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Theorem dvrunz 36817
Description: In a division ring the ring unit is different from the zero. (Contributed by FL, 14-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
dvrunz.1 𝐺 = (1st𝑅)
dvrunz.2 𝐻 = (2nd𝑅)
dvrunz.3 𝑋 = ran 𝐺
dvrunz.4 𝑍 = (GId‘𝐺)
dvrunz.5 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
dvrunz (𝑅 ∈ DivRingOps → 𝑈𝑍)

Proof of Theorem dvrunz
StepHypRef Expression
1 dvrunz.4 . . . 4 𝑍 = (GId‘𝐺)
21fvexi 6905 . . 3 𝑍 ∈ V
32zrdivrng 36816 . 2 ¬ ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps
4 dvrunz.1 . . . . . . 7 𝐺 = (1st𝑅)
5 dvrunz.2 . . . . . . 7 𝐻 = (2nd𝑅)
6 dvrunz.3 . . . . . . 7 𝑋 = ran 𝐺
74, 5, 6, 1drngoi 36814 . . . . . 6 (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
87simpld 495 . . . . 5 (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps)
9 dvrunz.5 . . . . . 6 𝑈 = (GId‘𝐻)
104, 5, 1, 9, 6rngoueqz 36803 . . . . 5 (𝑅 ∈ RingOps → (𝑋 ≈ 1o𝑈 = 𝑍))
118, 10syl 17 . . . 4 (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o𝑈 = 𝑍))
124, 6, 1rngosn6 36789 . . . . . . 7 (𝑅 ∈ RingOps → (𝑋 ≈ 1o𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
138, 12syl 17 . . . . . 6 (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
14 eleq1 2821 . . . . . . 7 (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps ↔ ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
1514biimpd 228 . . . . . 6 (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
1613, 15syl6bi 252 . . . . 5 (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o → (𝑅 ∈ DivRingOps → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps)))
1716pm2.43a 54 . . . 4 (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
1811, 17sylbird 259 . . 3 (𝑅 ∈ DivRingOps → (𝑈 = 𝑍 → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
1918necon3bd 2954 . 2 (𝑅 ∈ DivRingOps → (¬ ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps → 𝑈𝑍))
203, 19mpi 20 1 (𝑅 ∈ DivRingOps → 𝑈𝑍)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1541  wcel 2106  wne 2940  cdif 3945  {csn 4628  cop 4634   class class class wbr 5148   × cxp 5674  ran crn 5677  cres 5678  cfv 6543  1st c1st 7972  2nd c2nd 7973  1oc1o 8458  cen 8935  GrpOpcgr 29737  GIdcgi 29738  RingOpscrngo 36757  DivRingOpscdrng 36811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-1st 7974  df-2nd 7975  df-1o 8465  df-en 8939  df-grpo 29741  df-gid 29742  df-ablo 29793  df-ass 36706  df-exid 36708  df-mgmOLD 36712  df-sgrOLD 36724  df-mndo 36730  df-rngo 36758  df-drngo 36812
This theorem is referenced by:  isdrngo2  36821  divrngpr  36916  isfldidl  36931
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