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Theorem dvrunz 36463
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 6860 . . 3 𝑍 ∈ V
32zrdivrng 36462 . 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 36460 . . . . . 6 (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
87simpld 496 . . . . 5 (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps)
9 dvrunz.5 . . . . . 6 𝑈 = (GId‘𝐻)
104, 5, 1, 9, 6rngoueqz 36449 . . . . 5 (𝑅 ∈ RingOps → (𝑋 ≈ 1o𝑈 = 𝑍))
118, 10syl 17 . . . 4 (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o𝑈 = 𝑍))
124, 6, 1rngosn6 36435 . . . . . . 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 2822 . . . . . . 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 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 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 1542  wcel 2107  wne 2940  cdif 3911  {csn 4590  cop 4596   class class class wbr 5109   × cxp 5635  ran crn 5638  cres 5639  cfv 6500  1st c1st 7923  2nd c2nd 7924  1oc1o 8409  cen 8886  GrpOpcgr 29480  GIdcgi 29481  RingOpscrngo 36403  DivRingOpscdrng 36457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-1st 7925  df-2nd 7926  df-1o 8416  df-en 8890  df-grpo 29484  df-gid 29485  df-ablo 29536  df-ass 36352  df-exid 36354  df-mgmOLD 36358  df-sgrOLD 36370  df-mndo 36376  df-rngo 36404  df-drngo 36458
This theorem is referenced by:  isdrngo2  36467  divrngpr  36562  isfldidl  36577
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