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Theorem issdrg 36786
Description: Property of a division subring. (Contributed by Stefan O'Rear, 3-Oct-2015.)
Assertion
Ref Expression
issdrg (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))

Proof of Theorem issdrg
Dummy variables 𝑤 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sdrg 36785 . . . . 5 SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing})
21dmmptss 5548 . . . 4 dom SubDRing ⊆ DivRing
3 elfvdm 6130 . . . 4 (𝑆 ∈ (SubDRing‘𝑅) → 𝑅 ∈ dom SubDRing)
42, 3sseldi 3566 . . 3 (𝑆 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing)
5 fveq2 6103 . . . . . . 7 (𝑤 = 𝑅 → (SubRing‘𝑤) = (SubRing‘𝑅))
6 oveq1 6556 . . . . . . . 8 (𝑤 = 𝑅 → (𝑤s 𝑠) = (𝑅s 𝑠))
76eleq1d 2672 . . . . . . 7 (𝑤 = 𝑅 → ((𝑤s 𝑠) ∈ DivRing ↔ (𝑅s 𝑠) ∈ DivRing))
85, 7rabeqbidv 3168 . . . . . 6 (𝑤 = 𝑅 → {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing} = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing})
9 fvex 6113 . . . . . . 7 (SubRing‘𝑅) ∈ V
109rabex 4740 . . . . . 6 {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing} ∈ V
118, 1, 10fvmpt 6191 . . . . 5 (𝑅 ∈ DivRing → (SubDRing‘𝑅) = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing})
1211eleq2d 2673 . . . 4 (𝑅 ∈ DivRing → (𝑆 ∈ (SubDRing‘𝑅) ↔ 𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing}))
13 oveq2 6557 . . . . . 6 (𝑠 = 𝑆 → (𝑅s 𝑠) = (𝑅s 𝑆))
1413eleq1d 2672 . . . . 5 (𝑠 = 𝑆 → ((𝑅s 𝑠) ∈ DivRing ↔ (𝑅s 𝑆) ∈ DivRing))
1514elrab 3331 . . . 4 (𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing} ↔ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))
1612, 15syl6bb 275 . . 3 (𝑅 ∈ DivRing → (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing)))
174, 16biadan2 672 . 2 (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing)))
18 3anass 1035 . 2 ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing)))
1917, 18bitr4i 266 1 (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  {crab 2900  dom cdm 5038  cfv 5804  (class class class)co 6549  s cress 15696  DivRingcdr 18570  SubRingcsubrg 18599  SubDRingcsdrg 36784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-sdrg 36785
This theorem is referenced by:  issdrg2  36787
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