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Theorem grpoidinv2 28302
Description: A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpoidval.1 𝑋 = ran 𝐺
grpoidval.2 𝑈 = (GId‘𝐺)
Assertion
Ref Expression
grpoidinv2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐺   𝑦,𝑈   𝑦,𝑋

Proof of Theorem grpoidinv2
Dummy variables 𝑥 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpoidval.1 . . . . . . 7 𝑋 = ran 𝐺
2 grpoidval.2 . . . . . . 7 𝑈 = (GId‘𝐺)
31, 2grpoidval 28300 . . . . . 6 (𝐺 ∈ GrpOp → 𝑈 = (𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥))
41grpoideu 28296 . . . . . . 7 (𝐺 ∈ GrpOp → ∃!𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
5 riotacl2 7113 . . . . . . 7 (∃!𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥 → (𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥) ∈ {𝑢𝑋 ∣ ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥})
64, 5syl 17 . . . . . 6 (𝐺 ∈ GrpOp → (𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥) ∈ {𝑢𝑋 ∣ ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥})
73, 6eqeltrd 2893 . . . . 5 (𝐺 ∈ GrpOp → 𝑈 ∈ {𝑢𝑋 ∣ ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥})
8 simpll 766 . . . . . . . . . . 11 ((((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → (𝑢𝐺𝑥) = 𝑥)
98ralimi 3131 . . . . . . . . . 10 (∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
109rgenw 3121 . . . . . . . . 9 𝑢𝑋 (∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
1110a1i 11 . . . . . . . 8 (𝐺 ∈ GrpOp → ∀𝑢𝑋 (∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥))
121grpoidinv 28295 . . . . . . . 8 (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
1311, 12, 43jca 1125 . . . . . . 7 (𝐺 ∈ GrpOp → (∀𝑢𝑋 (∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥) ∧ ∃𝑢𝑋𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) ∧ ∃!𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥))
14 reupick2 4244 . . . . . . 7 (((∀𝑢𝑋 (∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥) ∧ ∃𝑢𝑋𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) ∧ ∃!𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥) ∧ 𝑢𝑋) → (∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥 ↔ ∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))))
1513, 14sylan 583 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑢𝑋) → (∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥 ↔ ∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))))
1615rabbidva 3428 . . . . 5 (𝐺 ∈ GrpOp → {𝑢𝑋 ∣ ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥} = {𝑢𝑋 ∣ ∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))})
177, 16eleqtrd 2895 . . . 4 (𝐺 ∈ GrpOp → 𝑈 ∈ {𝑢𝑋 ∣ ∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))})
18 oveq1 7146 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
1918eqeq1d 2803 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
20 oveq2 7147 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑥𝐺𝑢) = (𝑥𝐺𝑈))
2120eqeq1d 2803 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑥𝐺𝑢) = 𝑥 ↔ (𝑥𝐺𝑈) = 𝑥))
2219, 21anbi12d 633 . . . . . . 7 (𝑢 = 𝑈 → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)))
23 eqeq2 2813 . . . . . . . . 9 (𝑢 = 𝑈 → ((𝑦𝐺𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑈))
24 eqeq2 2813 . . . . . . . . 9 (𝑢 = 𝑈 → ((𝑥𝐺𝑦) = 𝑢 ↔ (𝑥𝐺𝑦) = 𝑈))
2523, 24anbi12d 633 . . . . . . . 8 (𝑢 = 𝑈 → (((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢) ↔ ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈)))
2625rexbidv 3259 . . . . . . 7 (𝑢 = 𝑈 → (∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢) ↔ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈)))
2722, 26anbi12d 633 . . . . . 6 (𝑢 = 𝑈 → ((((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) ↔ (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈))))
2827ralbidv 3165 . . . . 5 (𝑢 = 𝑈 → (∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) ↔ ∀𝑥𝑋 (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈))))
2928elrab 3631 . . . 4 (𝑈 ∈ {𝑢𝑋 ∣ ∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))} ↔ (𝑈𝑋 ∧ ∀𝑥𝑋 (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈))))
3017, 29sylib 221 . . 3 (𝐺 ∈ GrpOp → (𝑈𝑋 ∧ ∀𝑥𝑋 (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈))))
3130simprd 499 . 2 (𝐺 ∈ GrpOp → ∀𝑥𝑋 (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈)))
32 oveq2 7147 . . . . . 6 (𝑥 = 𝐴 → (𝑈𝐺𝑥) = (𝑈𝐺𝐴))
33 id 22 . . . . . 6 (𝑥 = 𝐴𝑥 = 𝐴)
3432, 33eqeq12d 2817 . . . . 5 (𝑥 = 𝐴 → ((𝑈𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝐴) = 𝐴))
35 oveq1 7146 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐺𝑈) = (𝐴𝐺𝑈))
3635, 33eqeq12d 2817 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐺𝑈) = 𝑥 ↔ (𝐴𝐺𝑈) = 𝐴))
3734, 36anbi12d 633 . . . 4 (𝑥 = 𝐴 → (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ↔ ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴)))
38 oveq2 7147 . . . . . . 7 (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴))
3938eqeq1d 2803 . . . . . 6 (𝑥 = 𝐴 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑦𝐺𝐴) = 𝑈))
40 oveq1 7146 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
4140eqeq1d 2803 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝐺𝑦) = 𝑈 ↔ (𝐴𝐺𝑦) = 𝑈))
4239, 41anbi12d 633 . . . . 5 (𝑥 = 𝐴 → (((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈) ↔ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
4342rexbidv 3259 . . . 4 (𝑥 = 𝐴 → (∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈) ↔ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
4437, 43anbi12d 633 . . 3 (𝑥 = 𝐴 → ((((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈)) ↔ (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈))))
4544rspccva 3573 . 2 ((∀𝑥𝑋 (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈)) ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
4631, 45sylan 583 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  wrex 3110  ∃!wreu 3111  {crab 3113  ran crn 5524  cfv 6328  crio 7096  (class class class)co 7139  GrpOpcgr 28276  GIdcgi 28277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336  df-riota 7097  df-ov 7142  df-grpo 28280  df-gid 28281
This theorem is referenced by:  grpolid  28303  grporid  28304  grporcan  28305  grpoinveu  28306  grpoinv  28312
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