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Theorem drngoi 37959
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 4872 . . . . . 6 (𝑔 = (1st𝑅) → ⟨𝑔, ⟩ = ⟨(1st𝑅), ⟩)
21eleq1d 2825 . . . . 5 (𝑔 = (1st𝑅) → (⟨𝑔, ⟩ ∈ RingOps ↔ ⟨(1st𝑅), ⟩ ∈ RingOps))
3 id 22 . . . . . . . . . . . 12 (𝑔 = (1st𝑅) → 𝑔 = (1st𝑅))
4 drngi.1 . . . . . . . . . . . 12 𝐺 = (1st𝑅)
53, 4eqtr4di 2794 . . . . . . . . . . 11 (𝑔 = (1st𝑅) → 𝑔 = 𝐺)
65rneqd 5948 . . . . . . . . . 10 (𝑔 = (1st𝑅) → ran 𝑔 = ran 𝐺)
7 drngi.3 . . . . . . . . . 10 𝑋 = ran 𝐺
86, 7eqtr4di 2794 . . . . . . . . 9 (𝑔 = (1st𝑅) → ran 𝑔 = 𝑋)
95fveq2d 6909 . . . . . . . . . . 11 (𝑔 = (1st𝑅) → (GId‘𝑔) = (GId‘𝐺))
10 drngi.4 . . . . . . . . . . 11 𝑍 = (GId‘𝐺)
119, 10eqtr4di 2794 . . . . . . . . . 10 (𝑔 = (1st𝑅) → (GId‘𝑔) = 𝑍)
1211sneqd 4637 . . . . . . . . 9 (𝑔 = (1st𝑅) → {(GId‘𝑔)} = {𝑍})
138, 12difeq12d 4126 . . . . . . . 8 (𝑔 = (1st𝑅) → (ran 𝑔 ∖ {(GId‘𝑔)}) = (𝑋 ∖ {𝑍}))
1413sqxpeqd 5716 . . . . . . 7 (𝑔 = (1st𝑅) → ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)})) = ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
1514reseq2d 5996 . . . . . 6 (𝑔 = (1st𝑅) → ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) = ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
1615eleq1d 2825 . . . . 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 4873 . . . . . . 7 ( = (2nd𝑅) → ⟨(1st𝑅), ⟩ = ⟨(1st𝑅), (2nd𝑅)⟩)
1918eleq1d 2825 . . . . . 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 2795 . . . . . . . 8 ( = (2nd𝑅) → 𝐻 = )
2423reseq1d 5995 . . . . . . 7 ( = (2nd𝑅) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
2524eleq1d 2825 . . . . . 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 8088 . . 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 37957 . . 3 DivRingOps = {⟨𝑔, ⟩ ∣ (⟨𝑔, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
3028, 29eleq2s 2858 . 2 (𝑅 ∈ DivRingOps → (⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
3129relopabiv 5829 . . . . 5 Rel DivRingOps
32 1st2nd 8065 . . . . 5 ((Rel DivRingOps ∧ 𝑅 ∈ DivRingOps) → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
3331, 32mpan 690 . . . 4 (𝑅 ∈ DivRingOps → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
3433eleq1d 2825 . . 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 1539  wcel 2107  cdif 3947  {csn 4625  cop 4631  {copab 5204   × cxp 5682  ran crn 5685  cres 5686  Rel wrel 5689  cfv 6560  1st c1st 8013  2nd c2nd 8014  GrpOpcgr 30509  GIdcgi 30510  RingOpscrngo 37902  DivRingOpscdrng 37956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-iota 6513  df-fun 6562  df-fv 6568  df-1st 8015  df-2nd 8016  df-drngo 37957
This theorem is referenced by:  dvrunz  37962  fldcrngo  38012
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