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Theorem isgrpoi 29614
Description: Properties that determine a group operation. Read 𝑁 as 𝑁(𝑥). (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
isgrpoi.1 𝑋 ∈ V
isgrpoi.2 𝐺:(𝑋 × 𝑋)⟶𝑋
isgrpoi.3 ((𝑥𝑋𝑦𝑋𝑧𝑋) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
isgrpoi.4 𝑈𝑋
isgrpoi.5 (𝑥𝑋 → (𝑈𝐺𝑥) = 𝑥)
isgrpoi.6 (𝑥𝑋𝑁𝑋)
isgrpoi.7 (𝑥𝑋 → (𝑁𝐺𝑥) = 𝑈)
Assertion
Ref Expression
isgrpoi 𝐺 ∈ GrpOp
Distinct variable groups:   𝑥,𝑦,𝑧,𝐺   𝑥,𝑈,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑦,𝑁
Allowed substitution hints:   𝑁(𝑥,𝑧)

Proof of Theorem isgrpoi
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 isgrpoi.2 . 2 𝐺:(𝑋 × 𝑋)⟶𝑋
2 isgrpoi.3 . . 3 ((𝑥𝑋𝑦𝑋𝑧𝑋) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
32rgen3 3201 . 2 𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))
4 isgrpoi.4 . . 3 𝑈𝑋
5 isgrpoi.5 . . . . 5 (𝑥𝑋 → (𝑈𝐺𝑥) = 𝑥)
6 isgrpoi.6 . . . . . 6 (𝑥𝑋𝑁𝑋)
7 isgrpoi.7 . . . . . 6 (𝑥𝑋 → (𝑁𝐺𝑥) = 𝑈)
8 oveq1 7400 . . . . . . . 8 (𝑦 = 𝑁 → (𝑦𝐺𝑥) = (𝑁𝐺𝑥))
98eqeq1d 2733 . . . . . . 7 (𝑦 = 𝑁 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑁𝐺𝑥) = 𝑈))
109rspcev 3609 . . . . . 6 ((𝑁𝑋 ∧ (𝑁𝐺𝑥) = 𝑈) → ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)
116, 7, 10syl2anc 584 . . . . 5 (𝑥𝑋 → ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)
125, 11jca 512 . . . 4 (𝑥𝑋 → ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
1312rgen 3062 . . 3 𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)
14 oveq1 7400 . . . . . . 7 (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
1514eqeq1d 2733 . . . . . 6 (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
16 eqeq2 2743 . . . . . . 7 (𝑢 = 𝑈 → ((𝑦𝐺𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑈))
1716rexbidv 3177 . . . . . 6 (𝑢 = 𝑈 → (∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
1815, 17anbi12d 631 . . . . 5 (𝑢 = 𝑈 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
1918ralbidv 3176 . . . 4 (𝑢 = 𝑈 → (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
2019rspcev 3609 . . 3 ((𝑈𝑋 ∧ ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))
214, 13, 20mp2an 690 . 2 𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)
22 isgrpoi.1 . . . . 5 𝑋 ∈ V
2322, 22xpex 7723 . . . 4 (𝑋 × 𝑋) ∈ V
24 fex 7212 . . . 4 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ (𝑋 × 𝑋) ∈ V) → 𝐺 ∈ V)
251, 23, 24mp2an 690 . . 3 𝐺 ∈ V
265eqcomd 2737 . . . . . . . . 9 (𝑥𝑋𝑥 = (𝑈𝐺𝑥))
27 rspceov 7440 . . . . . . . . . 10 ((𝑈𝑋𝑥𝑋𝑥 = (𝑈𝐺𝑥)) → ∃𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
284, 27mp3an1 1448 . . . . . . . . 9 ((𝑥𝑋𝑥 = (𝑈𝐺𝑥)) → ∃𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
2926, 28mpdan 685 . . . . . . . 8 (𝑥𝑋 → ∃𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
3029rgen 3062 . . . . . . 7 𝑥𝑋𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧)
31 foov 7564 . . . . . . 7 (𝐺:(𝑋 × 𝑋)–onto𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧)))
321, 30, 31mpbir2an 709 . . . . . 6 𝐺:(𝑋 × 𝑋)–onto𝑋
33 forn 6795 . . . . . 6 (𝐺:(𝑋 × 𝑋)–onto𝑋 → ran 𝐺 = 𝑋)
3432, 33ax-mp 5 . . . . 5 ran 𝐺 = 𝑋
3534eqcomi 2740 . . . 4 𝑋 = ran 𝐺
3635isgrpo 29613 . . 3 (𝐺 ∈ V → (𝐺 ∈ GrpOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))))
3725, 36ax-mp 5 . 2 (𝐺 ∈ GrpOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
381, 3, 21, 37mpbir3an 1341 1 𝐺 ∈ GrpOp
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3060  wrex 3069  Vcvv 3473   × cxp 5667  ran crn 5670  wf 6528  ontowfo 6530  (class class class)co 7393  GrpOpcgr 29605
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-grpo 29609
This theorem is referenced by:  cnaddabloOLD  29697  hilablo  30276  hhssabloilem  30377  grposnOLD  36555
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