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Theorem isdrngo1 36762
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 36755 . . . 4 DivRingOps = {⟨𝑔, ⟩ ∣ (⟨𝑔, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
21relopabiv 5818 . . 3 Rel DivRingOps
3 1st2nd 8020 . . 3 ((Rel DivRingOps ∧ 𝑅 ∈ DivRingOps) → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
42, 3mpan 689 . 2 (𝑅 ∈ DivRingOps → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
5 relrngo 36702 . . . 4 Rel RingOps
6 1st2nd 8020 . . . 4 ((Rel RingOps ∧ 𝑅 ∈ RingOps) → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
75, 6mpan 689 . . 3 (𝑅 ∈ RingOps → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
87adantr 482 . 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 4877 . . . 4 𝐺, 𝐻⟩ = ⟨(1st𝑅), (2nd𝑅)⟩
1211eqeq2i 2746 . . 3 (𝑅 = ⟨𝐺, 𝐻⟩ ↔ 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
1310fvexi 6902 . . . . . 6 𝐻 ∈ V
14 isdivrngo 36756 . . . . . 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 4638 . . . . . . . . . 10 {𝑍} = {(GId‘𝐺)}
1916, 18difeq12i 4119 . . . . . . . . 9 (𝑋 ∖ {𝑍}) = (ran 𝐺 ∖ {(GId‘𝐺)})
2019, 19xpeq12i 5703 . . . . . . . 8 ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) = ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))
2120reseq2i 5976 . . . . . . 7 (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)})))
2221eleq1i 2825 . . . . . 6 ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp ↔ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)
2322anbi2i 624 . . . . 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 278 . . . 4 (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
25 eleq1 2822 . . . . 5 (𝑅 = ⟨𝐺, 𝐻⟩ → (𝑅 ∈ DivRingOps ↔ ⟨𝐺, 𝐻⟩ ∈ DivRingOps))
26 eleq1 2822 . . . . . 6 (𝑅 = ⟨𝐺, 𝐻⟩ → (𝑅 ∈ RingOps ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps))
2726anbi1d 631 . . . . 5 (𝑅 = ⟨𝐺, 𝐻⟩ → ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
2825, 27bibi12d 346 . . . 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 258 . . 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 380 1 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  cdif 3944  {csn 4627  cop 4633   × cxp 5673  ran crn 5676  cres 5677  Rel wrel 5680  cfv 6540  1st c1st 7968  2nd c2nd 7969  GrpOpcgr 29720  GIdcgi 29721  RingOpscrngo 36700  DivRingOpscdrng 36754
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-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7407  df-1st 7970  df-2nd 7971  df-rngo 36701  df-drngo 36755
This theorem is referenced by:  divrngcl  36763  isdrngo2  36764  divrngpr  36859
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