MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isgrpo Structured version   Visualization version   GIF version

Theorem isgrpo 27321
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 6013 . . . . . 6 (𝑔 = 𝐺 → (𝑔:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑡 × 𝑡)⟶𝑡))
2 oveq 6641 . . . . . . . . . 10 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝑔𝑦)𝐺𝑧))
3 oveq 6641 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
43oveq1d 6650 . . . . . . . . . 10 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝐺𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
52, 4eqtrd 2654 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
6 oveq 6641 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝑔𝑧)))
7 oveq 6641 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
87oveq2d 6651 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥𝐺(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
96, 8eqtrd 2654 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
105, 9eqeq12d 2635 . . . . . . . 8 (𝑔 = 𝐺 → (((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
1110ralbidv 2983 . . . . . . 7 (𝑔 = 𝐺 → (∀𝑧𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
12112ralbidv 2986 . . . . . 6 (𝑔 = 𝐺 → (∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
13 oveq 6641 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑢𝑔𝑥) = (𝑢𝐺𝑥))
1413eqeq1d 2622 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑢𝑔𝑥) = 𝑥 ↔ (𝑢𝐺𝑥) = 𝑥))
15 oveq 6641 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
1615eqeq1d 2622 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑢))
1716rexbidv 3048 . . . . . . . 8 (𝑔 = 𝐺 → (∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢 ↔ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢))
1814, 17anbi12d 746 . . . . . . 7 (𝑔 = 𝐺 → (((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)))
1918rexralbidv 3054 . . . . . 6 (𝑔 = 𝐺 → (∃𝑢𝑡𝑥𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢) ↔ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)))
201, 12, 193anbi123d 1397 . . . . 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 1848 . . . 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 27317 . . . 4 GrpOp = {𝑔 ∣ ∃𝑡(𝑔:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢))}
2321, 22elab2g 3347 . . 3 (𝐺𝐴 → (𝐺 ∈ GrpOp ↔ ∃𝑡(𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢))))
24 simpl 473 . . . . . . . . . . . . . 14 (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → (𝑢𝐺𝑥) = 𝑥)
2524ralimi 2949 . . . . . . . . . . . . 13 (∀𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → ∀𝑥𝑡 (𝑢𝐺𝑥) = 𝑥)
26 oveq2 6643 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (𝑢𝐺𝑥) = (𝑢𝐺𝑧))
27 id 22 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧𝑥 = 𝑧)
2826, 27eqeq12d 2635 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑧 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑧) = 𝑧))
29 eqcom 2627 . . . . . . . . . . . . . . . 16 ((𝑢𝐺𝑧) = 𝑧𝑧 = (𝑢𝐺𝑧))
3028, 29syl6bb 276 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → ((𝑢𝐺𝑥) = 𝑥𝑧 = (𝑢𝐺𝑧)))
3130rspcv 3300 . . . . . . . . . . . . . 14 (𝑧𝑡 → (∀𝑥𝑡 (𝑢𝐺𝑥) = 𝑥𝑧 = (𝑢𝐺𝑧)))
32 oveq2 6643 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝑢𝐺𝑦) = (𝑢𝐺𝑧))
3332eqeq2d 2630 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (𝑧 = (𝑢𝐺𝑦) ↔ 𝑧 = (𝑢𝐺𝑧)))
3433rspcev 3304 . . . . . . . . . . . . . . 15 ((𝑧𝑡𝑧 = (𝑢𝐺𝑧)) → ∃𝑦𝑡 𝑧 = (𝑢𝐺𝑦))
3534ex 450 . . . . . . . . . . . . . 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 3013 . . . . . . . . . . 11 (𝑧𝑡 → (∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → ∃𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦)))
3938impcom 446 . . . . . . . . . 10 ((∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) ∧ 𝑧𝑡) → ∃𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦))
4039ralrimiva 2963 . . . . . . . . 9 (∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) → ∀𝑧𝑡𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦))
4140anim2i 592 . . . . . . . 8 ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)) → (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑧𝑡𝑢𝑡𝑦𝑡 𝑧 = (𝑢𝐺𝑦)))
42 foov 6793 . . . . . . . 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 6105 . . . . . . . 8 (𝐺:(𝑡 × 𝑡)–onto𝑡 → ran 𝐺 = 𝑡)
4544eqcomd 2626 . . . . . . 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 1078 . . . . 5 ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢)) → 𝑡 = ran 𝐺)
4847pm4.71ri 664 . . . 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 1772 . . 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 7083 . . 3 (𝐺𝐴 → ran 𝐺 ∈ V)
52 isgrp.1 . . . . . 6 𝑋 = ran 𝐺
5352eqeq2i 2632 . . . . 5 (𝑡 = 𝑋𝑡 = ran 𝐺)
54 xpeq1 5118 . . . . . . . . 9 (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑡))
55 xpeq2 5119 . . . . . . . . 9 (𝑡 = 𝑋 → (𝑋 × 𝑡) = (𝑋 × 𝑋))
5654, 55eqtrd 2654 . . . . . . . 8 (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑋))
5756feq2d 6018 . . . . . . 7 (𝑡 = 𝑋 → (𝐺:(𝑡 × 𝑡)⟶𝑡𝐺:(𝑋 × 𝑋)⟶𝑡))
58 feq3 6015 . . . . . . 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 3133 . . . . . . . 8 (𝑡 = 𝑋 → (∀𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
6160raleqbi1dv 3141 . . . . . . 7 (𝑡 = 𝑋 → (∀𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
6261raleqbi1dv 3141 . . . . . 6 (𝑡 = 𝑋 → (∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
63 rexeq 3134 . . . . . . . . 9 (𝑡 = 𝑋 → (∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))
6463anbi2d 739 . . . . . . . 8 (𝑡 = 𝑋 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
6564raleqbi1dv 3141 . . . . . . 7 (𝑡 = 𝑋 → (∀𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
6665rexeqbi1dv 3142 . . . . . 6 (𝑡 = 𝑋 → (∃𝑢𝑡𝑥𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢)))
6759, 62, 663anbi123d 1397 . . . . 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 3329 . . 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 384  w3a 1036   = wceq 1481  wex 1702  wcel 1988  wral 2909  wrex 2910  Vcvv 3195   × cxp 5102  ran crn 5105  wf 5872  ontowfo 5874  (class class class)co 6635  GrpOpcgr 27313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fo 5882  df-fv 5884  df-ov 6638  df-grpo 27317
This theorem is referenced by:  isgrpoi  27322  grpofo  27323  grpolidinv  27325  grpoass  27327  grpomndo  33645  isgrpda  33725
  Copyright terms: Public domain W3C validator