dmncan1 - Mathbox for Jeff Madsen

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Theorem dmncan1 38070

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 38051	. . . . . 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 2729	. . . . . . 7 ⊢ ( /_𝑔 ‘𝐺) = ( /_𝑔 ‘𝐺)
6	2, 3, 4, 5	rngosubdi 37939	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)))
7	1, 6	sylan 580	. . . . 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 2731	. . 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 37904	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
11	1, 10	syl 17	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Dmn → 𝐺 ∈ GrpOp)
12	4, 5	grpodivcl 30468	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵( /_𝑔 ‘𝐺)𝐶) ∈ 𝑋)
13	12	3expb 1120	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵( /_𝑔 ‘𝐺)𝐶) ∈ 𝑋)
14	11, 13	sylan 580	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Dmn ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵( /_𝑔 ‘𝐺)𝐶) ∈ 𝑋)
15	14	adantlr 715	. . . . . . . . 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 38069	. . . . . . . . . . 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 1355	. . . . . . . . . 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 591	. . . . . . . 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 1350	. . . . . 6 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍)))
23		neor 3017	. . . . . 6 ⊢ ((𝐴 = 𝑍 ∨ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍) ↔ (𝐴 ≠ 𝑍 → (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
24	22, 23	imbitrdi 251	. . . . 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 260	. 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 37895	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
30	29	3adant3r3 1185	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
31	1, 30	sylan 580	. . . 4 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
32	2, 3, 4	rngocl 37895	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
33	32	3adant3r2 1184	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
34	1, 33	sylan 580	. . . 4 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
35	4, 16, 5	grpoeqdivid 37875	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝐵) ∈ 𝑋 ∧ (𝐴𝐻𝐶) ∈ 𝑋) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)) = 𝑍))
36	28, 31, 34, 35	syl3anc 1373	. . 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 37875	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
39	38	3expb 1120	. . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
40	11, 39	sylan 580	. . . 4 ⊢ ((𝑅 ∈ Dmn ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
41	40	3adantr1 1170	. . 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 294	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ran crn 5639 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 2^nd c2nd 7967 GrpOpcgr 30418 GIdcgi 30419 /_𝑔 cgs 30421 RingOpscrngo 37888 Dmncdmn 38041

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-1st 7968 df-2nd 7969 df-1o 8434 df-en 8919 df-grpo 30422 df-gid 30423 df-ginv 30424 df-gdiv 30425 df-ablo 30474 df-ass 37837 df-exid 37839 df-mgmOLD 37843 df-sgrOLD 37855 df-mndo 37861 df-rngo 37889 df-com2 37984 df-crngo 37988 df-idl 38004 df-pridl 38005 df-prrngo 38042 df-dmn 38043 df-igen 38054

This theorem is referenced by: dmncan2 38071

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