dmncan1 - Mathbox for Jeff Madsen

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Theorem dmncan1 38990

Description: Obsolete theorem, use idomcanl 49413 instead. Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

**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 38971	. . . . . 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 2761	. . . . . . 7 ⊢ ( /_𝑔 ‘𝐺) = ( /_𝑔 ‘𝐺)
6	2, 3, 4, 5	rngosubdi 38859	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)))
7	1, 6	sylan 592	. . . . 5 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)))
8	7	adantr 486	. . . 4 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → (𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)))
9	8	eqeq1d 2763	. . 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 38824	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
11	1, 10	syl 18	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Dmn → 𝐺 ∈ GrpOp)
12	4, 5	grpodivcl 31134	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵( /_𝑔 ‘𝐺)𝐶) ∈ 𝑋)
13	12	3expb 1138	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵( /_𝑔 ‘𝐺)𝐶) ∈ 𝑋)
14	11, 13	sylan 592	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Dmn ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵( /_𝑔 ‘𝐺)𝐶) ∈ 𝑋)
15	14	adantlr 728	. . . . . . . . 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 38989	. . . . . . . . . . 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 1373	. . . . . . . . . 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 423	. . . . . . . . 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 603	. . . . . . . 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 442	. . . . . . 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 1368	. . . . . 6 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍)))
23		neor 3048	. . . . . 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 412	. . 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 486	. . . 4 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐺 ∈ GrpOp)
29	2, 3, 4	rngocl 38815	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
30	29	3adant3r3 1203	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
31	1, 30	sylan 592	. . . 4 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
32	2, 3, 4	rngocl 38815	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
33	32	3adant3r2 1202	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
34	1, 33	sylan 592	. . . 4 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
35	4, 16, 5	grpoeqdivid 38795	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝐵) ∈ 𝑋 ∧ (𝐴𝐻𝐶) ∈ 𝑋) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)) = 𝑍))
36	28, 31, 34, 35	syl3anc 1398	. . 3 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)) = 𝑍))
37	36	adantr 486	. 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)) = 𝑍))
38	4, 16, 5	grpoeqdivid 38795	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
39	38	3expb 1138	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
40	11, 39	sylan 592	. . . 4 ⊢ ((𝑅 ∈ Dmn ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
41	40	3adantr1 1188	. . 3 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
42	41	adantr 486	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ran crn 5652 ‘cfv 6537 (class class class)co 7418 1^st c1st 7997 2^nd c2nd 7998 GrpOpcgr 31084 GIdcgi 31085 /_𝑔 cgs 31087 RingOpscrngo 38808 Dmncdmn 38961

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 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 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-1st 7999 df-2nd 8000 df-1o 8469 df-en 8967 df-grpo 31088 df-gid 31089 df-ginv 31090 df-gdiv 31091 df-ablo 31140 df-ass 38757 df-exid 38759 df-mgmOLD 38763 df-sgrOLD 38775 df-mndo 38781 df-rngo 38809 df-com2 38904 df-crngo 38908 df-idl 38924 df-pridl 38925 df-prrngo 38962 df-dmn 38963 df-igen 38974

This theorem is used by: dmncan2 38991

W3C validator