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Theorem isgrpoi 28288
 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 3169 . 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 7142 . . . . . . . 8 (𝑦 = 𝑁 → (𝑦𝐺𝑥) = (𝑁𝐺𝑥))
98eqeq1d 2800 . . . . . . 7 (𝑦 = 𝑁 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑁𝐺𝑥) = 𝑈))
109rspcev 3571 . . . . . 6 ((𝑁𝑋 ∧ (𝑁𝐺𝑥) = 𝑈) → ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)
116, 7, 10syl2anc 587 . . . . 5 (𝑥𝑋 → ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)
125, 11jca 515 . . . 4 (𝑥𝑋 → ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
1312rgen 3116 . . 3 𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)
14 oveq1 7142 . . . . . . 7 (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
1514eqeq1d 2800 . . . . . 6 (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
16 eqeq2 2810 . . . . . . 7 (𝑢 = 𝑈 → ((𝑦𝐺𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑈))
1716rexbidv 3256 . . . . . 6 (𝑢 = 𝑈 → (∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
1815, 17anbi12d 633 . . . . 5 (𝑢 = 𝑈 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
1918ralbidv 3162 . . . 4 (𝑢 = 𝑈 → (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
2019rspcev 3571 . . 3 ((𝑈𝑋 ∧ ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))
214, 13, 20mp2an 691 . 2 𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)
22 isgrpoi.1 . . . . 5 𝑋 ∈ V
2322, 22xpex 7458 . . . 4 (𝑋 × 𝑋) ∈ V
24 fex 6966 . . . 4 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ (𝑋 × 𝑋) ∈ V) → 𝐺 ∈ V)
251, 23, 24mp2an 691 . . 3 𝐺 ∈ V
265eqcomd 2804 . . . . . . . . 9 (𝑥𝑋𝑥 = (𝑈𝐺𝑥))
27 rspceov 7182 . . . . . . . . . 10 ((𝑈𝑋𝑥𝑋𝑥 = (𝑈𝐺𝑥)) → ∃𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
284, 27mp3an1 1445 . . . . . . . . 9 ((𝑥𝑋𝑥 = (𝑈𝐺𝑥)) → ∃𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
2926, 28mpdan 686 . . . . . . . 8 (𝑥𝑋 → ∃𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
3029rgen 3116 . . . . . . 7 𝑥𝑋𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧)
31 foov 7303 . . . . . . 7 (𝐺:(𝑋 × 𝑋)–onto𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧)))
321, 30, 31mpbir2an 710 . . . . . 6 𝐺:(𝑋 × 𝑋)–onto𝑋
33 forn 6568 . . . . . 6 (𝐺:(𝑋 × 𝑋)–onto𝑋 → ran 𝐺 = 𝑋)
3432, 33ax-mp 5 . . . . 5 ran 𝐺 = 𝑋
3534eqcomi 2807 . . . 4 𝑋 = ran 𝐺
3635isgrpo 28287 . . 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 1338 1 𝐺 ∈ GrpOp
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441   × cxp 5517  ran crn 5520  ⟶wf 6320  –onto→wfo 6322  (class class class)co 7135  GrpOpcgr 28279 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-grpo 28283 This theorem is referenced by:  cnaddabloOLD  28371  hilablo  28950  hhssabloilem  29051  grposnOLD  35336
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