Theorem dmncan2 38827

Description: Obsolete theorem, use idomcanr 49263 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 = (1st ‘𝑅)
dmncan.2	⊢ 𝐻 = (2nd ‘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 38806	. . . 4 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)
2		dmncan.1	. . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅)
3		dmncan.2	. . . . . . 7 ⊢ 𝐻 = (2nd ‘𝑅)
4		dmncan.3	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
5	2, 3, 4	crngocom 38751	. . . . . 6 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐻𝐶) = (𝐶𝐻𝐴))
6	5	3adant3r2 1202	. . . . 5 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) = (𝐶𝐻𝐴))
7	2, 3, 4	crngocom 38751	. . . . . 6 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵))
8	7	3adant3r1 1201	. . . . 5 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵))
9	6, 8	eqeq12d 2776	. . . 4 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵)))
10	1, 9	sylan 592	. . 3 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵)))
11	10	adantr 486	. 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵)))
12		3anrot 1117	. . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋))
13	12	biimpri 231	. . 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 38826	. . 3 ⊢ (((𝑅 ∈ Dmn ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐶𝐻𝐴) = (𝐶𝐻𝐵) → 𝐴 = 𝐵))
16	13, 15	sylanl2 694	. 2 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐶𝐻𝐴) = (𝐶𝐻𝐵) → 𝐴 = 𝐵))
17	11, 16	sylbid 243	1 ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) → 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ran crn 5656  ‘cfv 6533  (class class class)co 7413  1^st c1st 7984  2^nd c2nd 7985  GIdcgi 30971  CRingOpsccring 38743  Dmncdmn 38797

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 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  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 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-1o 8455  df-en 8953  df-grpo 30974  df-gid 30975  df-ginv 30976  df-gdiv 30977  df-ablo 31026  df-ass 38593  df-exid 38595  df-mgmOLD 38599  df-sgrOLD 38611  df-mndo 38617  df-rngo 38645  df-com2 38740  df-crngo 38744  df-idl 38760  df-pridl 38761  df-prrngo 38798  df-dmn 38799  df-igen 38810

This theorem is used by: (None)