Theorem isgrpoi 31157

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 3211	. 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 7458	. . . . . . . 8 ⊢ (𝑦 = 𝑁 → (𝑦𝐺𝑥) = (𝑁𝐺𝑥))
9	8	eqeq1d 2765	. . . . . . 7 ⊢ (𝑦 = 𝑁 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑁𝐺𝑥) = 𝑈))
10	9	rspcev 3580	. . . . . 6 ⊢ ((𝑁 ∈ 𝑋 ∧ (𝑁𝐺𝑥) = 𝑈) → ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)
11	6, 7, 10	syl2anc 596	. . . . 5 ⊢ (𝑥 ∈ 𝑋 → ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)
12	5, 11	jca 521	. . . 4 ⊢ (𝑥 ∈ 𝑋 → ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
13	12	rgen 3082	. . 3 ⊢ ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)
14		oveq1 7458	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
15	14	eqeq1d 2765	. . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
16		eqeq2 2775	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑦𝐺𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑈))
17	16	rexbidv 3190	. . . . . 6 ⊢ (𝑢 = 𝑈 → (∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
18	15, 17	anbi12d 644	. . . . 5 ⊢ (𝑢 = 𝑈 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
19	18	ralbidv 3189	. . . 4 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
20	19	rspcev 3580	. . 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 705	. 2 ⊢ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)
22		isgrpoi.1	. . . . 5 ⊢ 𝑋 ∈ V
23	22, 22	xpex 7799	. . . 4 ⊢ (𝑋 × 𝑋) ∈ V
24		fex 7263	. . . 4 ⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ (𝑋 × 𝑋) ∈ V) → 𝐺 ∈ V)
25	1, 23, 24	mp2an 705	. . 3 ⊢ 𝐺 ∈ V
26	5	eqcomd 2769	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 → 𝑥 = (𝑈𝐺𝑥))
27		rspceov 7500	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑥 = (𝑈𝐺𝑥)) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
28	4, 27	mp3an1 1478	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 = (𝑈𝐺𝑥)) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
29	26, 28	mpdan 700	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
30	29	rgen 3082	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧)
31		foov 7627	. . . . . . 7 ⊢ (𝐺:(𝑋 × 𝑋)⟶onto𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧)))
32	1, 30, 31	mpbir2an 724	. . . . . 6 ⊢ 𝐺:(𝑋 × 𝑋)⟶onto𝑋
33		forn 6829	. . . . . 6 ⊢ (𝐺:(𝑋 × 𝑋)⟶onto𝑋 → ran 𝐺 = 𝑋)
34	32, 33	ax-mp 5	. . . . 5 ⊢ ran 𝐺 = 𝑋
35	34	eqcomi 2772	. . . 4 ⊢ 𝑋 = ran 𝐺
36	35	isgrpo 31156	. . 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 1360	1 ⊢ 𝐺 ∈ GrpOp

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   × cxp 5656  ran crn 5659  ⟶wf 6562  ⟶ontowfo 6564  (class class class)co 7451  GrpOpcgr 31148

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-grpo 31152

This theorem is used by:  cnaddabloOLD  31240  hilablo  31819  hhssabloilem  31920  grposnOLD  39082