dmncan1 - Mathbox for Jeff Madsen

	Mathbox for Jeff Madsen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dmncan1		Structured version Visualization version GIF version

Theorem dmncan1 34886

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 34867	. . . . . 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 2795	. . . . . . 7 ⊢ ( /_𝑔 ‘𝐺) = ( /_𝑔 ‘𝐺)
6	2, 3, 4, 5	rngosubdi 34755	. . . . . 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 481	. . . 4 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → (𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)))
9	8	eqeq1d 2797	. . 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 34720	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
11	1, 10	syl 17	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Dmn → 𝐺 ∈ GrpOp)
12	4, 5	grpodivcl 28007	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵( /_𝑔 ‘𝐺)𝐶) ∈ 𝑋)
13	12	3expb 1113	. . . . . . . . . . 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 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 34885	. . . . . . . . . . 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 1347	. . . . . . . . . 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 418	. . . . . . . . 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 437	. . . . . . 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 1342	. . . . . 6 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻(𝐵( /_𝑔 ‘𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍)))
23		neor 3076	. . . . . 6 ⊢ ((𝐴 = 𝑍 ∨ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍) ↔ (𝐴 ≠ 𝑍 → (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
24	22, 23	syl6ib 252	. . . . 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 407	. . 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 261	. 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → (((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)) = 𝑍 → (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
28	11	adantr 481	. . . 4 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐺 ∈ GrpOp)
29	2, 3, 4	rngocl 34711	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
30	29	3adant3r3 1177	. . . . 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 34711	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
33	32	3adant3r2 1176	. . . . 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 34691	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝐵) ∈ 𝑋 ∧ (𝐴𝐻𝐶) ∈ 𝑋) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)) = 𝑍))
36	28, 31, 34, 35	syl3anc 1364	. . 3 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)) = 𝑍))
37	36	adantr 481	. 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /_𝑔 ‘𝐺)(𝐴𝐻𝐶)) = 𝑍))
38	4, 16, 5	grpoeqdivid 34691	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
39	38	3expb 1113	. . . . 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 1162	. . 3 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
42	41	adantr 481	. 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → (𝐵 = 𝐶 ↔ (𝐵( /_𝑔 ‘𝐺)𝐶) = 𝑍))
43	27, 37, 42	3imtr4d 295	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 207 ∧ wa 396 ∨ wo 842 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ran crn 5444 ‘cfv 6225 (class class class)co 7016 1^st c1st 7543 2^nd c2nd 7544 GrpOpcgr 27957 GIdcgi 27958 /_𝑔 cgs 27960 RingOpscrngo 34704 Dmncdmn 34857

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-1o 7953 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-grpo 27961 df-gid 27962 df-ginv 27963 df-gdiv 27964 df-ablo 28013 df-ass 34653 df-exid 34655 df-mgmOLD 34659 df-sgrOLD 34671 df-mndo 34677 df-rngo 34705 df-com2 34800 df-crngo 34804 df-idl 34820 df-pridl 34821 df-prrngo 34858 df-dmn 34859 df-igen 34870

This theorem is referenced by: dmncan2 34887

W3C validator