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Theorem drngoi 36819
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 4874 . . . . . 6 (𝑔 = (1st𝑅) → ⟨𝑔, ⟩ = ⟨(1st𝑅), ⟩)
21eleq1d 2819 . . . . 5 (𝑔 = (1st𝑅) → (⟨𝑔, ⟩ ∈ RingOps ↔ ⟨(1st𝑅), ⟩ ∈ RingOps))
3 id 22 . . . . . . . . . . . 12 (𝑔 = (1st𝑅) → 𝑔 = (1st𝑅))
4 drngi.1 . . . . . . . . . . . 12 𝐺 = (1st𝑅)
53, 4eqtr4di 2791 . . . . . . . . . . 11 (𝑔 = (1st𝑅) → 𝑔 = 𝐺)
65rneqd 5938 . . . . . . . . . 10 (𝑔 = (1st𝑅) → ran 𝑔 = ran 𝐺)
7 drngi.3 . . . . . . . . . 10 𝑋 = ran 𝐺
86, 7eqtr4di 2791 . . . . . . . . 9 (𝑔 = (1st𝑅) → ran 𝑔 = 𝑋)
95fveq2d 6896 . . . . . . . . . . 11 (𝑔 = (1st𝑅) → (GId‘𝑔) = (GId‘𝐺))
10 drngi.4 . . . . . . . . . . 11 𝑍 = (GId‘𝐺)
119, 10eqtr4di 2791 . . . . . . . . . 10 (𝑔 = (1st𝑅) → (GId‘𝑔) = 𝑍)
1211sneqd 4641 . . . . . . . . 9 (𝑔 = (1st𝑅) → {(GId‘𝑔)} = {𝑍})
138, 12difeq12d 4124 . . . . . . . 8 (𝑔 = (1st𝑅) → (ran 𝑔 ∖ {(GId‘𝑔)}) = (𝑋 ∖ {𝑍}))
1413sqxpeqd 5709 . . . . . . 7 (𝑔 = (1st𝑅) → ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)})) = ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
1514reseq2d 5982 . . . . . 6 (𝑔 = (1st𝑅) → ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) = ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
1615eleq1d 2819 . . . . 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 4875 . . . . . . 7 ( = (2nd𝑅) → ⟨(1st𝑅), ⟩ = ⟨(1st𝑅), (2nd𝑅)⟩)
1918eleq1d 2819 . . . . . 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 2792 . . . . . . . 8 ( = (2nd𝑅) → 𝐻 = )
2423reseq1d 5981 . . . . . . 7 ( = (2nd𝑅) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
2524eleq1d 2819 . . . . . 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 8048 . . 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 36817 . . 3 DivRingOps = {⟨𝑔, ⟩ ∣ (⟨𝑔, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
3028, 29eleq2s 2852 . 2 (𝑅 ∈ DivRingOps → (⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
3129relopabiv 5821 . . . . 5 Rel DivRingOps
32 1st2nd 8025 . . . . 5 ((Rel DivRingOps ∧ 𝑅 ∈ DivRingOps) → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
3331, 32mpan 689 . . . 4 (𝑅 ∈ DivRingOps → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
3433eleq1d 2819 . . 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 397   = wceq 1542  wcel 2107  cdif 3946  {csn 4629  cop 4635  {copab 5211   × cxp 5675  ran crn 5678  cres 5679  Rel wrel 5682  cfv 6544  1st c1st 7973  2nd c2nd 7974  GrpOpcgr 29742  GIdcgi 29743  RingOpscrngo 36762  DivRingOpscdrng 36816
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 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
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 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fv 6552  df-1st 7975  df-2nd 7976  df-drngo 36817
This theorem is referenced by:  dvrunz  36822  fldcrngo  36872
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