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Theorem drngoi 37938
Description: The properties of a division ring. (Contributed by NM, 4-Apr-2009.) (New usage is discouraged.)
Hypotheses
Ref Expression
drngi.1 𝐺 = (1st𝑅)
drngi.2 𝐻 = (2nd𝑅)
drngi.3 𝑋 = ran 𝐺
drngi.4 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
drngoi (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))

Proof of Theorem drngoi
Dummy variables 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 4878 . . . . . 6 (𝑔 = (1st𝑅) → ⟨𝑔, ⟩ = ⟨(1st𝑅), ⟩)
21eleq1d 2824 . . . . 5 (𝑔 = (1st𝑅) → (⟨𝑔, ⟩ ∈ RingOps ↔ ⟨(1st𝑅), ⟩ ∈ RingOps))
3 id 22 . . . . . . . . . . . 12 (𝑔 = (1st𝑅) → 𝑔 = (1st𝑅))
4 drngi.1 . . . . . . . . . . . 12 𝐺 = (1st𝑅)
53, 4eqtr4di 2793 . . . . . . . . . . 11 (𝑔 = (1st𝑅) → 𝑔 = 𝐺)
65rneqd 5952 . . . . . . . . . 10 (𝑔 = (1st𝑅) → ran 𝑔 = ran 𝐺)
7 drngi.3 . . . . . . . . . 10 𝑋 = ran 𝐺
86, 7eqtr4di 2793 . . . . . . . . 9 (𝑔 = (1st𝑅) → ran 𝑔 = 𝑋)
95fveq2d 6911 . . . . . . . . . . 11 (𝑔 = (1st𝑅) → (GId‘𝑔) = (GId‘𝐺))
10 drngi.4 . . . . . . . . . . 11 𝑍 = (GId‘𝐺)
119, 10eqtr4di 2793 . . . . . . . . . 10 (𝑔 = (1st𝑅) → (GId‘𝑔) = 𝑍)
1211sneqd 4643 . . . . . . . . 9 (𝑔 = (1st𝑅) → {(GId‘𝑔)} = {𝑍})
138, 12difeq12d 4137 . . . . . . . 8 (𝑔 = (1st𝑅) → (ran 𝑔 ∖ {(GId‘𝑔)}) = (𝑋 ∖ {𝑍}))
1413sqxpeqd 5721 . . . . . . 7 (𝑔 = (1st𝑅) → ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)})) = ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
1514reseq2d 6000 . . . . . 6 (𝑔 = (1st𝑅) → ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) = ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
1615eleq1d 2824 . . . . 5 (𝑔 = (1st𝑅) → (( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp ↔ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
172, 16anbi12d 632 . . . 4 (𝑔 = (1st𝑅) → ((⟨𝑔, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp) ↔ (⟨(1st𝑅), ⟩ ∈ RingOps ∧ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
18 opeq2 4879 . . . . . . 7 ( = (2nd𝑅) → ⟨(1st𝑅), ⟩ = ⟨(1st𝑅), (2nd𝑅)⟩)
1918eleq1d 2824 . . . . . 6 ( = (2nd𝑅) → (⟨(1st𝑅), ⟩ ∈ RingOps ↔ ⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps))
2019anbi1d 631 . . . . 5 ( = (2nd𝑅) → ((⟨(1st𝑅), ⟩ ∈ RingOps ∧ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
21 drngi.2 . . . . . . . . 9 𝐻 = (2nd𝑅)
22 id 22 . . . . . . . . 9 ( = (2nd𝑅) → = (2nd𝑅))
2321, 22eqtr4id 2794 . . . . . . . 8 ( = (2nd𝑅) → 𝐻 = )
2423reseq1d 5999 . . . . . . 7 ( = (2nd𝑅) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
2524eleq1d 2824 . . . . . 6 ( = (2nd𝑅) → ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp ↔ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
2625anbi2d 630 . . . . 5 ( = (2nd𝑅) → ((⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
2720, 26bitr4d 282 . . . 4 ( = (2nd𝑅) → ((⟨(1st𝑅), ⟩ ∈ RingOps ∧ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
2817, 27elopabi 8086 . . 3 (𝑅 ∈ {⟨𝑔, ⟩ ∣ (⟨𝑔, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)} → (⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
29 df-drngo 37936 . . 3 DivRingOps = {⟨𝑔, ⟩ ∣ (⟨𝑔, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
3028, 29eleq2s 2857 . 2 (𝑅 ∈ DivRingOps → (⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
3129relopabiv 5833 . . . . 5 Rel DivRingOps
32 1st2nd 8063 . . . . 5 ((Rel DivRingOps ∧ 𝑅 ∈ DivRingOps) → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
3331, 32mpan 690 . . . 4 (𝑅 ∈ DivRingOps → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
3433eleq1d 2824 . . 3 (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ↔ ⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps))
3534anbi1d 631 . 2 (𝑅 ∈ DivRingOps → ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
3630, 35mpbird 257 1 (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cdif 3960  {csn 4631  cop 4637  {copab 5210   × cxp 5687  ran crn 5690  cres 5691  Rel wrel 5694  cfv 6563  1st c1st 8011  2nd c2nd 8012  GrpOpcgr 30518  GIdcgi 30519  RingOpscrngo 37881  DivRingOpscdrng 37935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014  df-drngo 37936
This theorem is referenced by:  dvrunz  37941  fldcrngo  37991
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