Theorem isgrpda 37949

Description: Properties that determine a group operation. (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.)

Hypotheses

Ref	Expression
isgrpda.1	⊢ (𝜑 → 𝑋 ∈ V)
isgrpda.2	⊢ (𝜑 → 𝐺:(𝑋 × 𝑋)⟶𝑋)
isgrpda.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
isgrpda.4	⊢ (𝜑 → 𝑈 ∈ 𝑋)
isgrpda.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑈𝐺𝑥) = 𝑥)
isgrpda.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑛 ∈ 𝑋 (𝑛𝐺𝑥) = 𝑈)

Assertion

Ref	Expression
isgrpda	⊢ (𝜑 → 𝐺 ∈ GrpOp)

Distinct variable groups:   𝜑,𝑥,𝑦,𝑧   𝑛,𝐺,𝑥,𝑦,𝑧   𝑛,𝑋,𝑥,𝑦,𝑧   𝑈,𝑛,𝑥,𝑦,𝑧

Allowed substitution hint:   𝜑(𝑛)

Proof of Theorem isgrpda

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isgrpda.2	. . 3 ⊢ (𝜑 → 𝐺:(𝑋 × 𝑋)⟶𝑋)
2		isgrpda.3	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
3	2	ralrimivvva 3183	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
4		isgrpda.4	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑋)
5		isgrpda.5	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑈𝐺𝑥) = 𝑥)
6		isgrpda.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑛 ∈ 𝑋 (𝑛𝐺𝑥) = 𝑈)
7		oveq1 7394	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → (𝑦𝐺𝑥) = (𝑛𝐺𝑥))
8	7	eqeq1d 2731	. . . . . . . 8 ⊢ (𝑦 = 𝑛 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑛𝐺𝑥) = 𝑈))
9	8	cbvrexvw 3216	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈 ↔ ∃𝑛 ∈ 𝑋 (𝑛𝐺𝑥) = 𝑈)
10	6, 9	sylibr 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)
11	5, 10	jca 511	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
12	11	ralrimiva 3125	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
13		oveq1 7394	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
14	13	eqeq1d 2731	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
15		eqeq2 2741	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → ((𝑦𝐺𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑈))
16	15	rexbidv 3157	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
17	14, 16	anbi12d 632	. . . . . 6 ⊢ (𝑢 = 𝑈 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
18	17	ralbidv 3156	. . . . 5 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
19	18	rspcev 3588	. . . 4 ⊢ ((𝑈 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)) → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))
20	4, 12, 19	syl2anc 584	. . 3 ⊢ (𝜑 → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))
21	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑈 ∈ 𝑋)
22		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
23	5	eqcomd 2735	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 = (𝑈𝐺𝑥))
24		rspceov 7436	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑥 = (𝑈𝐺𝑥)) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
25	21, 22, 23, 24	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
26	25	ralrimiva 3125	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
27		foov 7563	. . . . . . . 8 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧)))
28	1, 26, 27	sylanbrc 583	. . . . . . 7 ⊢ (𝜑 → 𝐺:(𝑋 × 𝑋)–onto→𝑋)
29		forn 6775	. . . . . . 7 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 → ran 𝐺 = 𝑋)
30	28, 29	syl 17	. . . . . 6 ⊢ (𝜑 → ran 𝐺 = 𝑋)
31	30	sqxpeqd 5670	. . . . 5 ⊢ (𝜑 → (ran 𝐺 × ran 𝐺) = (𝑋 × 𝑋))
32	31, 30	feq23d 6683	. . . 4 ⊢ (𝜑 → (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
33	30	raleqdv 3299	. . . . . 6 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
34	30, 33	raleqbidv 3319	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
35	30, 34	raleqbidv 3319	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
36	30	rexeqdv 3300	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))
37	36	anbi2d 630	. . . . . 6 ⊢ (𝜑 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))
38	30, 37	raleqbidv 3319	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))
39	30, 38	rexeqbidv 3320	. . . 4 ⊢ (𝜑 → (∃𝑢 ∈ ran 𝐺∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢) ↔ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))
40	32, 35, 39	3anbi123d 1438	. . 3 ⊢ (𝜑 → ((𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ ran 𝐺∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢)) ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))))
41	1, 3, 20, 40	mpbir3and 1343	. 2 ⊢ (𝜑 → (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ ran 𝐺∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢)))
42		isgrpda.1	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ V)
43	42, 42	xpexd 7727	. . . 4 ⊢ (𝜑 → (𝑋 × 𝑋) ∈ V)
44	1, 43	fexd 7201	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
45		eqid 2729	. . . 4 ⊢ ran 𝐺 = ran 𝐺
46	45	isgrpo 30426	. . 3 ⊢ (𝐺 ∈ V → (𝐺 ∈ GrpOp ↔ (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ ran 𝐺∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢))))
47	44, 46	syl 17	. 2 ⊢ (𝜑 → (𝐺 ∈ GrpOp ↔ (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢 ∈ ran 𝐺∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢))))
48	41, 47	mpbird 257	1 ⊢ (𝜑 → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3447   × cxp 5636  ran crn 5639  ⟶wf 6507  –onto→wfo 6509  (class class class)co 7387  GrpOpcgr 30418

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-grpo 30422

This theorem is referenced by:  isdrngo2  37952