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Theorem isgrpo 30526
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 6717 . . . . . 6 (𝑔 = 𝐺 → (𝑔:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑡 × 𝑡)⟶𝑡))
2 oveq 7437 . . . . . . . . . 10 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝑔𝑦)𝐺𝑧))
3 oveq 7437 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
43oveq1d 7446 . . . . . . . . . 10 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝐺𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
52, 4eqtrd 2775 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
6 oveq 7437 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝑔𝑧)))
7 oveq 7437 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
87oveq2d 7447 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥𝐺(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
96, 8eqtrd 2775 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
105, 9eqeq12d 2751 . . . . . . . 8 (𝑔 = 𝐺 → (((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
1110ralbidv 3176 . . . . . . 7 (𝑔 = 𝐺 → (∀𝑧𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
12112ralbidv 3219 . . . . . 6 (𝑔 = 𝐺 → (∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
13 oveq 7437 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑢𝑔𝑥) = (𝑢𝐺𝑥))
1413eqeq1d 2737 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑢𝑔𝑥) = 𝑥 ↔ (𝑢𝐺𝑥) = 𝑥))
15 oveq 7437 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
1615eqeq1d 2737 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑢))
1716rexbidv 3177 . . . . . . . 8 (𝑔 = 𝐺 → (∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢 ↔ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢))
1814, 17anbi12d 632 . . . . . . 7 (𝑔 = 𝐺 → (((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)))
1918rexralbidv 3221 . . . . . 6 (𝑔 = 𝐺 → (∃𝑢𝑡𝑥𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢) ↔ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)))
201, 12, 193anbi123d 1435 . . . . 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 1919 . . . 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 30522 . . . 4 GrpOp = {𝑔 ∣ ∃𝑡(𝑔:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢))}
2321, 22elab2g 3683 . . 3 (𝐺𝐴 → (𝐺 ∈ GrpOp ↔ ∃𝑡(𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢))))
24 simpl 482 . . . . . . . . . . . . . 14 (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → (𝑢𝐺𝑥) = 𝑥)
2524ralimi 3081 . . . . . . . . . . . . 13 (∀𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → ∀𝑥𝑡 (𝑢𝐺𝑥) = 𝑥)
26 oveq2 7439 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑢𝐺𝑥) = (𝑢𝐺𝑧))
27 id 22 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧𝑥 = 𝑧)
2826, 27eqeq12d 2751 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑧) = 𝑧))
29 eqcom 2742 . . . . . . . . . . . . . . . 16 ((𝑢𝐺𝑧) = 𝑧𝑧 = (𝑢𝐺𝑧))
3028, 29bitrdi 287 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ((𝑢𝐺𝑥) = 𝑥𝑧 = (𝑢𝐺𝑧)))
3130rspcv 3618 . . . . . . . . . . . . . 14 (𝑧𝑡 → (∀𝑥𝑡 (𝑢𝐺𝑥) = 𝑥𝑧 = (𝑢𝐺𝑧)))
32 oveq2 7439 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (𝑢𝐺𝑦) = (𝑢𝐺𝑧))
3332rspceeqv 3645 . . . . . . . . . . . . . . 15 ((𝑧𝑡𝑧 = (𝑢𝐺𝑧)) → ∃𝑦𝑡 𝑧 = (𝑢𝐺𝑦))
3433ex 412 . . . . . . . . . . . . . 14 (𝑧𝑡 → (𝑧 = (𝑢𝐺𝑧) → ∃𝑦𝑡 𝑧 = (𝑢𝐺𝑦)))
3531, 34syld 47 . . . . . . . . . . . . 13 (𝑧𝑡 → (∀𝑥𝑡 (𝑢𝐺𝑥) = 𝑥 → ∃𝑦𝑡 𝑧 = (𝑢𝐺𝑦)))
3625, 35syl5 34 . . . . . . . . . . . 12 (𝑧𝑡 → (∀𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → ∃𝑦𝑡 𝑧 = (𝑢𝐺𝑦)))
3736reximdv 3168 . . . . . . . . . . 11 (𝑧𝑡 → (∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → ∃𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦)))
3837impcom 407 . . . . . . . . . 10 ((∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) ∧ 𝑧𝑡) → ∃𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦))
3938ralrimiva 3144 . . . . . . . . 9 (∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → ∀𝑧𝑡𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦))
4039anim2i 617 . . . . . . . 8 ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)) → (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑧𝑡𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦)))
41 foov 7607 . . . . . . . 8 (𝐺:(𝑡 × 𝑡)–onto𝑡 ↔ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑧𝑡𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦)))
4240, 41sylibr 234 . . . . . . 7 ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)) → 𝐺:(𝑡 × 𝑡)–onto𝑡)
43 forn 6824 . . . . . . . 8 (𝐺:(𝑡 × 𝑡)–onto𝑡 → ran 𝐺 = 𝑡)
4443eqcomd 2741 . . . . . . 7 (𝐺:(𝑡 × 𝑡)–onto𝑡𝑡 = ran 𝐺)
4542, 44syl 17 . . . . . 6 ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)) → 𝑡 = ran 𝐺)
46453adant2 1130 . . . . 5 ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)) → 𝑡 = ran 𝐺)
4746pm4.71ri 560 . . . 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))))
4847exbii 1845 . . 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))))
4923, 48bitrdi 287 . 2 (𝐺𝐴 → (𝐺 ∈ GrpOp ↔ ∃𝑡(𝑡 = ran 𝐺 ∧ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)))))
50 rnexg 7925 . . 3 (𝐺𝐴 → ran 𝐺 ∈ V)
51 isgrp.1 . . . . . 6 𝑋 = ran 𝐺
5251eqeq2i 2748 . . . . 5 (𝑡 = 𝑋𝑡 = ran 𝐺)
53 xpeq1 5703 . . . . . . . . 9 (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑡))
54 xpeq2 5710 . . . . . . . . 9 (𝑡 = 𝑋 → (𝑋 × 𝑡) = (𝑋 × 𝑋))
5553, 54eqtrd 2775 . . . . . . . 8 (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑋))
5655feq2d 6723 . . . . . . 7 (𝑡 = 𝑋 → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑋 × 𝑋)⟶𝑡))
57 feq3 6719 . . . . . . 7 (𝑡 = 𝑋 → (𝐺:(𝑋 × 𝑋)⟶𝑡𝐺:(𝑋 × 𝑋)⟶𝑋))
5856, 57bitrd 279 . . . . . 6 (𝑡 = 𝑋 → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑋 × 𝑋)⟶𝑋))
59 raleq 3321 . . . . . . . 8 (𝑡 = 𝑋 → (∀𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
6059raleqbi1dv 3336 . . . . . . 7 (𝑡 = 𝑋 → (∀𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
6160raleqbi1dv 3336 . . . . . 6 (𝑡 = 𝑋 → (∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
62 rexeq 3320 . . . . . . . . 9 (𝑡 = 𝑋 → (∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))
6362anbi2d 630 . . . . . . . 8 (𝑡 = 𝑋 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
6463raleqbi1dv 3336 . . . . . . 7 (𝑡 = 𝑋 → (∀𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
6564rexeqbi1dv 3337 . . . . . 6 (𝑡 = 𝑋 → (∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
6658, 61, 653anbi123d 1435 . . . . 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))))
6752, 66sylbir 235 . . . 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))))
6867ceqsexgv 3654 . . 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))))
6950, 68syl 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))))
7049, 69bitrd 279 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 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  Vcvv 3478   × cxp 5687  ran crn 5690  wf 6559  ontowfo 6561  (class class class)co 7431  GrpOpcgr 30518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-ov 7434  df-grpo 30522
This theorem is referenced by:  isgrpoi  30527  grpofo  30528  grpolidinv  30530  grpoass  30532  grpomndo  37862  isgrpda  37942
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