Theorem grpoidinv2 27983

Description: A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpoidval.1	⊢ 𝑋 = ran 𝐺
grpoidval.2	⊢ 𝑈 = (GId‘𝐺)

Assertion

Ref	Expression
grpoidinv2	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐺   𝑦,𝑈   𝑦,𝑋

Proof of Theorem grpoidinv2

Dummy variables 𝑥 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpoidval.1	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
2		grpoidval.2	. . . . . . 7 ⊢ 𝑈 = (GId‘𝐺)
3	1, 2	grpoidval 27981	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥))
4	1	grpoideu 27977	. . . . . . 7 ⊢ (𝐺 ∈ GrpOp → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
5		riotacl2 6990	. . . . . . 7 ⊢ (∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 → (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥) ∈ {𝑢 ∈ 𝑋 ∣ ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥})
6	4, 5	syl 17	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥) ∈ {𝑢 ∈ 𝑋 ∣ ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥})
7	3, 6	eqeltrd 2883	. . . . 5 ⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ {𝑢 ∈ 𝑋 ∣ ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥})
8		simpll 763	. . . . . . . . . . 11 ⊢ ((((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → (𝑢𝐺𝑥) = 𝑥)
9	8	ralimi 3127	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
10	9	rgenw 3117	. . . . . . . . 9 ⊢ ∀𝑢 ∈ 𝑋 (∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
11	10	a1i 11	. . . . . . . 8 ⊢ (𝐺 ∈ GrpOp → ∀𝑢 ∈ 𝑋 (∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥))
12	1	grpoidinv 27976	. . . . . . . 8 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
13	11, 12, 4	3jca 1121	. . . . . . 7 ⊢ (𝐺 ∈ GrpOp → (∀𝑢 ∈ 𝑋 (∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) ∧ ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥))
14		reupick2 4209	. . . . . . 7 ⊢ (((∀𝑢 ∈ 𝑋 (∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) ∧ ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥) ∧ 𝑢 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ↔ ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))))
15	13, 14	sylan 580	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ↔ ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))))
16	15	rabbidva 3424	. . . . 5 ⊢ (𝐺 ∈ GrpOp → {𝑢 ∈ 𝑋 ∣ ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥} = {𝑢 ∈ 𝑋 ∣ ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))})
17	7, 16	eleqtrd 2885	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ {𝑢 ∈ 𝑋 ∣ ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))})
18		oveq1 7023	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
19	18	eqeq1d 2797	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
20		oveq2 7024	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → (𝑥𝐺𝑢) = (𝑥𝐺𝑈))
21	20	eqeq1d 2797	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → ((𝑥𝐺𝑢) = 𝑥 ↔ (𝑥𝐺𝑈) = 𝑥))
22	19, 21	anbi12d 630	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥)))
23		eqeq2 2806	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → ((𝑦𝐺𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑈))
24		eqeq2 2806	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → ((𝑥𝐺𝑦) = 𝑢 ↔ (𝑥𝐺𝑦) = 𝑈))
25	23, 24	anbi12d 630	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢) ↔ ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈)))
26	25	rexbidv 3260	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢) ↔ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈)))
27	22, 26	anbi12d 630	. . . . . 6 ⊢ (𝑢 = 𝑈 → ((((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) ↔ (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈))))
28	27	ralbidv 3164	. . . . 5 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) ↔ ∀𝑥 ∈ 𝑋 (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈))))
29	28	elrab 3618	. . . 4 ⊢ (𝑈 ∈ {𝑢 ∈ 𝑋 ∣ ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))} ↔ (𝑈 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈))))
30	17, 29	sylib 219	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝑈 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈))))
31	30	simprd 496	. 2 ⊢ (𝐺 ∈ GrpOp → ∀𝑥 ∈ 𝑋 (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈)))
32		oveq2 7024	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑈𝐺𝑥) = (𝑈𝐺𝐴))
33		id 22	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
34	32, 33	eqeq12d 2810	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑈𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝐴) = 𝐴))
35		oveq1 7023	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑈) = (𝐴𝐺𝑈))
36	35, 33	eqeq12d 2810	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑈) = 𝑥 ↔ (𝐴𝐺𝑈) = 𝐴))
37	34, 36	anbi12d 630	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ↔ ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴)))
38		oveq2 7024	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴))
39	38	eqeq1d 2797	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑦𝐺𝐴) = 𝑈))
40		oveq1 7023	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
41	40	eqeq1d 2797	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦) = 𝑈 ↔ (𝐴𝐺𝑦) = 𝑈))
42	39, 41	anbi12d 630	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈) ↔ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
43	42	rexbidv 3260	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈) ↔ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
44	37, 43	anbi12d 630	. . 3 ⊢ (𝑥 = 𝐴 → ((((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈)) ↔ (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈))))
45	44	rspccva 3558	. 2 ⊢ ((∀𝑥 ∈ 𝑋 (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈)) ∧ 𝐴 ∈ 𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
46	31, 45	sylan 580	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  ∀wral 3105  ∃wrex 3106  ∃!wreu 3107  {crab 3109  ran crn 5444  ‘cfv 6225  ℩crio 6976  (class class class)co 7016  GrpOpcgr 27957  GIdcgi 27958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fo 6231  df-fv 6233  df-riota 6977  df-ov 7019  df-grpo 27961  df-gid 27962

This theorem is referenced by:  grpolid  27984  grporid  27985  grporcan  27986  grpoinveu  27987  grpoinv  27993