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Theorem dmncan1 38062
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 38043 . . . . . 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 2734 . . . . . . 7 ( /𝑔𝐺) = ( /𝑔𝐺)
62, 3, 4, 5rngosubdi 37931 . . . . . 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 2736 . . 3 (((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) ∧ 𝐴𝑍) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 ↔ ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍))
102rngogrpo 37896 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
111, 10syl 17 . . . . . . . . . . 11 (𝑅 ∈ Dmn → 𝐺 ∈ GrpOp)
124, 5grpodivcl 30567 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋)
13123expb 1119 . . . . . . . . . . 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 38061 . . . . . . . . . . 11 ((𝑅 ∈ Dmn ∧ (𝐴𝑋 ∧ (𝐵( /𝑔𝐺)𝐶) ∈ 𝑋 ∧ (𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍)) → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
18173exp2 1353 . . . . . . . . . 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 1348 . . . . . 6 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻(𝐵( /𝑔𝐺)𝐶)) = 𝑍 → (𝐴 = 𝑍 ∨ (𝐵( /𝑔𝐺)𝐶) = 𝑍)))
23 neor 3031 . . . . . 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 37887 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
30293adant3r3 1183 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
311, 30sylan 580 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
322, 3, 4rngocl 37887 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
33323adant3r2 1182 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
341, 33sylan 580 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
354, 16, 5grpoeqdivid 37867 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝐻𝐵) ∈ 𝑋 ∧ (𝐴𝐻𝐶) ∈ 𝑋) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) ↔ ((𝐴𝐻𝐵)( /𝑔𝐺)(𝐴𝐻𝐶)) = 𝑍))
3628, 31, 34, 35syl3anc 1370 . . 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 37867 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐵𝑋𝐶𝑋) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
39383expb 1119 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝐵𝑋𝐶𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
4011, 39sylan 580 . . . 4 ((𝑅 ∈ Dmn ∧ (𝐵𝑋𝐶𝑋)) → (𝐵 = 𝐶 ↔ (𝐵( /𝑔𝐺)𝐶) = 𝑍))
41403adantr1 1168 . . 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 847  w3a 1086   = wceq 1536  wcel 2105  wne 2937  ran crn 5689  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  GrpOpcgr 30517  GIdcgi 30518   /𝑔 cgs 30520  RingOpscrngo 37880  Dmncdmn 38033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-1o 8504  df-en 8984  df-grpo 30521  df-gid 30522  df-ginv 30523  df-gdiv 30524  df-ablo 30573  df-ass 37829  df-exid 37831  df-mgmOLD 37835  df-sgrOLD 37847  df-mndo 37853  df-rngo 37881  df-com2 37976  df-crngo 37980  df-idl 37996  df-pridl 37997  df-prrngo 38034  df-dmn 38035  df-igen 38046
This theorem is referenced by:  dmncan2  38063
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