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Theorem dmncan1 33045
 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
dmncan1 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) → 𝐵 = 𝐶))

Proof of Theorem dmncan1
StepHypRef Expression
1 dmnrngo 33026 . . . . . 6 (𝑅 ∈ Dmn → 𝑅 ∈ RingOps)
2 dmncan.1 . . . . . . 7 𝐺 = (1st𝑅)
3 dmncan.2 . . . . . . 7 𝐻 = (2nd𝑅)
4 dmncan.3 . . . . . . 7 𝑋 = ran 𝐺
5 eqid 2610 . . . . . . 7 ( /𝑔𝐺) = ( /𝑔𝐺)
62, 3, 4, 5rngosubdi 32914 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)))
71, 6sylan 487 . . . . 5 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)))
87adantr 480 . . . 4 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → (𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)))
98eqeq1d 2612 . . 3 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 ↔ ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍))
102rngogrpo 32879 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
111, 10syl 17 . . . . . . . . . . 11 (𝑅 ∈ Dmn → 𝐺 ∈ GrpOp)
124, 5grpodivcl 26777 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋)
13123expb 1258 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋)
1411, 13sylan 487 . . . . . . . . . 10 ((𝑅 ∈ Dmn ∧ (𝐵𝑋𝐶𝑋)) → (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋)
1514adantlr 747 . . . . . . . . 9 (((𝑅 ∈ Dmn ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋)
16 dmncan.4 . . . . . . . . . . . 12 𝑍 = (GId‘𝐺)
172, 3, 4, 16dmnnzd 33044 . . . . . . . . . . 11 ((𝑅 ∈ Dmn ∧ (𝐴𝑋 ∧ (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋 ∧ (𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍)) → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
18173exp2 1277 . . . . . . . . . 10 (𝑅 ∈ Dmn → (𝐴𝑋 → ((𝐵( /𝑔𝐺)𝐶) ∈ 𝑋 → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍)))))
1918imp31 447 . . . . . . . . 9 (((𝑅 ∈ Dmn ∧ 𝐴𝑋) ∧ (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍)))
2015, 19syldan 486 . . . . . . . 8 (((𝑅 ∈ Dmn ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍)))
2120exp43 638 . . . . . . 7 (𝑅 ∈ Dmn → (𝐴𝑋 → (𝐵𝑋 → (𝐶𝑋 → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍))))))
22213imp2 1274 . . . . . 6 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍)))
23 neor 2873 . . . . . 6 ((𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍) ↔ (𝐴𝑍 → (𝐵( /𝑔𝐺)𝐶) = 𝑍))
2422, 23syl6ib 240 . . . . 5 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴𝑍 → (𝐵( /𝑔𝐺)𝐶) = 𝑍)))
2524com23 84 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑍 → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐵( /𝑔𝐺)𝐶) = 𝑍)))
2625imp 444 . . 3 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐵( /𝑔𝐺)𝐶) = 𝑍))
279, 26sylbird 249 . 2 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → (((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍 → (𝐵( /𝑔𝐺)𝐶) = 𝑍))
2811adantr 480 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐺 ∈ GrpOp)
292, 3, 4rngocl 32870 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
30293adant3r3 1268 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
311, 30sylan 487 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
322, 3, 4rngocl 32870 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
33323adant3r2 1267 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
341, 33sylan 487 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
354, 16, 5grpoeqdivid 32850 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝐵) ∈ 𝑋 ∧ (𝐴𝐻𝐶) ∈ 𝑋) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍))
3628, 31, 34, 35syl3anc 1318 . . 3 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍))
3736adantr 480 . 2 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍))
384, 16, 5grpoeqdivid 32850 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
39383expb 1258 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
4011, 39sylan 487 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐵𝑋𝐶𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
41403adantr1 1213 . . 3 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
4241adantr 480 . 2 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
4327, 37, 423imtr4d 282 1 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) → 𝐵 = 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ran crn 5039  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  GrpOpcgr 26727  GIdcgi 26728   /𝑔 cgs 26730  RingOpscrngo 32863  Dmncdmn 33016 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-grpo 26731  df-gid 26732  df-ginv 26733  df-gdiv 26734  df-ablo 26783  df-ass 32812  df-exid 32814  df-mgmOLD 32818  df-sgrOLD 32830  df-mndo 32836  df-rngo 32864  df-com2 32959  df-crngo 32963  df-idl 32979  df-pridl 32980  df-prrngo 33017  df-dmn 33018  df-igen 33029 This theorem is referenced by:  dmncan2  33046
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