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Theorem dmncan2 36945
Description: Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
Hypotheses
Ref Expression
dmncan.1 𝐺 = (1st𝑅)
dmncan.2 𝐻 = (2nd𝑅)
dmncan.3 𝑋 = ran 𝐺
dmncan.4 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
dmncan2 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐶𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) → 𝐴 = 𝐵))

Proof of Theorem dmncan2
StepHypRef Expression
1 dmncrng 36924 . . . 4 (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)
2 dmncan.1 . . . . . . 7 𝐺 = (1st𝑅)
3 dmncan.2 . . . . . . 7 𝐻 = (2nd𝑅)
4 dmncan.3 . . . . . . 7 𝑋 = ran 𝐺
52, 3, 4crngocom 36869 . . . . . 6 ((𝑅 ∈ CRingOps ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐻𝐶) = (𝐶𝐻𝐴))
653adant3r2 1184 . . . . 5 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐶) = (𝐶𝐻𝐴))
72, 3, 4crngocom 36869 . . . . . 6 ((𝑅 ∈ CRingOps ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵))
873adant3r1 1183 . . . . 5 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵))
96, 8eqeq12d 2749 . . . 4 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵)))
101, 9sylan 581 . . 3 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵)))
1110adantr 482 . 2 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐶𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵)))
12 3anrot 1101 . . . 4 ((𝐶𝑋𝐴𝑋𝐵𝑋) ↔ (𝐴𝑋𝐵𝑋𝐶𝑋))
1312biimpri 227 . . 3 ((𝐴𝑋𝐵𝑋𝐶𝑋) → (𝐶𝑋𝐴𝑋𝐵𝑋))
14 dmncan.4 . . . 4 𝑍 = (GId‘𝐺)
152, 3, 4, 14dmncan1 36944 . . 3 (((𝑅 ∈ Dmn ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) ∧ 𝐶𝑍) → ((𝐶𝐻𝐴) = (𝐶𝐻𝐵) → 𝐴 = 𝐵))
1613, 15sylanl2 680 . 2 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐶𝑍) → ((𝐶𝐻𝐴) = (𝐶𝐻𝐵) → 𝐴 = 𝐵))
1711, 16sylbid 239 1 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐶𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2941  ran crn 5678  cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  GIdcgi 29743  CRingOpsccring 36861  Dmncdmn 36915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-1o 8466  df-en 8940  df-grpo 29746  df-gid 29747  df-ginv 29748  df-gdiv 29749  df-ablo 29798  df-ass 36711  df-exid 36713  df-mgmOLD 36717  df-sgrOLD 36729  df-mndo 36735  df-rngo 36763  df-com2 36858  df-crngo 36862  df-idl 36878  df-pridl 36879  df-prrngo 36916  df-dmn 36917  df-igen 36928
This theorem is referenced by: (None)
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