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Theorem dmncan1 37247
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 37228 . . . . . 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 2730 . . . . . . 7 ( /𝑔𝐺) = ( /𝑔𝐺)
62, 3, 4, 5rngosubdi 37116 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)))
71, 6sylan 578 . . . . 5 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)))
87adantr 479 . . . 4 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → (𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)))
98eqeq1d 2732 . . 3 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 ↔ ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍))
102rngogrpo 37081 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
111, 10syl 17 . . . . . . . . . . 11 (𝑅 ∈ Dmn → 𝐺 ∈ GrpOp)
124, 5grpodivcl 30059 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋)
13123expb 1118 . . . . . . . . . . 11 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋)
1411, 13sylan 578 . . . . . . . . . 10 ((𝑅 ∈ Dmn ∧ (𝐵𝑋𝐶𝑋)) → (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋)
1514adantlr 711 . . . . . . . . 9 (((𝑅 ∈ Dmn ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋)
16 dmncan.4 . . . . . . . . . . . 12 𝑍 = (GId‘𝐺)
172, 3, 4, 16dmnnzd 37246 . . . . . . . . . . 11 ((𝑅 ∈ Dmn ∧ (𝐴𝑋 ∧ (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋 ∧ (𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍)) → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
18173exp2 1352 . . . . . . . . . 10 (𝑅 ∈ Dmn → (𝐴𝑋 → ((𝐵( /𝑔𝐺)𝐶) ∈ 𝑋 → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍)))))
1918imp31 416 . . . . . . . . 9 (((𝑅 ∈ Dmn ∧ 𝐴𝑋) ∧ (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍)))
2015, 19syldan 589 . . . . . . . 8 (((𝑅 ∈ Dmn ∧ 𝐴𝑋) ∧ (𝐵𝑋𝐶𝑋)) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍)))
2120exp43 435 . . . . . . 7 (𝑅 ∈ Dmn → (𝐴𝑋 → (𝐵𝑋 → (𝐶𝑋 → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍))))))
22213imp2 1347 . . . . . 6 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍)))
23 neor 3032 . . . . . 6 ((𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍) ↔ (𝐴𝑍 → (𝐵( /𝑔𝐺)𝐶) = 𝑍))
2422, 23imbitrdi 250 . . . . 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 405 . . 3 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐵( /𝑔𝐺)𝐶) = 𝑍))
279, 26sylbird 259 . 2 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → (((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍 → (𝐵( /𝑔𝐺)𝐶) = 𝑍))
2811adantr 479 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → 𝐺 ∈ GrpOp)
292, 3, 4rngocl 37072 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
30293adant3r3 1182 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
311, 30sylan 578 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
322, 3, 4rngocl 37072 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
33323adant3r2 1181 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
341, 33sylan 578 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
354, 16, 5grpoeqdivid 37052 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝐵) ∈ 𝑋 ∧ (𝐴𝐻𝐶) ∈ 𝑋) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍))
3628, 31, 34, 35syl3anc 1369 . . 3 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍))
3736adantr 479 . 2 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍))
384, 16, 5grpoeqdivid 37052 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
39383expb 1118 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
4011, 39sylan 578 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐵𝑋𝐶𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
41403adantr1 1167 . . 3 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
4241adantr 479 . 2 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
4327, 37, 423imtr4d 293 1 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) → 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 843  w3a 1085   = wceq 1539  wcel 2104  wne 2938  ran crn 5676  cfv 6542  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  GrpOpcgr 30009  GIdcgi 30010   /𝑔 cgs 30012  RingOpscrngo 37065  Dmncdmn 37218
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-1o 8468  df-en 8942  df-grpo 30013  df-gid 30014  df-ginv 30015  df-gdiv 30016  df-ablo 30065  df-ass 37014  df-exid 37016  df-mgmOLD 37020  df-sgrOLD 37032  df-mndo 37038  df-rngo 37066  df-com2 37161  df-crngo 37165  df-idl 37181  df-pridl 37182  df-prrngo 37219  df-dmn 37220  df-igen 37231
This theorem is referenced by:  dmncan2  37248
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