Theorem isgrpo 26735

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 𝑔 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		feq1 5939	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(𝑡 × 𝑡)⟶𝑡))
2		oveq 6555	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝑔𝑦)𝐺𝑧))
3		oveq 6555	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
4	3	oveq1d 6564	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝐺𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
5	2, 4	eqtrd 2644	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
6		oveq 6555	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝑔𝑧)))
7		oveq 6555	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
8	7	oveq2d 6565	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥𝐺(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
9	6, 8	eqtrd 2644	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
10	5, 9	eqeq12d 2625	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
11	10	ralbidv 2969	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (∀𝑧 ∈ 𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
12	11	2ralbidv 2972	. . . . . 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 6555	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑢𝑔𝑥) = (𝑢𝐺𝑥))
14	13	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑢𝑔𝑥) = 𝑥 ↔ (𝑢𝐺𝑥) = 𝑥))
15		oveq 6555	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥))
16	15	eqeq1d 2612	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((𝑦𝑔𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑢))
17	16	rexbidv 3034	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (∃𝑦 ∈ 𝑡 (𝑦𝑔𝑥) = 𝑢 ↔ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢))
18	14, 17	anbi12d 743	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝑔𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢)))
19	18	rexralbidv 3040	. . . . . 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 1391	. . . . 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 1837	. . . 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 26731	. . . 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 3322	. . 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 472	. . . . . . . . . . . . . 14 ⊢ (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) → (𝑢𝐺𝑥) = 𝑥)
25	24	ralimi 2936	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) → ∀𝑥 ∈ 𝑡 (𝑢𝐺𝑥) = 𝑥)
26		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (𝑢𝐺𝑥) = (𝑢𝐺𝑧))
27		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
28	26, 27	eqeq12d 2625	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑢𝐺𝑧) = 𝑧))
29		eqcom 2617	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢𝐺𝑧) = 𝑧 ↔ 𝑧 = (𝑢𝐺𝑧))
30	28, 29	syl6bb 275	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ((𝑢𝐺𝑥) = 𝑥 ↔ 𝑧 = (𝑢𝐺𝑧)))
31	30	rspcv 3278	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑡 → (∀𝑥 ∈ 𝑡 (𝑢𝐺𝑥) = 𝑥 → 𝑧 = (𝑢𝐺𝑧)))
32		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝑢𝐺𝑦) = (𝑢𝐺𝑧))
33	32	eqeq2d 2620	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → (𝑧 = (𝑢𝐺𝑦) ↔ 𝑧 = (𝑢𝐺𝑧)))
34	33	rspcev 3282	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑡 ∧ 𝑧 = (𝑢𝐺𝑧)) → ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦))
35	34	ex 449	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑡 → (𝑧 = (𝑢𝐺𝑧) → ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦)))
36	31, 35	syld 46	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑡 → (∀𝑥 ∈ 𝑡 (𝑢𝐺𝑥) = 𝑥 → ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦)))
37	25, 36	syl5 33	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑡 → (∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) → ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦)))
38	37	reximdv 2999	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑡 → (∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) → ∃𝑢 ∈ 𝑡 ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦)))
39	38	impcom 445	. . . . . . . . . 10 ⊢ ((∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) ∧ 𝑧 ∈ 𝑡) → ∃𝑢 ∈ 𝑡 ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦))
40	39	ralrimiva 2949	. . . . . . . . 9 ⊢ (∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) → ∀𝑧 ∈ 𝑡 ∃𝑢 ∈ 𝑡 ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦))
41	40	anim2i 591	. . . . . . . 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)))
42		foov 6706	. . . . . . . 8 ⊢ (𝐺:(𝑡 × 𝑡)–onto→𝑡 ↔ (𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑧 ∈ 𝑡 ∃𝑢 ∈ 𝑡 ∃𝑦 ∈ 𝑡 𝑧 = (𝑢𝐺𝑦)))
43	41, 42	sylibr 223	. . . . . . 7 ⊢ ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢)) → 𝐺:(𝑡 × 𝑡)–onto→𝑡)
44		forn 6031	. . . . . . . 8 ⊢ (𝐺:(𝑡 × 𝑡)–onto→𝑡 → ran 𝐺 = 𝑡)
45	44	eqcomd 2616	. . . . . . 7 ⊢ (𝐺:(𝑡 × 𝑡)–onto→𝑡 → 𝑡 = ran 𝐺)
46	43, 45	syl 17	. . . . . 6 ⊢ ((𝐺:(𝑡 × 𝑡)⟶𝑡 ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢)) → 𝑡 = ran 𝐺)
47	46	3adant2 1073	. . . . 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)
48	47	pm4.71ri 663	. . . 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))))
49	48	exbii 1764	. . 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))))
50	23, 49	syl6bb 275	. 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)))))
51		rnexg 6990	. . 3 ⊢ (𝐺 ∈ 𝐴 → ran 𝐺 ∈ V)
52		isgrp.1	. . . . . 6 ⊢ 𝑋 = ran 𝐺
53	52	eqeq2i 2622	. . . . 5 ⊢ (𝑡 = 𝑋 ↔ 𝑡 = ran 𝐺)
54		xpeq1 5052	. . . . . . . . 9 ⊢ (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑡))
55		xpeq2 5053	. . . . . . . . 9 ⊢ (𝑡 = 𝑋 → (𝑋 × 𝑡) = (𝑋 × 𝑋))
56	54, 55	eqtrd 2644	. . . . . . . 8 ⊢ (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑋))
57	56	feq2d 5944	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(𝑋 × 𝑋)⟶𝑡))
58		feq3 5941	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (𝐺:(𝑋 × 𝑋)⟶𝑡 ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
59	57, 58	bitrd 267	. . . . . 6 ⊢ (𝑡 = 𝑋 → (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
60		raleq 3115	. . . . . . . 8 ⊢ (𝑡 = 𝑋 → (∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
61	60	raleqbi1dv 3123	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
62	61	raleqbi1dv 3123	. . . . . 6 ⊢ (𝑡 = 𝑋 → (∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
63		rexeq 3116	. . . . . . . . 9 ⊢ (𝑡 = 𝑋 → (∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))
64	63	anbi2d 736	. . . . . . . 8 ⊢ (𝑡 = 𝑋 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))
65	64	raleqbi1dv 3123	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))
66	65	rexeqbi1dv 3124	. . . . . 6 ⊢ (𝑡 = 𝑋 → (∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝐺𝑥) = 𝑢) ↔ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))
67	59, 62, 66	3anbi123d 1391	. . . . 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))))
68	53, 67	sylbir 224	. . . 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))))
69	68	ceqsexgv 3305	. . 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))))
70	51, 69	syl 17	. 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))))
71	50, 70	bitrd 267	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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   × cxp 5036  ran crn 5039  ⟶wf 5800  –onto→wfo 5802  (class class class)co 6549  GrpOpcgr 26727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-ov 6552  df-grpo 26731

This theorem is referenced by:  isgrpoi  26736  grpofo  26737  grpolidinv  26739  grpoass  26741  grpomndo  32844  isgrpda  32924