Theorem isgrpo 30886

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	⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ GrpOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (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.

Step	Hyp	Ref	Expression
1		feq1 6690	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(𝑡 × 𝑡)⟶𝑡))
2		oveq 7429	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝑔𝑦)𝐺𝑧))
3		oveq 7429	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
4	3	oveq1d 7438	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝐺𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
5	2, 4	eqtrd 2801	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
6		oveq 7429	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝑔𝑧)))
7		oveq 7429	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
8	7	oveq2d 7439	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥𝐺(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
9	6, 8	eqtrd 2801	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
10	5, 9	eqeq12d 2782	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
11	10	ralbidv 3191	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (∀𝑧 ∈ 𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
12	11	2ralbidv 3232	. . . . . 6 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
13		oveq 7429	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑢𝑔𝑥) = (𝑢𝐺𝑥))
14	13	eqeq1d 2768	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑢𝑔𝑥) = 𝑥 ↔ (𝑢𝐺𝑥) = 𝑥))
15		oveq 7429	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
16	15	eqeq1d 2768	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑢))
17	16	rexbidv 3192	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (∃𝑦 ∈ 𝑡 (𝑦𝑔𝑥) = 𝑢 ↔ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢))
18	14, 17	anbi12d 644	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝑔𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢)))
19	18	rexralbidv 3234	. . . . . 6 ⊢ (𝑔 = 𝐺 → (∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝑔𝑥) = 𝑢) ↔ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢)))
20	1, 12, 19	3anbi123d 1464	. . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑔:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝑔𝑥) = 𝑢)) ↔ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢))))
21	20	exbidv 1954	. . . 4 ⊢ (𝑔 = 𝐺 → (∃𝑡(𝑔:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝑔𝑥) = 𝑢)) ↔ ∃𝑡(𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢))))
22		df-grpo 30882	. . . 4 ⊢ GrpOp = {𝑔 ∣ ∃𝑡(𝑔:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝑔𝑥) = 𝑢))}
23	21, 22	elab2g 3642	. . 3 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ GrpOp ↔ ∃𝑡(𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢))))
24		simpl 488	. . . . . . . . . . . . . 14 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) → (𝑢𝐺𝑥) = 𝑥)
25	24	ralimi 3105	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) → ∀𝑥 ∈ 𝑡 (𝑢𝐺𝑥) = 𝑥)
26		oveq2 7431	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (𝑢𝐺𝑥) = (𝑢𝐺𝑧))
27		id 23	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
28	26, 27	eqeq12d 2782	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑧) = 𝑧))
29		eqcom 2773	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢𝐺𝑧) = 𝑧 ↔ 𝑧 = (𝑢𝐺𝑧))
30	28, 29	bitrdi 290	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ((𝑢𝐺𝑥) = 𝑥 ↔ 𝑧 = (𝑢𝐺𝑧)))
31	30	rspcv 3580	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑡 → (∀𝑥 ∈ 𝑡 (𝑢𝐺𝑥) = 𝑥 → 𝑧 = (𝑢𝐺𝑧)))
32		oveq2 7431	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → (𝑢𝐺𝑦) = (𝑢𝐺𝑧))
33	32	rspceeqv 3607	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑡 ∧ 𝑧 = (𝑢𝐺𝑧)) → ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦))
34	33	ex 418	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑡 → (𝑧 = (𝑢𝐺𝑧) → ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦)))
35	31, 34	syld 48	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑡 → (∀𝑥 ∈ 𝑡 (𝑢𝐺𝑥) = 𝑥 → ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦)))
36	25, 35	syl5 35	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑡 → (∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) → ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦)))
37	36	reximdv 3183	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑡 → (∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) → ∃𝑢 ∈ 𝑡 ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦)))
38	37	impcom 413	. . . . . . . . . 10 ⊢ ((∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) ∧ 𝑧 ∈ 𝑡) → ∃𝑢 ∈ 𝑡 ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦))
39	38	ralrimiva 3160	. . . . . . . . 9 ⊢ (∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) → ∀𝑧 ∈ 𝑡 ∃𝑢 ∈ 𝑡 ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦))
40	39	anim2i 629	. . . . . . . 8 ⊢ ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢)) → (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑧 ∈ 𝑡 ∃𝑢 ∈ 𝑡 ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦)))
41		foov 7597	. . . . . . . 8 ⊢ (𝐺:(𝑡 × 𝑡)–onto→𝑡 ↔ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑧 ∈ 𝑡 ∃𝑢 ∈ 𝑡 ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦)))
42	40, 41	sylibr 237	. . . . . . 7 ⊢ ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢)) → 𝐺:(𝑡 × 𝑡)–onto→𝑡)
43		forn 6802	. . . . . . . 8 ⊢ (𝐺:(𝑡 × 𝑡)–onto→𝑡 → ran 𝐺 = 𝑡)
44	43	eqcomd 2772	. . . . . . 7 ⊢ (𝐺:(𝑡 × 𝑡)–onto→𝑡 → 𝑡 = ran 𝐺)
45	42, 44	syl 18	. . . . . 6 ⊢ ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢)) → 𝑡 = ran 𝐺)
46	45	3adant2 1149	. . . . 5 ⊢ ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢)) → 𝑡 = ran 𝐺)
47	46	pm4.71ri 570	. . . 4 ⊢ ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢)) ↔ (𝑡 = ran 𝐺 ∧ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢))))
48	47	exbii 1881	. . 3 ⊢ (∃𝑡(𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢)) ↔ ∃𝑡(𝑡 = ran 𝐺 ∧ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢))))
49	23, 48	bitrdi 290	. 2 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ GrpOp ↔ ∃𝑡(𝑡 = ran 𝐺 ∧ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢)))))
50		rnexg 7908	. . 3 ⊢ (𝐺 ∈ 𝐴 → ran 𝐺 ∈ V)
51		isgrp.1	. . . . . 6 ⊢ 𝑋 = ran 𝐺
52	51	eqeq2i 2779	. . . . 5 ⊢ (𝑡 = 𝑋 ↔ 𝑡 = ran 𝐺)
53		xpeq1 5680	. . . . . . . . 9 ⊢ (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑡))
54		xpeq2 5687	. . . . . . . . 9 ⊢ (𝑡 = 𝑋 → (𝑋 × 𝑡) = (𝑋 × 𝑋))
55	53, 54	eqtrd 2801	. . . . . . . 8 ⊢ (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑋))
56	55	feq2d 6696	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(𝑋 × 𝑋)⟶𝑡))
57		feq3 6692	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (𝐺:(𝑋 × 𝑋)⟶𝑡 ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
58	56, 57	bitrd 282	. . . . . 6 ⊢ (𝑡 = 𝑋 → (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
59		raleq 3323	. . . . . . . 8 ⊢ (𝑡 = 𝑋 → (∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
60	59	raleqbi1dv 3336	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
61	60	raleqbi1dv 3336	. . . . . 6 ⊢ (𝑡 = 𝑋 → (∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
62		rexeq 3322	. . . . . . . . 9 ⊢ (𝑡 = 𝑋 → (∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))
63	62	anbi2d 642	. . . . . . . 8 ⊢ (𝑡 = 𝑋 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))
64	63	raleqbi1dv 3336	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))
65	64	rexeqbi1dv 3337	. . . . . 6 ⊢ (𝑡 = 𝑋 → (∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))
66	58, 61, 65	3anbi123d 1464	. . . . 5 ⊢ (𝑡 = 𝑋 → ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢)) ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))))
67	52, 66	sylbir 238	. . . 4 ⊢ (𝑡 = ran 𝐺 → ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢)) ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))))
68	67	ceqsexgv 3616	. . 3 ⊢ (ran 𝐺 ∈ V → (∃𝑡(𝑡 = ran 𝐺 ∧ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢))) ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))))
69	50, 68	syl 18	. 2 ⊢ (𝐺 ∈ 𝐴 → (∃𝑡(𝑡 = ran 𝐺 ∧ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢))) ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))))
70	49, 69	bitrd 282	1 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ GrpOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092  Vcvv 3458   × cxp 5664  ran crn 5667  ⟶wf 6539  –onto→wfo 6541  (class class class)co 7423  GrpOpcgr 30878

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-ov 7426  df-grpo 30882

This theorem is used by:  isgrpoi  30887  grpofo  30888  grpolidinv  30890  grpoass  30892  grpomndo  38567  isgrpda  38647