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Theorem eqgen 19212
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 2735 . 2 (𝑋 / ) = (𝑋 / )
2 breq2 5152 . 2 ([𝑥] = 𝐴 → (𝑌 ≈ [𝑥] 𝑌𝐴))
3 simpl 482 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝑌 ∈ (SubGrp‘𝐺))
4 subgrcl 19162 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5 eqger.x . . . . . . . 8 𝑋 = (Base‘𝐺)
65subgss 19158 . . . . . . 7 (𝑌 ∈ (SubGrp‘𝐺) → 𝑌𝑋)
74, 6jca 511 . . . . . 6 (𝑌 ∈ (SubGrp‘𝐺) → (𝐺 ∈ Grp ∧ 𝑌𝑋))
8 eqger.r . . . . . . . 8 = (𝐺 ~QG 𝑌)
9 eqid 2735 . . . . . . . 8 (+g𝐺) = (+g𝐺)
105, 8, 9eqglact 19210 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝑥𝑋) → [𝑥] = ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
11103expa 1117 . . . . . 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 8748 . . . . . 6 ( ∈ V → [𝑥] ∈ V)
1513, 14ax-mp 5 . . . . 5 [𝑥] ∈ V
1612, 15eqeltrrdi 2848 . . . 4 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌) ∈ V)
17 eqid 2735 . . . . . . . . 9 (𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧))) = (𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))
1817, 5, 9grplactf1o 19075 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → ((𝑦𝑋 ↦ (𝑧𝑋 ↦ (𝑦(+g𝐺)𝑧)))‘𝑥):𝑋1-1-onto𝑋)
1917, 5grplactfval 19072 . . . . . . . . . 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 6844 . . . . . . . 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 6848 . . . . . 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 6863 . . . . 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 9008 . . . 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 1370 . . 3 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝑌 ≈ ((𝑧𝑋 ↦ (𝑥(+g𝐺)𝑧)) “ 𝑌))
3130, 12breqtrrd 5176 . 2 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑋) → 𝑌 ≈ [𝑥] )
321, 2, 31ectocld 8823 1 ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (𝑋 / )) → 𝑌𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963   class class class wbr 5148  cmpt 5231  cres 5691  cima 5692  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  [cec 8742   / cqs 8743  cen 8981  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964  SubGrpcsubg 19151   ~QG cqg 19153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-ec 8746  df-qs 8750  df-en 8985  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-subg 19154  df-eqg 19156
This theorem is referenced by:  lagsubg2  19225  sylow2blem1  19653
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