Theorem grpoidinv2 30367

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 30365	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → 𝑈 = (℩𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥))
4	1	grpoideu 30361	. . . . . . 7 ⊢ (𝐺 ∈ GrpOp → ∃!𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
5		riotacl2 7388	. . . . . . 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 2825	. . . . 5 ⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ {𝑢 ∈ 𝑋 ∣ ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥})
8		simpll 765	. . . . . . . . . . 11 ⊢ ((((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → (𝑢𝐺𝑥) = 𝑥)
9	8	ralimi 3073	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → ∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥)
10	9	rgenw 3055	. . . . . . . . 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 30360	. . . . . . . 8 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
13	11, 12, 4	3jca 1125	. . . . . . 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 4316	. . . . . . 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 578	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 (𝑢𝐺𝑥) = 𝑥 ↔ ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))))
16	15	rabbidva 3426	. . . . 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 2827	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ {𝑢 ∈ 𝑋 ∣ ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))})
18		oveq1 7422	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
19	18	eqeq1d 2727	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
20		oveq2 7423	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → (𝑥𝐺𝑢) = (𝑥𝐺𝑈))
21	20	eqeq1d 2727	. . . . . . . 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 2737	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → ((𝑦𝐺𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑈))
24		eqeq2 2737	. . . . . . . . 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 3169	. . . . . . 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 3168	. . . . 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 3675	. . . 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 217	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝑈 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈))))
31	30	simprd 494	. 2 ⊢ (𝐺 ∈ GrpOp → ∀𝑥 ∈ 𝑋 (((𝑈𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑈) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑈 ∧ (𝑥𝐺𝑦) = 𝑈)))
32		oveq2 7423	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑈𝐺𝑥) = (𝑈𝐺𝐴))
33		id 22	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
34	32, 33	eqeq12d 2741	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑈𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝐴) = 𝐴))
35		oveq1 7422	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑈) = (𝐴𝐺𝑈))
36	35, 33	eqeq12d 2741	. . . . 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 7423	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴))
39	38	eqeq1d 2727	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑦𝐺𝐴) = 𝑈))
40		oveq1 7422	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
41	40	eqeq1d 2727	. . . . . 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 3169	. . . 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 3601	. 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 578	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3051  ∃wrex 3060  ∃!wreu 3362  {crab 3419  ran crn 5673  ‘cfv 6542  ℩crio 7370  (class class class)co 7415  GrpOpcgr 30341  GIdcgi 30342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-riota 7371  df-ov 7418  df-grpo 30345  df-gid 30346

This theorem is referenced by:  grpolid  30368  grporid  30369  grporcan  30370  grpoinveu  30371  grpoinv  30377