o 19062 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ((𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))‘𝑥):𝑋1-1-onto𝑋)
1917, 5grplactfval 19059 . . . . . . . . . 10 (𝑥𝑋 → ((𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))‘𝑥) = (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)))
2019adantl 481 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ((𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))‘𝑥) = (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)))
2120f1oeq1d 6843 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → (((𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))‘𝑥):𝑋1-1-onto𝑋 ↔ (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1-onto𝑋))
2218, 21mpbid 232 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1-onto𝑋)
234, 22sylan 580 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1-onto𝑋)
24 f1of1 6847 . . . . . 6 ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1-onto𝑋 → (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1𝑋)
2523, 24syl 17 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1𝑋)
266adantr 480 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝑌𝑋)
27 f1ores 6862 . . . . 5 (((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1𝑋𝑌𝑋) → ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) ↾ 𝑌):𝑌1-1-onto→((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
2825, 26, 27syl2anc 584 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) ↾ 𝑌):𝑌1-1-onto→((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
29 f1oen2g 9009 . . . 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))
303, 16, 28, 29syl3anc 1373 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝑌 ≈ ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
3130, 12breqtrrd 5171 . 2 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝑌 ≈ [𝑥] )
321, 2, 31ectocld 8824 1 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (𝑋 / )) → 𝑌𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951   class class class wbr 5143  cmpt 5225  cres 5687  cima 5688  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  [cec 8743   / cqs 8744  cen 8982  Basecbs 17247  +gcplusg 17297  Grpcgrp 18951  SubGrpcsubg 19138   ~QG cqg 19140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-ec 8747  df-qs 8751  df-en 8986  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-subg 19141  df-eqg 19143
This theorem is referenced by:  lagsubg2  19212  sylow2blem1  19638
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