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Theorem dmncan2 35347
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 35326 . . . 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 35271 . . . . . 6 ((𝑅 ∈ CRingOps ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐻𝐶) = (𝐶𝐻𝐴))
653adant3r2 1177 . . . . 5 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐶) = (𝐶𝐻𝐴))
72, 3, 4crngocom 35271 . . . . . 6 ((𝑅 ∈ CRingOps ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵))
873adant3r1 1176 . . . . 5 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵))
96, 8eqeq12d 2835 . . . 4 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵)))
101, 9sylan 582 . . 3 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵)))
1110adantr 483 . 2 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐶𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) ↔ (𝐶𝐻𝐴) = (𝐶𝐻𝐵)))
12 3anrot 1094 . . . 4 ((𝐶𝑋𝐴𝑋𝐵𝑋) ↔ (𝐴𝑋𝐵𝑋𝐶𝑋))
1312biimpri 230 . . 3 ((𝐴𝑋𝐵𝑋𝐶𝑋) → (𝐶𝑋𝐴𝑋𝐵𝑋))
14 dmncan.4 . . . 4 𝑍 = (GId‘𝐺)
152, 3, 4, 14dmncan1 35346 . . 3 (((𝑅 ∈ Dmn ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) ∧ 𝐶𝑍) → ((𝐶𝐻𝐴) = (𝐶𝐻𝐵) → 𝐴 = 𝐵))
1613, 15sylanl2 679 . 2 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐶𝑍) → ((𝐶𝐻𝐴) = (𝐶𝐻𝐵) → 𝐴 = 𝐵))
1711, 16sylbid 242 1 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐶𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  wne 3014  ran crn 5549  cfv 6348  (class class class)co 7148  1st c1st 7679  2nd c2nd 7680  GIdcgi 28259  CRingOpsccring 35263  Dmncdmn 35317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-1o 8094  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-grpo 28262  df-gid 28263  df-ginv 28264  df-gdiv 28265  df-ablo 28314  df-ass 35113  df-exid 35115  df-mgmOLD 35119  df-sgrOLD 35131  df-mndo 35137  df-rngo 35165  df-com2 35260  df-crngo 35264  df-idl 35280  df-pridl 35281  df-prrngo 35318  df-dmn 35319  df-igen 35330
This theorem is referenced by: (None)
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