dmncan1 - Mathbox for Jeff Madsen

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Theorem dmncan1 38614

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 38595	. . . . . 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 2769	. . . . . . 7 ⊢ ( /_𝑔 ‘𝐺) = ( /_𝑔 ‘𝐺)
6	2, 3, 4, 5	rngosubdi 38483	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)))
7	1, 6	sylan 591	. . . . 5 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)))
8	7	adantr 485	. . . 4 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → (𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)))
9	8	eqeq1d 2771	. . 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 38448	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
11	1, 10	syl 18	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Dmn → 𝐺 ∈ GrpOp)
12	4, 5	grpodivcl 30831	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵( /_𝑔 ‘𝐺)𝐶) ∈ 𝑋)
13	12	3expb 1136	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵( /_𝑔 ‘𝐺)𝐶) ∈ 𝑋)
14	11, 13	sylan 591	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Dmn ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵( /_𝑔 ‘𝐺)𝐶) ∈ 𝑋)
15	14	adantlr 727	. . . . . . . . 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 38613	. . . . . . . . . . 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 1371	. . . . . . . . . 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 422	. . . . . . . . 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 602	. . . . . . . 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 441	. . . . . . 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 1366	. . . . . 6 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍)))
23		neor 3056	. . . . . 6 ⊢ ((𝐴 = 𝑍 ∨ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍) ↔ (𝐴 ≠ 𝑍 → (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
24	22, 23	imbitrdi 254	. . . . 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 87	. . . 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 411	. . 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 263	. 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → (((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)) = 𝑍 → (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
28	11	adantr 485	. . . 4 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐺 ∈ GrpOp)
29	2, 3, 4	rngocl 38439	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
30	29	3adant3r3 1201	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
31	1, 30	sylan 591	. . . 4 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
32	2, 3, 4	rngocl 38439	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
33	32	3adant3r2 1200	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
34	1, 33	sylan 591	. . . 4 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
35	4, 16, 5	grpoeqdivid 38419	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝐵) ∈ 𝑋 ∧ (𝐴𝐻𝐶) ∈ 𝑋) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)) = 𝑍))
36	28, 31, 34, 35	syl3anc 1396	. . 3 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)) = 𝑍))
37	36	adantr 485	. 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)) = 𝑍))
38	4, 16, 5	grpoeqdivid 38419	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
39	38	3expb 1136	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
40	11, 39	sylan 591	. . . 4 ⊢ ((𝑅 ∈ Dmn ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
41	40	3adantr1 1186	. . 3 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
42	41	adantr 485	. 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
43	27, 37, 42	3imtr4d 297	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 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ran crn 5663 ‘cfv 6537 (class class class)co 7411 1^st c1st 7983 2^nd c2nd 7984 GrpOpcgr 30781 GIdcgi 30782 /_𝑔 cgs 30784 RingOpscrngo 38432 Dmncdmn 38585

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-id 5557 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7985 df-2nd 7986 df-1o 8452 df-en 8943 df-grpo 30785 df-gid 30786 df-ginv 30787 df-gdiv 30788 df-ablo 30837 df-ass 38381 df-exid 38383 df-mgmOLD 38387 df-sgrOLD 38399 df-mndo 38405 df-rngo 38433 df-com2 38528 df-crngo 38532 df-idl 38548 df-pridl 38549 df-prrngo 38586 df-dmn 38587 df-igen 38598

This theorem is referenced by: dmncan2 38615

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