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Theorem grpoidinv2 27497
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 27495 . . . . . 6 (𝐺 ∈ GrpOp → 𝑈 = (𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥))
41grpoideu 27491 . . . . . . 7 (𝐺 ∈ GrpOp → ∃!𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
5 riotacl2 6664 . . . . . . 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 2730 . . . . 5 (𝐺 ∈ GrpOp → 𝑈 ∈ {𝑢𝑋 ∣ ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥})
8 simpll 805 . . . . . . . . . . 11 ((((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → (𝑢𝐺𝑥) = 𝑥)
98ralimi 2981 . . . . . . . . . 10 (∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
109rgenw 2953 . . . . . . . . 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 27490 . . . . . . . 8 (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
1311, 12, 43jca 1261 . . . . . . 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 3946 . . . . . . 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 487 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑢𝑋) → (∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥 ↔ ∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))))
1615rabbidva 3219 . . . . 5 (𝐺 ∈ GrpOp → {𝑢𝑋 ∣ ∀𝑥𝑋 (𝑢𝐺𝑥) = 𝑥} = {𝑢𝑋 ∣ ∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))})
177, 16eleqtrd 2732 . . . 4 (𝐺 ∈ GrpOp → 𝑈 ∈ {𝑢𝑋 ∣ ∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))})
18 oveq1 6697 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
1918eqeq1d 2653 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
20 oveq2 6698 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑥𝐺𝑢) = (𝑥𝐺𝑈))
2120eqeq1d 2653 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑥𝐺𝑢) = 𝑥 ↔ (𝑥𝐺𝑈) = 𝑥))
2219, 21anbi12d 747 . . . . . . 7 (𝑢 = 𝑈 → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)))
23 eqeq2 2662 . . . . . . . . 9 (𝑢 = 𝑈 → ((𝑦𝐺𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑈))
24 eqeq2 2662 . . . . . . . . 9 (𝑢 = 𝑈 → ((𝑥𝐺𝑦) = 𝑢 ↔ (𝑥𝐺𝑦) = 𝑈))
2523, 24anbi12d 747 . . . . . . . 8 (𝑢 = 𝑈 → (((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢) ↔ ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈)))
2625rexbidv 3081 . . . . . . 7 (𝑢 = 𝑈 → (∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢) ↔ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈)))
2722, 26anbi12d 747 . . . . . 6 (𝑢 = 𝑈 → ((((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) ↔ (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈))))
2827ralbidv 3015 . . . . 5 (𝑢 = 𝑈 → (∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) ↔ ∀𝑥𝑋 (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈))))
2928elrab 3396 . . . 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 208 . . 3 (𝐺 ∈ GrpOp → (𝑈𝑋 ∧ ∀𝑥𝑋 (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈))))
3130simprd 478 . 2 (𝐺 ∈ GrpOp → ∀𝑥𝑋 (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈)))
32 oveq2 6698 . . . . . 6 (𝑥 = 𝐴 → (𝑈𝐺𝑥) = (𝑈𝐺𝐴))
33 id 22 . . . . . 6 (𝑥 = 𝐴𝑥 = 𝐴)
3432, 33eqeq12d 2666 . . . . 5 (𝑥 = 𝐴 → ((𝑈𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝐴) = 𝐴))
35 oveq1 6697 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐺𝑈) = (𝐴𝐺𝑈))
3635, 33eqeq12d 2666 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐺𝑈) = 𝑥 ↔ (𝐴𝐺𝑈) = 𝐴))
3734, 36anbi12d 747 . . . 4 (𝑥 = 𝐴 → (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ↔ ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴)))
38 oveq2 6698 . . . . . . 7 (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴))
3938eqeq1d 2653 . . . . . 6 (𝑥 = 𝐴 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑦𝐺𝐴) = 𝑈))
40 oveq1 6697 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
4140eqeq1d 2653 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝐺𝑦) = 𝑈 ↔ (𝐴𝐺𝑦) = 𝑈))
4239, 41anbi12d 747 . . . . 5 (𝑥 = 𝐴 → (((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈) ↔ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
4342rexbidv 3081 . . . 4 (𝑥 = 𝐴 → (∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈) ↔ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
4437, 43anbi12d 747 . . 3 (𝑥 = 𝐴 → ((((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈)) ↔ (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈))))
4544rspccva 3339 . 2 ((∀𝑥𝑋 (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈)) ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
4631, 45sylan 487 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  ∃!wreu 2943  {crab 2945  ran crn 5144  cfv 5926  crio 6650  (class class class)co 6690  GrpOpcgr 27471  GIdcgi 27472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-riota 6651  df-ov 6693  df-grpo 27475  df-gid 27476
This theorem is referenced by:  grpolid  27498  grporid  27499  grporcan  27500  grpoinveu  27501  grpoinv  27507
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