Theorem isgrpo 30758

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 6673	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(𝑡 × 𝑡)⟶𝑡))
2		oveq 7406	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝑔𝑦)𝐺𝑧))
3		oveq 7406	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
4	3	oveq1d 7415	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝐺𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
5	2, 4	eqtrd 2800	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
6		oveq 7406	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝑔𝑧)))
7		oveq 7406	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
8	7	oveq2d 7416	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥𝐺(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
9	6, 8	eqtrd 2800	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
10	5, 9	eqeq12d 2781	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
11	10	ralbidv 3188	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (∀𝑧 ∈ 𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
12	11	2ralbidv 3229	. . . . . 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 7406	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑢𝑔𝑥) = (𝑢𝐺𝑥))
14	13	eqeq1d 2767	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑢𝑔𝑥) = 𝑥 ↔ (𝑢𝐺𝑥) = 𝑥))
15		oveq 7406	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
16	15	eqeq1d 2767	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑢))
17	16	rexbidv 3189	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (∃𝑦 ∈ 𝑡 (𝑦𝑔𝑥) = 𝑢 ↔ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢))
18	14, 17	anbi12d 643	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝑔𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢)))
19	18	rexralbidv 3231	. . . . . 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 1460	. . . . 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 1944	. . . 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 30754	. . . 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 487	. . . . . . . . . . . . . 14 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) → (𝑢𝐺𝑥) = 𝑥)
25	24	ralimi 3102	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) → ∀𝑥 ∈ 𝑡 (𝑢𝐺𝑥) = 𝑥)
26		oveq2 7408	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (𝑢𝐺𝑥) = (𝑢𝐺𝑧))
27		id 23	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
28	26, 27	eqeq12d 2781	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑧) = 𝑧))
29		eqcom 2772	. . . . . . . . . . . . . . . 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 7408	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → (𝑢𝐺𝑦) = (𝑢𝐺𝑧))
33	32	rspceeqv 3607	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑡 ∧ 𝑧 = (𝑢𝐺𝑧)) → ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦))
34	33	ex 417	. . . . . . . . . . . . . 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 3180	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑡 → (∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) → ∃𝑢 ∈ 𝑡 ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦)))
38	37	impcom 412	. . . . . . . . . 10 ⊢ ((∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) ∧ 𝑧 ∈ 𝑡) → ∃𝑢 ∈ 𝑡 ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦))
39	38	ralrimiva 3157	. . . . . . . . 9 ⊢ (∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) → ∀𝑧 ∈ 𝑡 ∃𝑢 ∈ 𝑡 ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦))
40	39	anim2i 628	. . . . . . . 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 7574	. . . . . . . 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 6785	. . . . . . . 8 ⊢ (𝐺:(𝑡 × 𝑡)–onto→𝑡 → ran 𝐺 = 𝑡)
44	43	eqcomd 2771	. . . . . . 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 1147	. . . . 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 569	. . . 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 1871	. . 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 7887	. . 3 ⊢ (𝐺 ∈ 𝐴 → ran 𝐺 ∈ V)
51		isgrp.1	. . . . . 6 ⊢ 𝑋 = ran 𝐺
52	51	eqeq2i 2778	. . . . 5 ⊢ (𝑡 = 𝑋 ↔ 𝑡 = ran 𝐺)
53		xpeq1 5666	. . . . . . . . 9 ⊢ (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑡))
54		xpeq2 5673	. . . . . . . . 9 ⊢ (𝑡 = 𝑋 → (𝑋 × 𝑡) = (𝑋 × 𝑋))
55	53, 54	eqtrd 2800	. . . . . . . 8 ⊢ (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑋))
56	55	feq2d 6679	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(𝑋 × 𝑋)⟶𝑡))
57		feq3 6675	. . . . . . 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 3320	. . . . . . . 8 ⊢ (𝑡 = 𝑋 → (∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
60	59	raleqbi1dv 3333	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
61	60	raleqbi1dv 3333	. . . . . 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 3319	. . . . . . . . 9 ⊢ (𝑡 = 𝑋 → (∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))
63	62	anbi2d 641	. . . . . . . 8 ⊢ (𝑡 = 𝑋 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))
64	63	raleqbi1dv 3333	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))
65	64	rexeqbi1dv 3334	. . . . . 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 1460	. . . . 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))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  Vcvv 3457   × cxp 5650  ran crn 5653  ⟶wf 6521  –onto→wfo 6523  (class class class)co 7400  GrpOpcgr 30750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fo 6531  df-fv 6533  df-ov 7403  df-grpo 30754

This theorem is referenced by:  isgrpoi  30759  grpofo  30760  grpolidinv  30762  grpoass  30764  grpomndo  38386  isgrpda  38466