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Theorem isgrpoi 27878
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 3157 . 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 6885 . . . . . . . 8 (𝑦 = 𝑁 → (𝑦𝐺𝑥) = (𝑁𝐺𝑥))
98eqeq1d 2801 . . . . . . 7 (𝑦 = 𝑁 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑁𝐺𝑥) = 𝑈))
109rspcev 3497 . . . . . 6 ((𝑁𝑋 ∧ (𝑁𝐺𝑥) = 𝑈) → ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)
116, 7, 10syl2anc 580 . . . . 5 (𝑥𝑋 → ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)
125, 11jca 508 . . . 4 (𝑥𝑋 → ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
1312rgen 3103 . . 3 𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)
14 oveq1 6885 . . . . . . 7 (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
1514eqeq1d 2801 . . . . . 6 (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
16 eqeq2 2810 . . . . . . 7 (𝑢 = 𝑈 → ((𝑦𝐺𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑈))
1716rexbidv 3233 . . . . . 6 (𝑢 = 𝑈 → (∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
1815, 17anbi12d 625 . . . . 5 (𝑢 = 𝑈 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
1918ralbidv 3167 . . . 4 (𝑢 = 𝑈 → (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
2019rspcev 3497 . . 3 ((𝑈𝑋 ∧ ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))
214, 13, 20mp2an 684 . 2 𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)
22 isgrpoi.1 . . . . 5 𝑋 ∈ V
2322, 22xpex 7196 . . . 4 (𝑋 × 𝑋) ∈ V
24 fex 6718 . . . 4 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ (𝑋 × 𝑋) ∈ V) → 𝐺 ∈ V)
251, 23, 24mp2an 684 . . 3 𝐺 ∈ V
265eqcomd 2805 . . . . . . . . 9 (𝑥𝑋𝑥 = (𝑈𝐺𝑥))
27 rspceov 6924 . . . . . . . . . 10 ((𝑈𝑋𝑥𝑋𝑥 = (𝑈𝐺𝑥)) → ∃𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
284, 27mp3an1 1573 . . . . . . . . 9 ((𝑥𝑋𝑥 = (𝑈𝐺𝑥)) → ∃𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
2926, 28mpdan 679 . . . . . . . 8 (𝑥𝑋 → ∃𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
3029rgen 3103 . . . . . . 7 𝑥𝑋𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧)
31 foov 7042 . . . . . . 7 (𝐺:(𝑋 × 𝑋)–onto𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧)))
321, 30, 31mpbir2an 703 . . . . . 6 𝐺:(𝑋 × 𝑋)–onto𝑋
33 forn 6334 . . . . . 6 (𝐺:(𝑋 × 𝑋)–onto𝑋 → ran 𝐺 = 𝑋)
3432, 33ax-mp 5 . . . . 5 ran 𝐺 = 𝑋
3534eqcomi 2808 . . . 4 𝑋 = ran 𝐺
3635isgrpo 27877 . . 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 1442 1 𝐺 ∈ GrpOp
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3089  wrex 3090  Vcvv 3385   × cxp 5310  ran crn 5313  wf 6097  ontowfo 6099  (class class class)co 6878  GrpOpcgr 27869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-grpo 27873
This theorem is referenced by:  cnaddabloOLD  27961  hilablo  28542  hhssabloilem  28643  grposnOLD  34168
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