MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  issdrg Structured version   Visualization version   GIF version

Theorem issdrg 19839
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 19838 . . . 4 SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing})
21mptrcl 6827 . . 3 (𝑆 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing)
3 fveq2 6717 . . . . . . 7 (𝑤 = 𝑅 → (SubRing‘𝑤) = (SubRing‘𝑅))
4 oveq1 7220 . . . . . . . 8 (𝑤 = 𝑅 → (𝑤s 𝑠) = (𝑅s 𝑠))
54eleq1d 2822 . . . . . . 7 (𝑤 = 𝑅 → ((𝑤s 𝑠) ∈ DivRing ↔ (𝑅s 𝑠) ∈ DivRing))
63, 5rabeqbidv 3396 . . . . . 6 (𝑤 = 𝑅 → {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing} = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing})
7 fvex 6730 . . . . . . 7 (SubRing‘𝑅) ∈ V
87rabex 5225 . . . . . 6 {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing} ∈ V
96, 1, 8fvmpt 6818 . . . . 5 (𝑅 ∈ DivRing → (SubDRing‘𝑅) = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing})
109eleq2d 2823 . . . 4 (𝑅 ∈ DivRing → (𝑆 ∈ (SubDRing‘𝑅) ↔ 𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing}))
11 oveq2 7221 . . . . . 6 (𝑠 = 𝑆 → (𝑅s 𝑠) = (𝑅s 𝑆))
1211eleq1d 2822 . . . . 5 (𝑠 = 𝑆 → ((𝑅s 𝑠) ∈ DivRing ↔ (𝑅s 𝑆) ∈ DivRing))
1312elrab 3602 . . . 4 (𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing} ↔ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))
1410, 13bitrdi 290 . . 3 (𝑅 ∈ DivRing → (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing)))
152, 14biadanii 822 . 2 (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing)))
16 3anass 1097 . 2 ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing)))
1715, 16bitr4i 281 1 (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  {crab 3065  cfv 6380  (class class class)co 7213  s cress 16784  DivRingcdr 19767  SubRingcsubrg 19796  SubDRingcsdrg 19837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fv 6388  df-ov 7216  df-sdrg 19838
This theorem is referenced by:  sdrgid  19840  sdrgss  19841  issdrg2  19842  sdrgint  19848  primefld  19849  primefld0cl  19850  primefld1cl  19851  primefldchr  31212
  Copyright terms: Public domain W3C validator