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Theorem dmncan1 38083
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 38064 . . . . . 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 2737 . . . . . . 7 ( /𝑔𝐺) = ( /𝑔𝐺)
62, 3, 4, 5rngosubdi 37952 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)))
71, 6sylan 580 . . . . 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 2739 . . 3 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 ↔ ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍))
102rngogrpo 37917 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
111, 10syl 17 . . . . . . . . . . 11 (𝑅 ∈ Dmn → 𝐺 ∈ GrpOp)
124, 5grpodivcl 30558 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋)
13123expb 1121 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋)
1411, 13sylan 580 . . . . . . . . . 10 ((𝑅 ∈ Dmn ∧ (𝐵𝑋𝐶𝑋)) → (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋)
1514adantlr 715 . . . . . . . . 9 (((𝑅 ∈ Dmn ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋)
16 dmncan.4 . . . . . . . . . . . 12 𝑍 = (GId‘𝐺)
172, 3, 4, 16dmnnzd 38082 . . . . . . . . . . 11 ((𝑅 ∈ Dmn ∧ (𝐴𝑋 ∧ (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋 ∧ (𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍)) → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
18173exp2 1355 . . . . . . . . . 10 (𝑅 ∈ Dmn → (𝐴𝑋 → ((𝐵( /𝑔𝐺)𝐶) ∈ 𝑋 → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍)))))
1918imp31 417 . . . . . . . . 9 (((𝑅 ∈ Dmn ∧ 𝐴𝑋) ∧ (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍)))
2015, 19syldan 591 . . . . . . . 8 (((𝑅 ∈ Dmn ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍)))
2120exp43 436 . . . . . . 7 (𝑅 ∈ Dmn → (𝐴𝑋 → (𝐵𝑋 → (𝐶𝑋 → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍))))))
22213imp2 1350 . . . . . 6 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍)))
23 neor 3034 . . . . . 6 ((𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍) ↔ (𝐴𝑍 → (𝐵( /𝑔𝐺)𝐶) = 𝑍))
2422, 23imbitrdi 251 . . . . 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 406 . . 3 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐵( /𝑔𝐺)𝐶) = 𝑍))
279, 26sylbird 260 . 2 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → (((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍 → (𝐵( /𝑔𝐺)𝐶) = 𝑍))
2811adantr 480 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐺 ∈ GrpOp)
292, 3, 4rngocl 37908 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
30293adant3r3 1185 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
311, 30sylan 580 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
322, 3, 4rngocl 37908 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
33323adant3r2 1184 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
341, 33sylan 580 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
354, 16, 5grpoeqdivid 37888 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝐵) ∈ 𝑋 ∧ (𝐴𝐻𝐶) ∈ 𝑋) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍))
3628, 31, 34, 35syl3anc 1373 . . 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 37888 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
39383expb 1121 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
4011, 39sylan 580 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐵𝑋𝐶𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
41403adantr1 1170 . . 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 294 1 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) → 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  ran crn 5686  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  GrpOpcgr 30508  GIdcgi 30509   /𝑔 cgs 30511  RingOpscrngo 37901  Dmncdmn 38054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-1o 8506  df-en 8986  df-grpo 30512  df-gid 30513  df-ginv 30514  df-gdiv 30515  df-ablo 30564  df-ass 37850  df-exid 37852  df-mgmOLD 37856  df-sgrOLD 37868  df-mndo 37874  df-rngo 37902  df-com2 37997  df-crngo 38001  df-idl 38017  df-pridl 38018  df-prrngo 38055  df-dmn 38056  df-igen 38067
This theorem is referenced by:  dmncan2  38084
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