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Theorem isgrpda 35393
Description: Properties that determine a group operation. (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.)
Hypotheses
Ref Expression
isgrpda.1 (𝜑𝑋 ∈ V)
isgrpda.2 (𝜑𝐺:(𝑋 × 𝑋)⟶𝑋)
isgrpda.3 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
isgrpda.4 (𝜑𝑈𝑋)
isgrpda.5 ((𝜑𝑥𝑋) → (𝑈𝐺𝑥) = 𝑥)
isgrpda.6 ((𝜑𝑥𝑋) → ∃𝑛𝑋 (𝑛𝐺𝑥) = 𝑈)
Assertion
Ref Expression
isgrpda (𝜑𝐺 ∈ GrpOp)
Distinct variable groups:   𝜑,𝑥,𝑦,𝑧   𝑛,𝐺,𝑥,𝑦,𝑧   𝑛,𝑋,𝑥,𝑦,𝑧   𝑈,𝑛,𝑥,𝑦,𝑧
Allowed substitution hint:   𝜑(𝑛)

Proof of Theorem isgrpda
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 isgrpda.2 . . 3 (𝜑𝐺:(𝑋 × 𝑋)⟶𝑋)
2 isgrpda.3 . . . 4 ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
32ralrimivvva 3157 . . 3 (𝜑 → ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
4 isgrpda.4 . . . 4 (𝜑𝑈𝑋)
5 isgrpda.5 . . . . . 6 ((𝜑𝑥𝑋) → (𝑈𝐺𝑥) = 𝑥)
6 isgrpda.6 . . . . . . 7 ((𝜑𝑥𝑋) → ∃𝑛𝑋 (𝑛𝐺𝑥) = 𝑈)
7 oveq1 7142 . . . . . . . . 9 (𝑦 = 𝑛 → (𝑦𝐺𝑥) = (𝑛𝐺𝑥))
87eqeq1d 2800 . . . . . . . 8 (𝑦 = 𝑛 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑛𝐺𝑥) = 𝑈))
98cbvrexvw 3397 . . . . . . 7 (∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈 ↔ ∃𝑛𝑋 (𝑛𝐺𝑥) = 𝑈)
106, 9sylibr 237 . . . . . 6 ((𝜑𝑥𝑋) → ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)
115, 10jca 515 . . . . 5 ((𝜑𝑥𝑋) → ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
1211ralrimiva 3149 . . . 4 (𝜑 → ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
13 oveq1 7142 . . . . . . . 8 (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
1413eqeq1d 2800 . . . . . . 7 (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
15 eqeq2 2810 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑦𝐺𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑈))
1615rexbidv 3256 . . . . . . 7 (𝑢 = 𝑈 → (∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
1714, 16anbi12d 633 . . . . . 6 (𝑢 = 𝑈 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
1817ralbidv 3162 . . . . 5 (𝑢 = 𝑈 → (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
1918rspcev 3571 . . . 4 ((𝑈𝑋 ∧ ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))
204, 12, 19syl2anc 587 . . 3 (𝜑 → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))
214adantr 484 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑈𝑋)
22 simpr 488 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑥𝑋)
235eqcomd 2804 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑥 = (𝑈𝐺𝑥))
24 rspceov 7182 . . . . . . . . . 10 ((𝑈𝑋𝑥𝑋𝑥 = (𝑈𝐺𝑥)) → ∃𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
2521, 22, 23, 24syl3anc 1368 . . . . . . . . 9 ((𝜑𝑥𝑋) → ∃𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
2625ralrimiva 3149 . . . . . . . 8 (𝜑 → ∀𝑥𝑋𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
27 foov 7302 . . . . . . . 8 (𝐺:(𝑋 × 𝑋)–onto𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧)))
281, 26, 27sylanbrc 586 . . . . . . 7 (𝜑𝐺:(𝑋 × 𝑋)–onto𝑋)
29 forn 6568 . . . . . . 7 (𝐺:(𝑋 × 𝑋)–onto𝑋 → ran 𝐺 = 𝑋)
3028, 29syl 17 . . . . . 6 (𝜑 → ran 𝐺 = 𝑋)
3130sqxpeqd 5551 . . . . 5 (𝜑 → (ran 𝐺 × ran 𝐺) = (𝑋 × 𝑋))
3231, 30feq23d 6482 . . . 4 (𝜑 → (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺𝐺:(𝑋 × 𝑋)⟶𝑋))
3330raleqdv 3364 . . . . . 6 (𝜑 → (∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
3430, 33raleqbidv 3354 . . . . 5 (𝜑 → (∀𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
3530, 34raleqbidv 3354 . . . 4 (𝜑 → (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
3630rexeqdv 3365 . . . . . . 7 (𝜑 → (∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))
3736anbi2d 631 . . . . . 6 (𝜑 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
3830, 37raleqbidv 3354 . . . . 5 (𝜑 → (∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
3930, 38rexeqbidv 3355 . . . 4 (𝜑 → (∃𝑢 ∈ ran 𝐺𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢) ↔ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
4032, 35, 393anbi123d 1433 . . 3 (𝜑 → ((𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ ran 𝐺𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢)) ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))))
411, 3, 20, 40mpbir3and 1339 . 2 (𝜑 → (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ ran 𝐺𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢)))
42 isgrpda.1 . . . . 5 (𝜑𝑋 ∈ V)
4342, 42xpexd 7454 . . . 4 (𝜑 → (𝑋 × 𝑋) ∈ V)
44 fex 6966 . . . 4 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ (𝑋 × 𝑋) ∈ V) → 𝐺 ∈ V)
451, 43, 44syl2anc 587 . . 3 (𝜑𝐺 ∈ V)
46 eqid 2798 . . . 4 ran 𝐺 = ran 𝐺
4746isgrpo 28280 . . 3 (𝐺 ∈ V → (𝐺 ∈ GrpOp ↔ (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ ran 𝐺𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢))))
4845, 47syl 17 . 2 (𝜑 → (𝐺 ∈ GrpOp ↔ (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ ran 𝐺𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢))))
4941, 48mpbird 260 1 (𝜑𝐺 ∈ 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  ontowfo 6322  (class class class)co 7135  GrpOpcgr 28272
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 7441
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 28276
This theorem is referenced by:  isdrngo2  35396
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