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Theorem isdrngo1 36819
Description: The predicate "is a division ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
Hypotheses
Ref Expression
isdivrng1.1 𝐺 = (1st𝑅)
isdivrng1.2 𝐻 = (2nd𝑅)
isdivrng1.3 𝑍 = (GId‘𝐺)
isdivrng1.4 𝑋 = ran 𝐺
Assertion
Ref Expression
isdrngo1 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))

Proof of Theorem isdrngo1
Dummy variables 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-drngo 36812 . . . 4 DivRingOps = {⟨𝑔, ⟩ ∣ (⟨𝑔, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
21relopabiv 5820 . . 3 Rel DivRingOps
3 1st2nd 8024 . . 3 ((Rel DivRingOps ∧ 𝑅 ∈ DivRingOps) → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
42, 3mpan 688 . 2 (𝑅 ∈ DivRingOps → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
5 relrngo 36759 . . . 4 Rel RingOps
6 1st2nd 8024 . . . 4 ((Rel RingOps ∧ 𝑅 ∈ RingOps) → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
75, 6mpan 688 . . 3 (𝑅 ∈ RingOps → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
87adantr 481 . 2 ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
9 isdivrng1.1 . . . . 5 𝐺 = (1st𝑅)
10 isdivrng1.2 . . . . 5 𝐻 = (2nd𝑅)
119, 10opeq12i 4878 . . . 4 𝐺, 𝐻⟩ = ⟨(1st𝑅), (2nd𝑅)⟩
1211eqeq2i 2745 . . 3 (𝑅 = ⟨𝐺, 𝐻⟩ ↔ 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
1310fvexi 6905 . . . . . 6 𝐻 ∈ V
14 isdivrngo 36813 . . . . . 6 (𝐻 ∈ V → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
1513, 14ax-mp 5 . . . . 5 (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp))
16 isdivrng1.4 . . . . . . . . . 10 𝑋 = ran 𝐺
17 isdivrng1.3 . . . . . . . . . . 11 𝑍 = (GId‘𝐺)
1817sneqi 4639 . . . . . . . . . 10 {𝑍} = {(GId‘𝐺)}
1916, 18difeq12i 4120 . . . . . . . . 9 (𝑋 ∖ {𝑍}) = (ran 𝐺 ∖ {(GId‘𝐺)})
2019, 19xpeq12i 5704 . . . . . . . 8 ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) = ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))
2120reseq2i 5978 . . . . . . 7 (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)})))
2221eleq1i 2824 . . . . . 6 ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp ↔ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)
2322anbi2i 623 . . . . 5 ((⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp))
2415, 23bitr4i 277 . . . 4 (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
25 eleq1 2821 . . . . 5 (𝑅 = ⟨𝐺, 𝐻⟩ → (𝑅 ∈ DivRingOps ↔ ⟨𝐺, 𝐻⟩ ∈ DivRingOps))
26 eleq1 2821 . . . . . 6 (𝑅 = ⟨𝐺, 𝐻⟩ → (𝑅 ∈ RingOps ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps))
2726anbi1d 630 . . . . 5 (𝑅 = ⟨𝐺, 𝐻⟩ → ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
2825, 27bibi12d 345 . . . 4 (𝑅 = ⟨𝐺, 𝐻⟩ → ((𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)) ↔ (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))))
2924, 28mpbiri 257 . . 3 (𝑅 = ⟨𝐺, 𝐻⟩ → (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
3012, 29sylbir 234 . 2 (𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩ → (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
314, 8, 30pm5.21nii 379 1 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  cdif 3945  {csn 4628  cop 4634   × cxp 5674  ran crn 5677  cres 5678  Rel wrel 5681  cfv 6543  1st c1st 7972  2nd c2nd 7973  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-sep 5299  ax-nul 5306  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-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  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-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-1st 7974  df-2nd 7975  df-rngo 36758  df-drngo 36812
This theorem is referenced by:  divrngcl  36820  isdrngo2  36821  divrngpr  36916
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