Theorem isgrpda 36211

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 3196	. . 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 7336	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → (𝑦𝐺𝑥) = (𝑛𝐺𝑥))
8	7	eqeq1d 2738	. . . . . . . 8 ⊢ (𝑦 = 𝑛 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑛𝐺𝑥) = 𝑈))
9	8	cbvrexvw 3222	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈 ↔ ∃𝑛 ∈ 𝑋 (𝑛𝐺𝑥) = 𝑈)
10	6, 9	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)
11	5, 10	jca 512	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
12	11	ralrimiva 3139	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
13		oveq1 7336	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
14	13	eqeq1d 2738	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
15		eqeq2 2748	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → ((𝑦𝐺𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑈))
16	15	rexbidv 3171	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
17	14, 16	anbi12d 631	. . . . . 6 ⊢ (𝑢 = 𝑈 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
18	17	ralbidv 3170	. . . . 5 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
19	18	rspcev 3570	. . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑈 ∈ 𝑋)
22		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
23	5	eqcomd 2742	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 = (𝑈𝐺𝑥))
24		rspceov 7376	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑥 = (𝑈𝐺𝑥)) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
25	21, 22, 23, 24	syl3anc 1370	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
26	25	ralrimiva 3139	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
27		foov 7500	. . . . . . . 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 6736	. . . . . . 7 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 → ran 𝐺 = 𝑋)
30	28, 29	syl 17	. . . . . 6 ⊢ (𝜑 → ran 𝐺 = 𝑋)
31	30	sqxpeqd 5646	. . . . 5 ⊢ (𝜑 → (ran 𝐺 × ran 𝐺) = (𝑋 × 𝑋))
32	31, 30	feq23d 6640	. . . 4 ⊢ (𝜑 → (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
33	30	raleqdv 3309	. . . . . 6 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
34	30, 33	raleqbidv 3315	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
35	30, 34	raleqbidv 3315	. . . 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 3310	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))
37	36	anbi2d 629	. . . . . 6 ⊢ (𝜑 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))
38	30, 37	raleqbidv 3315	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))
39	30, 38	rexeqbidv 3316	. . . 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 1435	. . 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 1341	. 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 7655	. . . 4 ⊢ (𝜑 → (𝑋 × 𝑋) ∈ V)
44	1, 43	fexd 7153	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
45		eqid 2736	. . . 4 ⊢ ran 𝐺 = ran 𝐺
46	45	isgrpo 29060	. . 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 256	1 ⊢ (𝜑 → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3061  ∃wrex 3070  Vcvv 3441   × cxp 5612  ran crn 5615  ⟶wf 6469  –onto→wfo 6471  (class class class)co 7329  GrpOpcgr 29052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-ov 7332  df-grpo 29056

This theorem is referenced by:  isdrngo2  36214