dmncan2 - Mathbox for Jeff Madsen

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Theorem dmncan2 38076

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
dmncan2	⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) → 𝐴 = 𝐵))

**Proof of Theorem dmncan2**
Step	Hyp	Ref	Expression
1		dmncrng 38055	. . . 4 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)
2		dmncan.1	. . . . . . 7 ⊢ 𝐺 = (1^st ‘𝑅)
3		dmncan.2	. . . . . . 7 ⊢ 𝐻 = (2^nd ‘𝑅)
4		dmncan.3	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
5	2, 3, 4	crngocom 38000	. . . . . 6 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐻𝐶) = (𝐶𝐻𝐴))
6	5	3adant3r2 1184	. . . . 5 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) = (𝐶𝐻𝐴))
7	2, 3, 4	crngocom 38000	. . . . . 6 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵))
8	7	3adant3r1 1183	. . . . 5 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵))
9	6, 8	eqeq12d 2745	. . . 4 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵)))
10	1, 9	sylan 580	. . 3 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵)))
11	10	adantr 480	. 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵)))
12		3anrot 1099	. . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋))
13	12	biimpri 228	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
14		dmncan.4	. . . 4 ⊢ 𝑍 = (GId‘𝐺)
15	2, 3, 4, 14	dmncan1 38075	. . 3 ⊢ (((𝑅 ∈ Dmn ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐶𝐻𝐴) = (𝐶𝐻𝐵) → 𝐴 = 𝐵))
16	13, 15	sylanl2 681	. 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐶𝐻𝐴) = (𝐶𝐻𝐵) → 𝐴 = 𝐵))
17	11, 16	sylbid 240	1 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) → 𝐴 = 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ran crn 5624 ‘cfv 6486 (class class class)co 7353 1^st c1st 7929 2^nd c2nd 7930 GIdcgi 30453 CRingOpsccring 37992 Dmncdmn 38046

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 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675

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 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7931 df-2nd 7932 df-1o 8395 df-en 8880 df-grpo 30456 df-gid 30457 df-ginv 30458 df-gdiv 30459 df-ablo 30508 df-ass 37842 df-exid 37844 df-mgmOLD 37848 df-sgrOLD 37860 df-mndo 37866 df-rngo 37894 df-com2 37989 df-crngo 37993 df-idl 38009 df-pridl 38010 df-prrngo 38047 df-dmn 38048 df-igen 38059

This theorem is referenced by: (None)

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