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Theorem dmncan1 34005
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 33986 . . . . . 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 2651 . . . . . . 7 ( /𝑔𝐺) = ( /𝑔𝐺)
62, 3, 4, 5rngosubdi 33874 . . . . . 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 2653 . . 3 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 ↔ ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍))
102rngogrpo 33839 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
111, 10syl 17 . . . . . . . . . . 11 (𝑅 ∈ Dmn → 𝐺 ∈ GrpOp)
124, 5grpodivcl 27521 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋)
13123expb 1285 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋)
1411, 13sylan 487 . . . . . . . . . 10 ((𝑅 ∈ Dmn ∧ (𝐵𝑋𝐶𝑋)) → (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋)
1514adantlr 751 . . . . . . . . 9 (((𝑅 ∈ Dmn ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋)
16 dmncan.4 . . . . . . . . . . . 12 𝑍 = (GId‘𝐺)
172, 3, 4, 16dmnnzd 34004 . . . . . . . . . . 11 ((𝑅 ∈ Dmn ∧ (𝐴𝑋 ∧ (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋 ∧ (𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍)) → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
18173exp2 1307 . . . . . . . . . 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 639 . . . . . . 7 (𝑅 ∈ Dmn → (𝐴𝑋 → (𝐵𝑋 → (𝐶𝑋 → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍))))))
22213imp2 1304 . . . . . 6 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍)))
23 neor 2914 . . . . . 6 ((𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍) ↔ (𝐴𝑍 → (𝐵( /𝑔𝐺)𝐶) = 𝑍))
2422, 23syl6ib 241 . . . . 5 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴𝑍 → (𝐵( /𝑔𝐺)𝐶) = 𝑍)))
2524com23 86 . . . 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 250 . 2 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → (((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍 → (𝐵( /𝑔𝐺)𝐶) = 𝑍))
2811adantr 480 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐺 ∈ GrpOp)
292, 3, 4rngocl 33830 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
30293adant3r3 1297 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
311, 30sylan 487 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
322, 3, 4rngocl 33830 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
33323adant3r2 1296 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
341, 33sylan 487 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
354, 16, 5grpoeqdivid 33810 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝐵) ∈ 𝑋 ∧ (𝐴𝐻𝐶) ∈ 𝑋) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍))
3628, 31, 34, 35syl3anc 1366 . . 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 33810 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
39383expb 1285 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
4011, 39sylan 487 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐵𝑋𝐶𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
41403adantr1 1240 . . 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 283 1 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) → 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  ran crn 5144  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  GrpOpcgr 27471  GIdcgi 27472   /𝑔 cgs 27474  RingOpscrngo 33823  Dmncdmn 33976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-grpo 27475  df-gid 27476  df-ginv 27477  df-gdiv 27478  df-ablo 27527  df-ass 33772  df-exid 33774  df-mgmOLD 33778  df-sgrOLD 33790  df-mndo 33796  df-rngo 33824  df-com2 33919  df-crngo 33923  df-idl 33939  df-pridl 33940  df-prrngo 33977  df-dmn 33978  df-igen 33989
This theorem is referenced by:  dmncan2  34006
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