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Theorem issdrg 38286
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 38285 . . . . 5 SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing})
21dmmptss 5775 . . . 4 dom SubDRing ⊆ DivRing
3 elfvdm 6361 . . . 4 (𝑆 ∈ (SubDRing‘𝑅) → 𝑅 ∈ dom SubDRing)
42, 3sseldi 3748 . . 3 (𝑆 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing)
5 fveq2 6332 . . . . . . 7 (𝑤 = 𝑅 → (SubRing‘𝑤) = (SubRing‘𝑅))
6 oveq1 6799 . . . . . . . 8 (𝑤 = 𝑅 → (𝑤s 𝑠) = (𝑅s 𝑠))
76eleq1d 2834 . . . . . . 7 (𝑤 = 𝑅 → ((𝑤s 𝑠) ∈ DivRing ↔ (𝑅s 𝑠) ∈ DivRing))
85, 7rabeqbidv 3344 . . . . . 6 (𝑤 = 𝑅 → {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing} = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing})
9 fvex 6342 . . . . . . 7 (SubRing‘𝑅) ∈ V
109rabex 4943 . . . . . 6 {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing} ∈ V
118, 1, 10fvmpt 6424 . . . . 5 (𝑅 ∈ DivRing → (SubDRing‘𝑅) = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing})
1211eleq2d 2835 . . . 4 (𝑅 ∈ DivRing → (𝑆 ∈ (SubDRing‘𝑅) ↔ 𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing}))
13 oveq2 6800 . . . . . 6 (𝑠 = 𝑆 → (𝑅s 𝑠) = (𝑅s 𝑆))
1413eleq1d 2834 . . . . 5 (𝑠 = 𝑆 → ((𝑅s 𝑠) ∈ DivRing ↔ (𝑅s 𝑆) ∈ DivRing))
1514elrab 3513 . . . 4 (𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing} ↔ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))
1612, 15syl6bb 276 . . 3 (𝑅 ∈ DivRing → (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing)))
174, 16biadan2 802 . 2 (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing)))
18 3anass 1079 . 2 ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing)))
1917, 18bitr4i 267 1 (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  {crab 3064  dom cdm 5249  cfv 6031  (class class class)co 6792  s cress 16064  DivRingcdr 18956  SubRingcsubrg 18985  SubDRingcsdrg 38284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  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-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-sdrg 38285
This theorem is referenced by:  issdrg2  38287
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