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Theorem isgrpo 27691
Description: The predicate "is a group operation." Note that 𝑋 is the base set of the group. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
isgrp.1 𝑋 = ran 𝐺
Assertion
Ref Expression
isgrpo (𝐺𝐴 → (𝐺 ∈ GrpOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))))
Distinct variable groups:   𝑥,𝑢,𝑦,𝑧,𝐺   𝑢,𝑋,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧,𝑢)

Proof of Theorem isgrpo
Dummy variables 𝑡 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 feq1 6166 . . . . . 6 (𝑔 = 𝐺 → (𝑔:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑡 × 𝑡)⟶𝑡))
2 oveq 6799 . . . . . . . . . 10 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝑔𝑦)𝐺𝑧))
3 oveq 6799 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
43oveq1d 6808 . . . . . . . . . 10 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝐺𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
52, 4eqtrd 2805 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
6 oveq 6799 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝑔𝑧)))
7 oveq 6799 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
87oveq2d 6809 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥𝐺(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
96, 8eqtrd 2805 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
105, 9eqeq12d 2786 . . . . . . . 8 (𝑔 = 𝐺 → (((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
1110ralbidv 3135 . . . . . . 7 (𝑔 = 𝐺 → (∀𝑧𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
12112ralbidv 3138 . . . . . 6 (𝑔 = 𝐺 → (∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
13 oveq 6799 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑢𝑔𝑥) = (𝑢𝐺𝑥))
1413eqeq1d 2773 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑢𝑔𝑥) = 𝑥 ↔ (𝑢𝐺𝑥) = 𝑥))
15 oveq 6799 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
1615eqeq1d 2773 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑢))
1716rexbidv 3200 . . . . . . . 8 (𝑔 = 𝐺 → (∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢 ↔ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢))
1814, 17anbi12d 616 . . . . . . 7 (𝑔 = 𝐺 → (((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)))
1918rexralbidv 3206 . . . . . 6 (𝑔 = 𝐺 → (∃𝑢𝑡𝑥𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢) ↔ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)))
201, 12, 193anbi123d 1547 . . . . 5 (𝑔 = 𝐺 → ((𝑔:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢)) ↔ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢))))
2120exbidv 2002 . . . 4 (𝑔 = 𝐺 → (∃𝑡(𝑔:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢)) ↔ ∃𝑡(𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢))))
22 df-grpo 27687 . . . 4 GrpOp = {𝑔 ∣ ∃𝑡(𝑔:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢))}
2321, 22elab2g 3504 . . 3 (𝐺𝐴 → (𝐺 ∈ GrpOp ↔ ∃𝑡(𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢))))
24 simpl 468 . . . . . . . . . . . . . 14 (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → (𝑢𝐺𝑥) = 𝑥)
2524ralimi 3101 . . . . . . . . . . . . 13 (∀𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → ∀𝑥𝑡 (𝑢𝐺𝑥) = 𝑥)
26 oveq2 6801 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑢𝐺𝑥) = (𝑢𝐺𝑧))
27 id 22 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧𝑥 = 𝑧)
2826, 27eqeq12d 2786 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑧) = 𝑧))
29 eqcom 2778 . . . . . . . . . . . . . . . 16 ((𝑢𝐺𝑧) = 𝑧𝑧 = (𝑢𝐺𝑧))
3028, 29syl6bb 276 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ((𝑢𝐺𝑥) = 𝑥𝑧 = (𝑢𝐺𝑧)))
3130rspcv 3456 . . . . . . . . . . . . . 14 (𝑧𝑡 → (∀𝑥𝑡 (𝑢𝐺𝑥) = 𝑥𝑧 = (𝑢𝐺𝑧)))
32 oveq2 6801 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝑢𝐺𝑦) = (𝑢𝐺𝑧))
3332eqeq2d 2781 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (𝑧 = (𝑢𝐺𝑦) ↔ 𝑧 = (𝑢𝐺𝑧)))
3433rspcev 3460 . . . . . . . . . . . . . . 15 ((𝑧𝑡𝑧 = (𝑢𝐺𝑧)) → ∃𝑦𝑡 𝑧 = (𝑢𝐺𝑦))
3534ex 397 . . . . . . . . . . . . . 14 (𝑧𝑡 → (𝑧 = (𝑢𝐺𝑧) → ∃𝑦𝑡 𝑧 = (𝑢𝐺𝑦)))
3631, 35syld 47 . . . . . . . . . . . . 13 (𝑧𝑡 → (∀𝑥𝑡 (𝑢𝐺𝑥) = 𝑥 → ∃𝑦𝑡 𝑧 = (𝑢𝐺𝑦)))
3725, 36syl5 34 . . . . . . . . . . . 12 (𝑧𝑡 → (∀𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → ∃𝑦𝑡 𝑧 = (𝑢𝐺𝑦)))
3837reximdv 3164 . . . . . . . . . . 11 (𝑧𝑡 → (∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → ∃𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦)))
3938impcom 394 . . . . . . . . . 10 ((∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) ∧ 𝑧𝑡) → ∃𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦))
4039ralrimiva 3115 . . . . . . . . 9 (∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → ∀𝑧𝑡𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦))
4140anim2i 603 . . . . . . . 8 ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)) → (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑧𝑡𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦)))
42 foov 6955 . . . . . . . 8 (𝐺:(𝑡 × 𝑡)–onto𝑡 ↔ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑧𝑡𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦)))
4341, 42sylibr 224 . . . . . . 7 ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)) → 𝐺:(𝑡 × 𝑡)–onto𝑡)
44 forn 6259 . . . . . . . 8 (𝐺:(𝑡 × 𝑡)–onto𝑡 → ran 𝐺 = 𝑡)
4544eqcomd 2777 . . . . . . 7 (𝐺:(𝑡 × 𝑡)–onto𝑡𝑡 = ran 𝐺)
4643, 45syl 17 . . . . . 6 ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)) → 𝑡 = ran 𝐺)
47463adant2 1125 . . . . 5 ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)) → 𝑡 = ran 𝐺)
4847pm4.71ri 550 . . . 4 ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)) ↔ (𝑡 = ran 𝐺 ∧ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢))))
4948exbii 1924 . . 3 (∃𝑡(𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)) ↔ ∃𝑡(𝑡 = ran 𝐺 ∧ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢))))
5023, 49syl6bb 276 . 2 (𝐺𝐴 → (𝐺 ∈ GrpOp ↔ ∃𝑡(𝑡 = ran 𝐺 ∧ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)))))
51 rnexg 7245 . . 3 (𝐺𝐴 → ran 𝐺 ∈ V)
52 isgrp.1 . . . . . 6 𝑋 = ran 𝐺
5352eqeq2i 2783 . . . . 5 (𝑡 = 𝑋𝑡 = ran 𝐺)
54 xpeq1 5263 . . . . . . . . 9 (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑡))
55 xpeq2 5269 . . . . . . . . 9 (𝑡 = 𝑋 → (𝑋 × 𝑡) = (𝑋 × 𝑋))
5654, 55eqtrd 2805 . . . . . . . 8 (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑋))
5756feq2d 6171 . . . . . . 7 (𝑡 = 𝑋 → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑋 × 𝑋)⟶𝑡))
58 feq3 6168 . . . . . . 7 (𝑡 = 𝑋 → (𝐺:(𝑋 × 𝑋)⟶𝑡𝐺:(𝑋 × 𝑋)⟶𝑋))
5957, 58bitrd 268 . . . . . 6 (𝑡 = 𝑋 → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑋 × 𝑋)⟶𝑋))
60 raleq 3287 . . . . . . . 8 (𝑡 = 𝑋 → (∀𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
6160raleqbi1dv 3295 . . . . . . 7 (𝑡 = 𝑋 → (∀𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
6261raleqbi1dv 3295 . . . . . 6 (𝑡 = 𝑋 → (∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
63 rexeq 3288 . . . . . . . . 9 (𝑡 = 𝑋 → (∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))
6463anbi2d 614 . . . . . . . 8 (𝑡 = 𝑋 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
6564raleqbi1dv 3295 . . . . . . 7 (𝑡 = 𝑋 → (∀𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
6665rexeqbi1dv 3296 . . . . . 6 (𝑡 = 𝑋 → (∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
6759, 62, 663anbi123d 1547 . . . . 5 (𝑡 = 𝑋 → ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)) ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))))
6853, 67sylbir 225 . . . 4 (𝑡 = ran 𝐺 → ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)) ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))))
6968ceqsexgv 3485 . . 3 (ran 𝐺 ∈ V → (∃𝑡(𝑡 = ran 𝐺 ∧ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢))) ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))))
7051, 69syl 17 . 2 (𝐺𝐴 → (∃𝑡(𝑡 = ran 𝐺 ∧ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢))) ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))))
7150, 70bitrd 268 1 (𝐺𝐴 → (𝐺 ∈ GrpOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  wral 3061  wrex 3062  Vcvv 3351   × cxp 5247  ran crn 5250  wf 6027  ontowfo 6029  (class class class)co 6793  GrpOpcgr 27683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-ov 6796  df-grpo 27687
This theorem is referenced by:  isgrpoi  27692  grpofo  27693  grpolidinv  27695  grpoass  27697  grpomndo  34006  isgrpda  34086
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