isgrpda - Mathbox for Jeff Madsen

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Theorem isgrpda 37363

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 3198	. . 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 7421	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → (𝑦𝐺𝑥) = (𝑛𝐺𝑥))
8	7	eqeq1d 2729	. . . . . . . 8 ⊢ (𝑦 = 𝑛 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑛𝐺𝑥) = 𝑈))
9	8	cbvrexvw 3230	. . . . . . 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 511	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
12	11	ralrimiva 3141	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
13		oveq1 7421	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢𝐺𝑥) = (𝑈𝐺𝑥))
14	13	eqeq1d 2729	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢𝐺𝑥) = 𝑥 ↔ (𝑈𝐺𝑥) = 𝑥))
15		eqeq2 2739	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → ((𝑦𝐺𝑥) = 𝑢 ↔ (𝑦𝐺𝑥) = 𝑈))
16	15	rexbidv 3173	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈))
17	14, 16	anbi12d 630	. . . . . 6 ⊢ (𝑢 = 𝑈 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
18	17	ralbidv 3172	. . . . 5 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥 ∈ 𝑋 ((𝑈𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑈)))
19	18	rspcev 3607	. . . 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 583	. . 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 2733	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 = (𝑈𝐺𝑥))
24		rspceov 7461	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑥 = (𝑈𝐺𝑥)) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
25	21, 22, 23, 24	syl3anc 1369	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
26	25	ralrimiva 3141	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧))
27		foov 7589	. . . . . . . 8 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ 𝑋 𝑥 = (𝑦𝐺𝑧)))
28	1, 26, 27	sylanbrc 582	. . . . . . 7 ⊢ (𝜑 → 𝐺:(𝑋 × 𝑋)–onto→𝑋)
29		forn 6808	. . . . . . 7 ⊢ (𝐺:(𝑋 × 𝑋)–onto→𝑋 → ran 𝐺 = 𝑋)
30	28, 29	syl 17	. . . . . 6 ⊢ (𝜑 → ran 𝐺 = 𝑋)
31	30	sqxpeqd 5704	. . . . 5 ⊢ (𝜑 → (ran 𝐺 × ran 𝐺) = (𝑋 × 𝑋))
32	31, 30	feq23d 6711	. . . 4 ⊢ (𝜑 → (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
33	30	raleqdv 3320	. . . . . 6 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
34	30, 33	raleqbidv 3337	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
35	30, 34	raleqbidv 3337	. . . 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 3321	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢 ↔ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢))
37	36	anbi2d 628	. . . . . 6 ⊢ (𝜑 → (((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢) ↔ ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))
38	30, 37	raleqbidv 3337	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐺((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ ran 𝐺(𝑦𝐺𝑥) = 𝑢) ↔ ∀𝑥 ∈ 𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑋 (𝑦𝐺𝑥) = 𝑢)))
39	30, 38	rexeqbidv 3338	. . . 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 1433	. . 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 1340	. 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 7747	. . . 4 ⊢ (𝜑 → (𝑋 × 𝑋) ∈ V)
44	1, 43	fexd 7233	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
45		eqid 2727	. . . 4 ⊢ ran 𝐺 = ran 𝐺
46	45	isgrpo 30294	. . 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)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 Vcvv 3469 × cxp 5670 ran crn 5673 ⟶wf 6538 –onto→wfo 6540 (class class class)co 7414 GrpOpcgr 30286

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-grpo 30290

This theorem is referenced by: isdrngo2 37366

W3C validator