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Theorem dvrunz 35543
 Description: In a division ring the 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 6669 . . 3 𝑍 ∈ V
32zrdivrng 35542 . 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 35540 . . . . . 6 (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
87simpld 498 . . . . 5 (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps)
9 dvrunz.5 . . . . . 6 𝑈 = (GId‘𝐻)
104, 5, 1, 9, 6rngoueqz 35529 . . . . 5 (𝑅 ∈ RingOps → (𝑋 ≈ 1o𝑈 = 𝑍))
118, 10syl 17 . . . 4 (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o𝑈 = 𝑍))
124, 6, 1rngosn6 35515 . . . . . . 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 2877 . . . . . . 7 (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps ↔ ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
1514biimpd 232 . . . . . 6 (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
1613, 15syl6bi 256 . . . . 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 263 . . 3 (𝑅 ∈ DivRingOps → (𝑈 = 𝑍 → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
1918necon3bd 3001 . 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 209   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   ∖ cdif 3880  {csn 4528  ⟨cop 4534   class class class wbr 5034   × cxp 5521  ran crn 5524   ↾ cres 5525  ‘cfv 6332  1st c1st 7682  2nd c2nd 7683  1oc1o 8096   ≈ cen 8507  GrpOpcgr 28316  GIdcgi 28317  RingOpscrngo 35483  DivRingOpscdrng 35537 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-om 7574  df-1st 7684  df-2nd 7685  df-1o 8103  df-er 8290  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-grpo 28320  df-gid 28321  df-ablo 28372  df-ass 35432  df-exid 35434  df-mgmOLD 35438  df-sgrOLD 35450  df-mndo 35456  df-rngo 35484  df-drngo 35538 This theorem is referenced by:  isdrngo2  35547  divrngpr  35642  isfldidl  35657
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