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Theorem eqgen 19055
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 2733 . 2 (𝑋 / ) = (𝑋 / )
2 breq2 5151 . 2 ([𝑥] = 𝐴 → (𝑌 ≈ [𝑥] 𝑌𝐴))
3 simpl 484 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝑌 ∈ (SubGrp‘𝐺))
4 subgrcl 19005 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5 eqger.x . . . . . . . 8 𝑋 = (Base‘𝐺)
65subgss 19001 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
74, 6jca 513 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → (𝐺 ∈ Grp ∧ 𝑌𝑋))
8 eqger.r . . . . . . . 8 = (𝐺 ~QG 𝑌)
9 eqid 2733 . . . . . . . 8 (+g𝐺) = (+g𝐺)
105, 8, 9eqglact 19053 . . . . . . 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 581 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → [𝑥] = ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
138ovexi 7438 . . . . . 6 ∈ V
14 ecexg 8703 . . . . . 6 ( ∈ V → [𝑥] ∈ V)
1513, 14ax-mp 5 . . . . 5 [𝑥] ∈ V
1612, 15eqeltrrdi 2843 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌) ∈ V)
17 eqid 2733 . . . . . . . . 9 (𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧))) = (𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))
1817, 5, 9grplactf1o 18923 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ((𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))‘𝑥):𝑋1-1-onto𝑋)
1917, 5grplactfval 18920 . . . . . . . . . 10 (𝑥𝑋 → ((𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))‘𝑥) = (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)))
2019adantl 483 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ((𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))‘𝑥) = (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)))
2120f1oeq1d 6825 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → (((𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))‘𝑥):𝑋1-1-onto𝑋 ↔ (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1-onto𝑋))
2218, 21mpbid 231 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1-onto𝑋)
234, 22sylan 581 . . . . . 6 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → (𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1-onto𝑋)
24 f1of1 6829 . . . . . 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 482 . . . . 5 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝑌𝑋)
27 f1ores 6844 . . . . 5 (((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)):𝑋1-1𝑋𝑌𝑋) → ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) ↾ 𝑌):𝑌1-1-onto→((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
2825, 26, 27syl2anc 585 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) ↾ 𝑌):𝑌1-1-onto→((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
29 f1oen2g 8960 . . . 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 1372 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝑌 ≈ ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
3130, 12breqtrrd 5175 . 2 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝑌 ≈ [𝑥] )
321, 2, 31ectocld 8774 1 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (𝑋 / )) → 𝑌𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  wss 3947   class class class wbr 5147  cmpt 5230  cres 5677  cima 5678  1-1wf1 6537  1-1-ontowf1o 6539  cfv 6540  (class class class)co 7404  [cec 8697   / cqs 8698  cen 8932  Basecbs 17140  +gcplusg 17193  Grpcgrp 18815  SubGrpcsubg 18994   ~QG cqg 18996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-ec 8701  df-qs 8705  df-en 8936  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-subg 18997  df-eqg 18999
This theorem is referenced by:  lagsubg2  19065  sylow2blem1  19481
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