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Theorem isgrpoi 27582
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 3078 . 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 6772 . . . . . . . 8 (𝑦 = 𝑁 → (𝑦𝐺𝑥) = (𝑁𝐺𝑥))
98eqeq1d 2726 . . . . . . 7 (𝑦 = 𝑁 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑁𝐺𝑥) = 𝑈))
109rspcev 3413 . . . . . 6 ((𝑁𝑋 ∧ (𝑁𝐺𝑥) = 𝑈) → ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)
116, 7, 10syl2anc 696 . . . . 5 (𝑥𝑋 → ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)
125, 11jca 555 . . . 4 (𝑥𝑋 → ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
1312rgen 3024 . . 3 𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)
14 oveq1 6772 . . . . . . 7 (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
1514eqeq1d 2726 . . . . . 6 (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
16 eqeq2 2735 . . . . . . 7 (𝑢 = 𝑈 → ((𝑦𝐺𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑈))
1716rexbidv 3154 . . . . . 6 (𝑢 = 𝑈 → (∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
1815, 17anbi12d 749 . . . . 5 (𝑢 = 𝑈 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
1918ralbidv 3088 . . . 4 (𝑢 = 𝑈 → (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
2019rspcev 3413 . . 3 ((𝑈𝑋 ∧ ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))
214, 13, 20mp2an 710 . 2 𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)
22 isgrpoi.1 . . . . 5 𝑋 ∈ V
2322, 22xpex 7079 . . . 4 (𝑋 × 𝑋) ∈ V
24 fex 6605 . . . 4 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ (𝑋 × 𝑋) ∈ V) → 𝐺 ∈ V)
251, 23, 24mp2an 710 . . 3 𝐺 ∈ V
265eqcomd 2730 . . . . . . . . 9 (𝑥𝑋𝑥 = (𝑈𝐺𝑥))
27 rspceov 6807 . . . . . . . . . 10 ((𝑈𝑋𝑥𝑋𝑥 = (𝑈𝐺𝑥)) → ∃𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
284, 27mp3an1 1524 . . . . . . . . 9 ((𝑥𝑋𝑥 = (𝑈𝐺𝑥)) → ∃𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
2926, 28mpdan 705 . . . . . . . 8 (𝑥𝑋 → ∃𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
3029rgen 3024 . . . . . . 7 𝑥𝑋𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧)
31 foov 6925 . . . . . . 7 (𝐺:(𝑋 × 𝑋)–onto𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧)))
321, 30, 31mpbir2an 993 . . . . . 6 𝐺:(𝑋 × 𝑋)–onto𝑋
33 forn 6231 . . . . . 6 (𝐺:(𝑋 × 𝑋)–onto𝑋 → ran 𝐺 = 𝑋)
3432, 33ax-mp 5 . . . . 5 ran 𝐺 = 𝑋
3534eqcomi 2733 . . . 4 𝑋 = ran 𝐺
3635isgrpo 27581 . . 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 1381 1 𝐺 ∈ GrpOp
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1596  wcel 2103  wral 3014  wrex 3015  Vcvv 3304   × cxp 5216  ran crn 5219  wf 5997  ontowfo 5999  (class class class)co 6765  GrpOpcgr 27573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-grpo 27577
This theorem is referenced by:  cnaddabloOLD  27666  hilablo  28247  hhssabloilem  28348  grposnOLD  33913
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