Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  drngoi Structured version   Visualization version   GIF version

Theorem drngoi 34075
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 4539 . . . . . 6 (𝑔 = (1st𝑅) → ⟨𝑔, ⟩ = ⟨(1st𝑅), ⟩)
21eleq1d 2835 . . . . 5 (𝑔 = (1st𝑅) → (⟨𝑔, ⟩ ∈ RingOps ↔ ⟨(1st𝑅), ⟩ ∈ RingOps))
3 id 22 . . . . . . . . . . . 12 (𝑔 = (1st𝑅) → 𝑔 = (1st𝑅))
4 drngi.1 . . . . . . . . . . . 12 𝐺 = (1st𝑅)
53, 4syl6eqr 2823 . . . . . . . . . . 11 (𝑔 = (1st𝑅) → 𝑔 = 𝐺)
65rneqd 5489 . . . . . . . . . 10 (𝑔 = (1st𝑅) → ran 𝑔 = ran 𝐺)
7 drngi.3 . . . . . . . . . 10 𝑋 = ran 𝐺
86, 7syl6eqr 2823 . . . . . . . . 9 (𝑔 = (1st𝑅) → ran 𝑔 = 𝑋)
95fveq2d 6334 . . . . . . . . . . 11 (𝑔 = (1st𝑅) → (GId‘𝑔) = (GId‘𝐺))
10 drngi.4 . . . . . . . . . . 11 𝑍 = (GId‘𝐺)
119, 10syl6eqr 2823 . . . . . . . . . 10 (𝑔 = (1st𝑅) → (GId‘𝑔) = 𝑍)
1211sneqd 4328 . . . . . . . . 9 (𝑔 = (1st𝑅) → {(GId‘𝑔)} = {𝑍})
138, 12difeq12d 3880 . . . . . . . 8 (𝑔 = (1st𝑅) → (ran 𝑔 ∖ {(GId‘𝑔)}) = (𝑋 ∖ {𝑍}))
1413sqxpeqd 5280 . . . . . . 7 (𝑔 = (1st𝑅) → ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)})) = ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
1514reseq2d 5532 . . . . . 6 (𝑔 = (1st𝑅) → ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) = ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
1615eleq1d 2835 . . . . 5 (𝑔 = (1st𝑅) → (( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp ↔ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
172, 16anbi12d 616 . . . 4 (𝑔 = (1st𝑅) → ((⟨𝑔, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp) ↔ (⟨(1st𝑅), ⟩ ∈ RingOps ∧ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
18 opeq2 4540 . . . . . . 7 ( = (2nd𝑅) → ⟨(1st𝑅), ⟩ = ⟨(1st𝑅), (2nd𝑅)⟩)
1918eleq1d 2835 . . . . . 6 ( = (2nd𝑅) → (⟨(1st𝑅), ⟩ ∈ RingOps ↔ ⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps))
2019anbi1d 615 . . . . 5 ( = (2nd𝑅) → ((⟨(1st𝑅), ⟩ ∈ RingOps ∧ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
21 id 22 . . . . . . . . 9 ( = (2nd𝑅) → = (2nd𝑅))
22 drngi.2 . . . . . . . . 9 𝐻 = (2nd𝑅)
2321, 22syl6reqr 2824 . . . . . . . 8 ( = (2nd𝑅) → 𝐻 = )
2423reseq1d 5531 . . . . . . 7 ( = (2nd𝑅) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
2524eleq1d 2835 . . . . . 6 ( = (2nd𝑅) → ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp ↔ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
2625anbi2d 614 . . . . 5 ( = (2nd𝑅) → ((⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
2720, 26bitr4d 271 . . . 4 ( = (2nd𝑅) → ((⟨(1st𝑅), ⟩ ∈ RingOps ∧ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
2817, 27elopabi 7379 . . 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 34073 . . 3 DivRingOps = {⟨𝑔, ⟩ ∣ (⟨𝑔, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
3028, 29eleq2s 2868 . 2 (𝑅 ∈ DivRingOps → (⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
3129relopabi 5382 . . . . 5 Rel DivRingOps
32 1st2nd 7361 . . . . 5 ((Rel DivRingOps ∧ 𝑅 ∈ DivRingOps) → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
3331, 32mpan 670 . . . 4 (𝑅 ∈ DivRingOps → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
3433eleq1d 2835 . . 3 (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ↔ ⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps))
3534anbi1d 615 . 2 (𝑅 ∈ DivRingOps → ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
3630, 35mpbird 247 1 (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cdif 3720  {csn 4316  cop 4322  {copab 4846   × cxp 5247  ran crn 5250  cres 5251  Rel wrel 5254  cfv 6029  1st c1st 7311  2nd c2nd 7312  GrpOpcgr 27676  GIdcgi 27677  RingOpscrngo 34018  DivRingOpscdrng 34072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5992  df-fun 6031  df-fv 6037  df-1st 7313  df-2nd 7314  df-drngo 34073
This theorem is referenced by:  dvrunz  34078  fldcrng  34128
  Copyright terms: Public domain W3C validator