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Theorem dvrunz 37940
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 6920 . . 3 𝑍 ∈ V
32zrdivrng 37939 . 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 37937 . . . . . 6 (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
87simpld 494 . . . . 5 (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps)
9 dvrunz.5 . . . . . 6 𝑈 = (GId‘𝐻)
104, 5, 1, 9, 6rngoueqz 37926 . . . . 5 (𝑅 ∈ RingOps → (𝑋 ≈ 1o𝑈 = 𝑍))
118, 10syl 17 . . . 4 (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o𝑈 = 𝑍))
124, 6, 1rngosn6 37912 . . . . . . 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 2826 . . . . . . 7 (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps ↔ ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
1514biimpd 229 . . . . . 6 (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
1613, 15biimtrdi 253 . . . . 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 260 . . 3 (𝑅 ∈ DivRingOps → (𝑈 = 𝑍 → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
1918necon3bd 2951 . 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 206   = wceq 1536  wcel 2105  wne 2937  cdif 3959  {csn 4630  cop 4636   class class class wbr 5147   × cxp 5686  ran crn 5689  cres 5690  cfv 6562  1st c1st 8010  2nd c2nd 8011  1oc1o 8497  cen 8980  GrpOpcgr 30517  GIdcgi 30518  RingOpscrngo 37880  DivRingOpscdrng 37934
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-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-1st 8012  df-2nd 8013  df-1o 8504  df-en 8984  df-grpo 30521  df-gid 30522  df-ablo 30573  df-ass 37829  df-exid 37831  df-mgmOLD 37835  df-sgrOLD 37847  df-mndo 37853  df-rngo 37881  df-drngo 37935
This theorem is referenced by:  isdrngo2  37944  divrngpr  38039  isfldidl  38054
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