Theorem dmncan2 38761

Description: Obsolete theorem, use idomcanr 49146 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 38740	. . . 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 38685	. . . . . 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 38685	. . . . . 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 2782	. . . 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 38760	. . 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 2146   ≠ wne 2961  ran crn 5667  ‘cfv 6543  (class class class)co 7423  1^st c1st 7993  2^nd c2nd 7994  GIdcgi 30872  CRingOpsccring 38677  Dmncdmn 38731

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-1o 8462  df-en 8953  df-grpo 30875  df-gid 30876  df-ginv 30877  df-gdiv 30878  df-ablo 30927  df-ass 38527  df-exid 38529  df-mgmOLD 38533  df-sgrOLD 38545  df-mndo 38551  df-rngo 38579  df-com2 38674  df-crngo 38678  df-idl 38694  df-pridl 38695  df-prrngo 38732  df-dmn 38733  df-igen 38744

This theorem is used by: (None)