dmncan1 - Mathbox for Jeff Madsen

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Theorem dmncan1 36161

Description: Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)

**Hypotheses**
Ref	Expression
dmncan.1	⊢ 𝐺 = (1^st ‘𝑅)
dmncan.2	⊢ 𝐻 = (2^nd ‘𝑅)
dmncan.3	⊢ 𝑋 = ran 𝐺
dmncan.4	⊢ 𝑍 = (GId‘𝐺)

**Assertion**
Ref	Expression
dmncan1	⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) → 𝐵 = 𝐶))

**Proof of Theorem dmncan1**
Step	Hyp	Ref	Expression
1		dmnrngo 36142	. . . . . 6 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ RingOps)
2		dmncan.1	. . . . . . 7 ⊢ 𝐺 = (1^st ‘𝑅)
3		dmncan.2	. . . . . . 7 ⊢ 𝐻 = (2^nd ‘𝑅)
4		dmncan.3	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
5		eqid 2738	. . . . . . 7 ⊢ ( /_𝑔 ‘𝐺) = ( /_𝑔 ‘𝐺)
6	2, 3, 4, 5	rngosubdi 36030	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)))
7	1, 6	sylan 579	. . . . 5 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)))
8	7	adantr 480	. . . 4 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → (𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)))
9	8	eqeq1d 2740	. . 3 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → ((𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = 𝑍 ↔ ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)) = 𝑍))
10	2	rngogrpo 35995	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
11	1, 10	syl 17	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Dmn → 𝐺 ∈ GrpOp)
12	4, 5	grpodivcl 28802	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵( /_𝑔 ‘𝐺)𝐶) ∈ 𝑋)
13	12	3expb 1118	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵( /_𝑔 ‘𝐺)𝐶) ∈ 𝑋)
14	11, 13	sylan 579	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Dmn ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵( /_𝑔 ‘𝐺)𝐶) ∈ 𝑋)
15	14	adantlr 711	. . . . . . . . 9 ⊢ (((𝑅 ∈ Dmn ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵( /_𝑔 ‘𝐺)𝐶) ∈ 𝑋)
16		dmncan.4	. . . . . . . . . . . 12 ⊢ 𝑍 = (GId‘𝐺)
17	2, 3, 4, 16	dmnnzd 36160	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ (𝐵( /_𝑔 ‘𝐺)𝐶) ∈ 𝑋 ∧ (𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = 𝑍)) → (𝐴 = 𝑍 ∨ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
18	17	3exp2 1352	. . . . . . . . . 10 ⊢ (𝑅 ∈ Dmn → (𝐴 ∈ 𝑋 → ((𝐵( /_𝑔 ‘𝐺)𝐶) ∈ 𝑋 → ((𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍)))))
19	18	imp31 417	. . . . . . . . 9 ⊢ (((𝑅 ∈ Dmn ∧ 𝐴 ∈ 𝑋) ∧ (𝐵( /_𝑔 ‘𝐺)𝐶) ∈ 𝑋) → ((𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍)))
20	15, 19	syldan 590	. . . . . . . 8 ⊢ (((𝑅 ∈ Dmn ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍)))
21	20	exp43 436	. . . . . . 7 ⊢ (𝑅 ∈ Dmn → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐶 ∈ 𝑋 → ((𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))))))
22	21	3imp2 1347	. . . . . 6 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍)))
23		neor 3035	. . . . . 6 ⊢ ((𝐴 = 𝑍 ∨ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍) ↔ (𝐴 ≠ 𝑍 → (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
24	22, 23	syl6ib 250	. . . . 5 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = 𝑍 → (𝐴 ≠ 𝑍 → (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍)))
25	24	com23 86	. . . 4 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴 ≠ 𝑍 → ((𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = 𝑍 → (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍)))
26	25	imp 406	. . 3 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → ((𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = 𝑍 → (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
27	9, 26	sylbird 259	. 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → (((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)) = 𝑍 → (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
28	11	adantr 480	. . . 4 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐺 ∈ GrpOp)
29	2, 3, 4	rngocl 35986	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
30	29	3adant3r3 1182	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
31	1, 30	sylan 579	. . . 4 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
32	2, 3, 4	rngocl 35986	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
33	32	3adant3r2 1181	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
34	1, 33	sylan 579	. . . 4 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
35	4, 16, 5	grpoeqdivid 35966	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝐵) ∈ 𝑋 ∧ (𝐴𝐻𝐶) ∈ 𝑋) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)) = 𝑍))
36	28, 31, 34, 35	syl3anc 1369	. . 3 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)) = 𝑍))
37	36	adantr 480	. 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)) = 𝑍))
38	4, 16, 5	grpoeqdivid 35966	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
39	38	3expb 1118	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
40	11, 39	sylan 579	. . . 4 ⊢ ((𝑅 ∈ Dmn ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
41	40	3adantr1 1167	. . 3 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
42	41	adantr 480	. 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
43	27, 37, 42	3imtr4d 293	1 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) → 𝐵 = 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ran crn 5581 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 2^nd c2nd 7803 GrpOpcgr 28752 GIdcgi 28753 /_𝑔 cgs 28755 RingOpscrngo 35979 Dmncdmn 36132

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-1o 8267 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-grpo 28756 df-gid 28757 df-ginv 28758 df-gdiv 28759 df-ablo 28808 df-ass 35928 df-exid 35930 df-mgmOLD 35934 df-sgrOLD 35946 df-mndo 35952 df-rngo 35980 df-com2 36075 df-crngo 36079 df-idl 36095 df-pridl 36096 df-prrngo 36133 df-dmn 36134 df-igen 36145

This theorem is referenced by: dmncan2 36162

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