Theorem isgrpoi 30434

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 3183	. 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 7397	. . . . . . . 8 ⊢ (𝑦 = 𝑁 → (𝑦𝐺𝑥) = (𝑁𝐺𝑥))
9	8	eqeq1d 2732	. . . . . . 7 ⊢ (𝑦 = 𝑁 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑁𝐺𝑥) = 𝑈))
10	9	rspcev 3591	. . . . . 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 511	. . . 4 ⊢ (𝑥 ∈ 𝑋 → ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
13	12	rgen 3047	. . 3 ⊢ ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)
14		oveq1 7397	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
15	14	eqeq1d 2732	. . . . . 6 ⊢ (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
16		eqeq2 2742	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑦𝐺𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑈))
17	16	rexbidv 3158	. . . . . 6 ⊢ (𝑢 = 𝑈 → (∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
18	15, 17	anbi12d 632	. . . . 5 ⊢ (𝑢 = 𝑈 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
19	18	ralbidv 3157	. . . 4 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
20	19	rspcev 3591	. . 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 692	. 2 ⊢ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)
22		isgrpoi.1	. . . . 5 ⊢ 𝑋 ∈ V
23	22, 22	xpex 7732	. . . 4 ⊢ (𝑋 × 𝑋) ∈ V
24		fex 7203	. . . 4 ⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ (𝑋 × 𝑋) ∈ V) → 𝐺 ∈ V)
25	1, 23, 24	mp2an 692	. . 3 ⊢ 𝐺 ∈ V
26	5	eqcomd 2736	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 → 𝑥 = (𝑈𝐺𝑥))
27		rspceov 7439	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑥 = (𝑈𝐺𝑥)) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
28	4, 27	mp3an1 1450	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 = (𝑈𝐺𝑥)) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
29	26, 28	mpdan 687	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
30	29	rgen 3047	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧)
31		foov 7566	. . . . . . 7 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧)))
32	1, 30, 31	mpbir2an 711	. . . . . 6 ⊢ 𝐺:(𝑋 × 𝑋)–onto→𝑋
33		forn 6778	. . . . . 6 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 → ran 𝐺 = 𝑋)
34	32, 33	ax-mp 5	. . . . 5 ⊢ ran 𝐺 = 𝑋
35	34	eqcomi 2739	. . . 4 ⊢ 𝑋 = ran 𝐺
36	35	isgrpo 30433	. . 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 1342	1 ⊢ 𝐺 ∈ GrpOp

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450   × cxp 5639  ran crn 5642  ⟶wf 6510  –onto→wfo 6512  (class class class)co 7390  GrpOpcgr 30425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-grpo 30429

This theorem is referenced by:  cnaddabloOLD  30517  hilablo  31096  hhssabloilem  31197  grposnOLD  37883