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Theorem drngoi 35382
 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 4766 . . . . . 6 (𝑔 = (1st𝑅) → ⟨𝑔, ⟩ = ⟨(1st𝑅), ⟩)
21eleq1d 2877 . . . . 5 (𝑔 = (1st𝑅) → (⟨𝑔, ⟩ ∈ RingOps ↔ ⟨(1st𝑅), ⟩ ∈ RingOps))
3 id 22 . . . . . . . . . . . 12 (𝑔 = (1st𝑅) → 𝑔 = (1st𝑅))
4 drngi.1 . . . . . . . . . . . 12 𝐺 = (1st𝑅)
53, 4eqtr4di 2854 . . . . . . . . . . 11 (𝑔 = (1st𝑅) → 𝑔 = 𝐺)
65rneqd 5776 . . . . . . . . . 10 (𝑔 = (1st𝑅) → ran 𝑔 = ran 𝐺)
7 drngi.3 . . . . . . . . . 10 𝑋 = ran 𝐺
86, 7eqtr4di 2854 . . . . . . . . 9 (𝑔 = (1st𝑅) → ran 𝑔 = 𝑋)
95fveq2d 6653 . . . . . . . . . . 11 (𝑔 = (1st𝑅) → (GId‘𝑔) = (GId‘𝐺))
10 drngi.4 . . . . . . . . . . 11 𝑍 = (GId‘𝐺)
119, 10eqtr4di 2854 . . . . . . . . . 10 (𝑔 = (1st𝑅) → (GId‘𝑔) = 𝑍)
1211sneqd 4540 . . . . . . . . 9 (𝑔 = (1st𝑅) → {(GId‘𝑔)} = {𝑍})
138, 12difeq12d 4054 . . . . . . . 8 (𝑔 = (1st𝑅) → (ran 𝑔 ∖ {(GId‘𝑔)}) = (𝑋 ∖ {𝑍}))
1413sqxpeqd 5555 . . . . . . 7 (𝑔 = (1st𝑅) → ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)})) = ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
1514reseq2d 5822 . . . . . 6 (𝑔 = (1st𝑅) → ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) = ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
1615eleq1d 2877 . . . . 5 (𝑔 = (1st𝑅) → (( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp ↔ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
172, 16anbi12d 633 . . . 4 (𝑔 = (1st𝑅) → ((⟨𝑔, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp) ↔ (⟨(1st𝑅), ⟩ ∈ RingOps ∧ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
18 opeq2 4768 . . . . . . 7 ( = (2nd𝑅) → ⟨(1st𝑅), ⟩ = ⟨(1st𝑅), (2nd𝑅)⟩)
1918eleq1d 2877 . . . . . 6 ( = (2nd𝑅) → (⟨(1st𝑅), ⟩ ∈ RingOps ↔ ⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps))
2019anbi1d 632 . . . . 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 2855 . . . . . . . 8 ( = (2nd𝑅) → 𝐻 = )
2423reseq1d 5821 . . . . . . 7 ( = (2nd𝑅) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
2524eleq1d 2877 . . . . . 6 ( = (2nd𝑅) → ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp ↔ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
2625anbi2d 631 . . . . 5 ( = (2nd𝑅) → ((⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
2720, 26bitr4d 285 . . . 4 ( = (2nd𝑅) → ((⟨(1st𝑅), ⟩ ∈ RingOps ∧ ( ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
2817, 27elopabi 7746 . . 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 35380 . . 3 DivRingOps = {⟨𝑔, ⟩ ∣ (⟨𝑔, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
3028, 29eleq2s 2911 . 2 (𝑅 ∈ DivRingOps → (⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
3129relopabi 5662 . . . . 5 Rel DivRingOps
32 1st2nd 7724 . . . . 5 ((Rel DivRingOps ∧ 𝑅 ∈ DivRingOps) → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
3331, 32mpan 689 . . . 4 (𝑅 ∈ DivRingOps → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
3433eleq1d 2877 . . 3 (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ↔ ⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps))
3534anbi1d 632 . 2 (𝑅 ∈ DivRingOps → ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨(1st𝑅), (2nd𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
3630, 35mpbird 260 1 (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112   ∖ cdif 3881  {csn 4528  ⟨cop 4534  {copab 5095   × cxp 5521  ran crn 5524   ↾ cres 5525  Rel wrel 5528  ‘cfv 6328  1st c1st 7673  2nd c2nd 7674  GrpOpcgr 28275  GIdcgi 28276  RingOpscrngo 35325  DivRingOpscdrng 35379 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-iota 6287  df-fun 6330  df-fv 6336  df-1st 7675  df-2nd 7676  df-drngo 35380 This theorem is referenced by:  dvrunz  35385  fldcrng  35435
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