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Theorem isgrpda 34062
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 3159 . . 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 6878 . . . . . . . . 9 (𝑦 = 𝑛 → (𝑦𝐺𝑥) = (𝑛𝐺𝑥))
87eqeq1d 2807 . . . . . . . 8 (𝑦 = 𝑛 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑛𝐺𝑥) = 𝑈))
98cbvrexv 3360 . . . . . . 7 (∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈 ↔ ∃𝑛𝑋 (𝑛𝐺𝑥) = 𝑈)
106, 9sylibr 225 . . . . . 6 ((𝜑𝑥𝑋) → ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)
115, 10jca 503 . . . . 5 ((𝜑𝑥𝑋) → ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
1211ralrimiva 3153 . . . 4 (𝜑 → ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
13 oveq1 6878 . . . . . . . 8 (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
1413eqeq1d 2807 . . . . . . 7 (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
15 eqeq2 2816 . . . . . . . 8 (𝑢 = 𝑈 → ((𝑦𝐺𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑈))
1615rexbidv 3239 . . . . . . 7 (𝑢 = 𝑈 → (∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
1714, 16anbi12d 618 . . . . . 6 (𝑢 = 𝑈 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
1817ralbidv 3173 . . . . 5 (𝑢 = 𝑈 → (∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
1918rspcev 3501 . . . 4 ((𝑈𝑋 ∧ ∀𝑥𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))
204, 12, 19syl2anc 575 . . 3 (𝜑 → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))
214adantr 468 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑈𝑋)
22 simpr 473 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑥𝑋)
235eqcomd 2811 . . . . . . . . . 10 ((𝜑𝑥𝑋) → 𝑥 = (𝑈𝐺𝑥))
24 rspceov 6917 . . . . . . . . . 10 ((𝑈𝑋𝑥𝑋𝑥 = (𝑈𝐺𝑥)) → ∃𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
2521, 22, 23, 24syl3anc 1483 . . . . . . . . 9 ((𝜑𝑥𝑋) → ∃𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
2625ralrimiva 3153 . . . . . . . 8 (𝜑 → ∀𝑥𝑋𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧))
27 foov 7035 . . . . . . . 8 (𝐺:(𝑋 × 𝑋)–onto𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 𝑥 = (𝑦𝐺𝑧)))
281, 26, 27sylanbrc 574 . . . . . . 7 (𝜑𝐺:(𝑋 × 𝑋)–onto𝑋)
29 forn 6331 . . . . . . 7 (𝐺:(𝑋 × 𝑋)–onto𝑋 → ran 𝐺 = 𝑋)
3028, 29syl 17 . . . . . 6 (𝜑 → ran 𝐺 = 𝑋)
3130sqxpeqd 5339 . . . . 5 (𝜑 → (ran 𝐺 × ran 𝐺) = (𝑋 × 𝑋))
3231, 30feq23d 6248 . . . 4 (𝜑 → (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺𝐺:(𝑋 × 𝑋)⟶𝑋))
3330raleqdv 3332 . . . . . 6 (𝜑 → (∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
3430, 33raleqbidv 3340 . . . . 5 (𝜑 → (∀𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
3530, 34raleqbidv 3340 . . . 4 (𝜑 → (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
3630rexeqdv 3333 . . . . . . 7 (𝜑 → (∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))
3736anbi2d 616 . . . . . 6 (𝜑 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
3830, 37raleqbidv 3340 . . . . 5 (𝜑 → (∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
3930, 38rexeqbidv 3341 . . . 4 (𝜑 → (∃𝑢 ∈ ran 𝐺𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢) ↔ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
4032, 35, 393anbi123d 1553 . . 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 1435 . 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)
43 xpexg 7187 . . . . 5 ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑋 × 𝑋) ∈ V)
4442, 42, 43syl2anc 575 . . . 4 (𝜑 → (𝑋 × 𝑋) ∈ V)
45 fex 6711 . . . 4 ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ (𝑋 × 𝑋) ∈ V) → 𝐺 ∈ V)
461, 44, 45syl2anc 575 . . 3 (𝜑𝐺 ∈ V)
47 eqid 2805 . . . 4 ran 𝐺 = ran 𝐺
4847isgrpo 27676 . . 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))))
4946, 48syl 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))))
5041, 49mpbird 248 1 (𝜑𝐺 ∈ GrpOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2158  wral 3095  wrex 3096  Vcvv 3390   × cxp 5306  ran crn 5309  wf 6094  ontowfo 6096  (class class class)co 6871  GrpOpcgr 27668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-rep 4960  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-ral 3100  df-rex 3101  df-reu 3102  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4627  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-id 5216  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-ov 6874  df-grpo 27672
This theorem is referenced by:  isdrngo2  34065
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