Theorem isgrpoi 27966

Description: Properties that determine a group operation. Read 𝑁 as 𝑁(𝑥). (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
isgrpoi.1	⊢ 𝑋 ∈ V
isgrpoi.2	⊢ 𝐺:(𝑋 × 𝑋)⟶𝑋
isgrpoi.3	⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
isgrpoi.4	⊢ 𝑈 ∈ 𝑋
isgrpoi.5	⊢ (𝑥 ∈ 𝑋 → (𝑈𝐺𝑥) = 𝑥)
isgrpoi.6	⊢ (𝑥 ∈ 𝑋 → 𝑁 ∈ 𝑋)
isgrpoi.7	⊢ (𝑥 ∈ 𝑋 → (𝑁𝐺𝑥) = 𝑈)

Assertion

Ref	Expression
isgrpoi	⊢ 𝐺 ∈ GrpOp

Distinct variable groups:   𝑥,𝑦,𝑧,𝐺   𝑥,𝑈,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑦,𝑁

Allowed substitution hints:   𝑁(𝑥,𝑧)

Proof of Theorem isgrpoi

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isgrpoi.2	. 2 ⊢ 𝐺:(𝑋 × 𝑋)⟶𝑋
2		isgrpoi.3	. . 3 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
3	2	rgen3 3171	. 2 ⊢ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))
4		isgrpoi.4	. . 3 ⊢ 𝑈 ∈ 𝑋
5		isgrpoi.5	. . . . 5 ⊢ (𝑥 ∈ 𝑋 → (𝑈𝐺𝑥) = 𝑥)
6		isgrpoi.6	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → 𝑁 ∈ 𝑋)
7		isgrpoi.7	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → (𝑁𝐺𝑥) = 𝑈)
8		oveq1 7023	. . . . . . . 8 ⊢ (𝑦 = 𝑁 → (𝑦𝐺𝑥) = (𝑁𝐺𝑥))
9	8	eqeq1d 2797	. . . . . . 7 ⊢ (𝑦 = 𝑁 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑁𝐺𝑥) = 𝑈))
10	9	rspcev 3559	. . . . . 6 ⊢ ((𝑁 ∈ 𝑋 ∧ (𝑁𝐺𝑥) = 𝑈) → ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)
11	6, 7, 10	syl2anc 584	. . . . 5 ⊢ (𝑥 ∈ 𝑋 → ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)
12	5, 11	jca 512	. . . 4 ⊢ (𝑥 ∈ 𝑋 → ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
13	12	rgen 3115	. . 3 ⊢ ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)
14		oveq1 7023	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
15	14	eqeq1d 2797	. . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
16		eqeq2 2806	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑦𝐺𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑈))
17	16	rexbidv 3260	. . . . . 6 ⊢ (𝑢 = 𝑈 → (∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
18	15, 17	anbi12d 630	. . . . 5 ⊢ (𝑢 = 𝑈 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
19	18	ralbidv 3164	. . . 4 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
20	19	rspcev 3559	. . 3 ⊢ ((𝑈 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))
21	4, 13, 20	mp2an 688	. 2 ⊢ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)
22		isgrpoi.1	. . . . 5 ⊢ 𝑋 ∈ V
23	22, 22	xpex 7333	. . . 4 ⊢ (𝑋 × 𝑋) ∈ V
24		fex 6855	. . . 4 ⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ (𝑋 × 𝑋) ∈ V) → 𝐺 ∈ V)
25	1, 23, 24	mp2an 688	. . 3 ⊢ 𝐺 ∈ V
26	5	eqcomd 2801	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 → 𝑥 = (𝑈𝐺𝑥))
27		rspceov 7062	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑥 = (𝑈𝐺𝑥)) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
28	4, 27	mp3an1 1440	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 = (𝑈𝐺𝑥)) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
29	26, 28	mpdan 683	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
30	29	rgen 3115	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧)
31		foov 7178	. . . . . . 7 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧)))
32	1, 30, 31	mpbir2an 707	. . . . . 6 ⊢ 𝐺:(𝑋 × 𝑋)–onto→𝑋
33		forn 6461	. . . . . 6 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 → ran 𝐺 = 𝑋)
34	32, 33	ax-mp 5	. . . . 5 ⊢ ran 𝐺 = 𝑋
35	34	eqcomi 2804	. . . 4 ⊢ 𝑋 = ran 𝐺
36	35	isgrpo 27965	. . 3 ⊢ (𝐺 ∈ V → (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))))
37	25, 36	ax-mp 5	. 2 ⊢ (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)))
38	1, 3, 21, 37	mpbir3an 1334	1 ⊢ 𝐺 ∈ GrpOp

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  ∀wral 3105  ∃wrex 3106  Vcvv 3437   × cxp 5441  ran crn 5444  ⟶wf 6221  –onto→wfo 6223  (class class class)co 7016  GrpOpcgr 27957

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-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  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-ne 2985  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-pw 4455  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-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-grpo 27961

This theorem is referenced by:  cnaddabloOLD  28049  hilablo  28628  hhssabloilem  28729  grposnOLD  34692