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Theorem isgrpo 27808
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 6204 . . . . . 6 (𝑔 = 𝐺 → (𝑔:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑡 × 𝑡)⟶𝑡))
2 oveq 6848 . . . . . . . . . 10 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝑔𝑦)𝐺𝑧))
3 oveq 6848 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
43oveq1d 6857 . . . . . . . . . 10 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝐺𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
52, 4eqtrd 2799 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
6 oveq 6848 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝑔𝑧)))
7 oveq 6848 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
87oveq2d 6858 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥𝐺(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
96, 8eqtrd 2799 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
105, 9eqeq12d 2780 . . . . . . . 8 (𝑔 = 𝐺 → (((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
1110ralbidv 3133 . . . . . . 7 (𝑔 = 𝐺 → (∀𝑧𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
12112ralbidv 3136 . . . . . 6 (𝑔 = 𝐺 → (∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
13 oveq 6848 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑢𝑔𝑥) = (𝑢𝐺𝑥))
1413eqeq1d 2767 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑢𝑔𝑥) = 𝑥 ↔ (𝑢𝐺𝑥) = 𝑥))
15 oveq 6848 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
1615eqeq1d 2767 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑢))
1716rexbidv 3199 . . . . . . . 8 (𝑔 = 𝐺 → (∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢 ↔ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢))
1814, 17anbi12d 624 . . . . . . 7 (𝑔 = 𝐺 → (((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)))
1918rexralbidv 3205 . . . . . 6 (𝑔 = 𝐺 → (∃𝑢𝑡𝑥𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢) ↔ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)))
201, 12, 193anbi123d 1560 . . . . 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 2016 . . . 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 27804 . . . 4 GrpOp = {𝑔 ∣ ∃𝑡(𝑔:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢))}
2321, 22elab2g 3508 . . 3 (𝐺𝐴 → (𝐺 ∈ GrpOp ↔ ∃𝑡(𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢))))
24 simpl 474 . . . . . . . . . . . . . 14 (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → (𝑢𝐺𝑥) = 𝑥)
2524ralimi 3099 . . . . . . . . . . . . 13 (∀𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → ∀𝑥𝑡 (𝑢𝐺𝑥) = 𝑥)
26 oveq2 6850 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑢𝐺𝑥) = (𝑢𝐺𝑧))
27 id 22 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧𝑥 = 𝑧)
2826, 27eqeq12d 2780 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑧) = 𝑧))
29 eqcom 2772 . . . . . . . . . . . . . . . 16 ((𝑢𝐺𝑧) = 𝑧𝑧 = (𝑢𝐺𝑧))
3028, 29syl6bb 278 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ((𝑢𝐺𝑥) = 𝑥𝑧 = (𝑢𝐺𝑧)))
3130rspcv 3457 . . . . . . . . . . . . . 14 (𝑧𝑡 → (∀𝑥𝑡 (𝑢𝐺𝑥) = 𝑥𝑧 = (𝑢𝐺𝑧)))
32 oveq2 6850 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (𝑢𝐺𝑦) = (𝑢𝐺𝑧))
3332rspceeqv 3479 . . . . . . . . . . . . . . 15 ((𝑧𝑡𝑧 = (𝑢𝐺𝑧)) → ∃𝑦𝑡 𝑧 = (𝑢𝐺𝑦))
3433ex 401 . . . . . . . . . . . . . 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 3162 . . . . . . . . . . 11 (𝑧𝑡 → (∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → ∃𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦)))
3837impcom 396 . . . . . . . . . 10 ((∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) ∧ 𝑧𝑡) → ∃𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦))
3938ralrimiva 3113 . . . . . . . . 9 (∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → ∀𝑧𝑡𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦))
4039anim2i 610 . . . . . . . 8 ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)) → (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑧𝑡𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦)))
41 foov 7006 . . . . . . . 8 (𝐺:(𝑡 × 𝑡)–onto𝑡 ↔ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑧𝑡𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦)))
4240, 41sylibr 225 . . . . . . 7 ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)) → 𝐺:(𝑡 × 𝑡)–onto𝑡)
43 forn 6301 . . . . . . . 8 (𝐺:(𝑡 × 𝑡)–onto𝑡 → ran 𝐺 = 𝑡)
4443eqcomd 2771 . . . . . . 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 1161 . . . . 5 ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)) → 𝑡 = ran 𝐺)
4746pm4.71ri 556 . . . 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 1943 . . 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, 48syl6bb 278 . 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 7296 . . 3 (𝐺𝐴 → ran 𝐺 ∈ V)
51 isgrp.1 . . . . . 6 𝑋 = ran 𝐺
5251eqeq2i 2777 . . . . 5 (𝑡 = 𝑋𝑡 = ran 𝐺)
53 xpeq1 5291 . . . . . . . . 9 (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑡))
54 xpeq2 5298 . . . . . . . . 9 (𝑡 = 𝑋 → (𝑋 × 𝑡) = (𝑋 × 𝑋))
5553, 54eqtrd 2799 . . . . . . . 8 (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑋))
5655feq2d 6209 . . . . . . 7 (𝑡 = 𝑋 → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑋 × 𝑋)⟶𝑡))
57 feq3 6206 . . . . . . 7 (𝑡 = 𝑋 → (𝐺:(𝑋 × 𝑋)⟶𝑡𝐺:(𝑋 × 𝑋)⟶𝑋))
5856, 57bitrd 270 . . . . . 6 (𝑡 = 𝑋 → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑋 × 𝑋)⟶𝑋))
59 raleq 3286 . . . . . . . 8 (𝑡 = 𝑋 → (∀𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
6059raleqbi1dv 3294 . . . . . . 7 (𝑡 = 𝑋 → (∀𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
6160raleqbi1dv 3294 . . . . . 6 (𝑡 = 𝑋 → (∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
62 rexeq 3287 . . . . . . . . 9 (𝑡 = 𝑋 → (∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))
6362anbi2d 622 . . . . . . . 8 (𝑡 = 𝑋 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
6463raleqbi1dv 3294 . . . . . . 7 (𝑡 = 𝑋 → (∀𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
6564rexeqbi1dv 3295 . . . . . 6 (𝑡 = 𝑋 → (∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
6658, 61, 653anbi123d 1560 . . . . 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 226 . . . 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 3488 . . 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 270 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 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wral 3055  wrex 3056  Vcvv 3350   × cxp 5275  ran crn 5278  wf 6064  ontowfo 6066  (class class class)co 6842  GrpOpcgr 27800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fo 6074  df-fv 6076  df-ov 6845  df-grpo 27804
This theorem is referenced by:  isgrpoi  27809  grpofo  27810  grpolidinv  27812  grpoass  27814  grpomndo  34096  isgrpda  34176
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