Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvrunz Structured version   Visualization version   GIF version

Theorem dvrunz 33724
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)
2 fvex 6188 . . . 4 (GId‘𝐺) ∈ V
31, 2eqeltri 2695 . . 3 𝑍 ∈ V
43zrdivrng 33723 . 2 ¬ ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps
5 dvrunz.1 . . . . . . 7 𝐺 = (1st𝑅)
6 dvrunz.2 . . . . . . 7 𝐻 = (2nd𝑅)
7 dvrunz.3 . . . . . . 7 𝑋 = ran 𝐺
85, 6, 7, 1drngoi 33721 . . . . . 6 (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
98simpld 475 . . . . 5 (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps)
10 dvrunz.5 . . . . . 6 𝑈 = (GId‘𝐻)
115, 6, 1, 10, 7rngoueqz 33710 . . . . 5 (𝑅 ∈ RingOps → (𝑋 ≈ 1𝑜𝑈 = 𝑍))
129, 11syl 17 . . . 4 (𝑅 ∈ DivRingOps → (𝑋 ≈ 1𝑜𝑈 = 𝑍))
135, 7, 1rngosn6 33696 . . . . . . 7 (𝑅 ∈ RingOps → (𝑋 ≈ 1𝑜𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
149, 13syl 17 . . . . . 6 (𝑅 ∈ DivRingOps → (𝑋 ≈ 1𝑜𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩))
15 eleq1 2687 . . . . . . 7 (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps ↔ ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
1615biimpd 219 . . . . . 6 (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
1714, 16syl6bi 243 . . . . 5 (𝑅 ∈ DivRingOps → (𝑋 ≈ 1𝑜 → (𝑅 ∈ DivRingOps → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps)))
1817pm2.43a 54 . . . 4 (𝑅 ∈ DivRingOps → (𝑋 ≈ 1𝑜 → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
1912, 18sylbird 250 . . 3 (𝑅 ∈ DivRingOps → (𝑈 = 𝑍 → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps))
2019necon3bd 2805 . 2 (𝑅 ∈ DivRingOps → (¬ ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps → 𝑈𝑍))
214, 20mpi 20 1 (𝑅 ∈ DivRingOps → 𝑈𝑍)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1481  wcel 1988  wne 2791  Vcvv 3195  cdif 3564  {csn 4168  cop 4174   class class class wbr 4644   × cxp 5102  ran crn 5105  cres 5106  cfv 5876  1st c1st 7151  2nd c2nd 7152  1𝑜c1o 7538  cen 7937  GrpOpcgr 27313  GIdcgi 27314  RingOpscrngo 33664  DivRingOpscdrng 33718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-om 7051  df-1st 7153  df-2nd 7154  df-1o 7545  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-grpo 27317  df-gid 27318  df-ablo 27369  df-ass 33613  df-exid 33615  df-mgmOLD 33619  df-sgrOLD 33631  df-mndo 33637  df-rngo 33665  df-drngo 33719
This theorem is referenced by:  isdrngo2  33728  divrngpr  33823  isfldidl  33838
  Copyright terms: Public domain W3C validator