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

Theorem issdrg 20805
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 20804 . . . 4 SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing})
21mptrcl 7024 . . 3 (𝑆 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing)
3 fveq2 6906 . . . . . . 7 (𝑤 = 𝑅 → (SubRing‘𝑤) = (SubRing‘𝑅))
4 oveq1 7437 . . . . . . . 8 (𝑤 = 𝑅 → (𝑤s 𝑠) = (𝑅s 𝑠))
54eleq1d 2823 . . . . . . 7 (𝑤 = 𝑅 → ((𝑤s 𝑠) ∈ DivRing ↔ (𝑅s 𝑠) ∈ DivRing))
63, 5rabeqbidv 3451 . . . . . 6 (𝑤 = 𝑅 → {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing} = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing})
7 fvex 6919 . . . . . . 7 (SubRing‘𝑅) ∈ V
87rabex 5344 . . . . . 6 {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing} ∈ V
96, 1, 8fvmpt 7015 . . . . 5 (𝑅 ∈ DivRing → (SubDRing‘𝑅) = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing})
109eleq2d 2824 . . . 4 (𝑅 ∈ DivRing → (𝑆 ∈ (SubDRing‘𝑅) ↔ 𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing}))
11 oveq2 7438 . . . . . 6 (𝑠 = 𝑆 → (𝑅s 𝑠) = (𝑅s 𝑆))
1211eleq1d 2823 . . . . 5 (𝑠 = 𝑆 → ((𝑅s 𝑠) ∈ DivRing ↔ (𝑅s 𝑆) ∈ DivRing))
1312elrab 3694 . . . 4 (𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅s 𝑠) ∈ DivRing} ↔ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))
1410, 13bitrdi 287 . . 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 1094 . 2 ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing)))
1715, 16bitr4i 278 1 (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  {crab 3432  cfv 6562  (class class class)co 7430  s cress 17273  SubRingcsubrg 20585  DivRingcdr 20745  SubDRingcsdrg 20803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-sdrg 20804
This theorem is referenced by:  sdrgrcl  20806  sdrgdrng  20807  sdrgsubrg  20808  sdrgid  20809  sdrgss  20810  issdrg2  20812  fldsdrgfld  20815  sdrgint  20821  primefld  20822  primefld0cl  20823  primefld1cl  20824  sdrgdvcl  33280  sdrginvcl  33281  primefldchr  33282  fldgensdrg  33295  fldgenssp  33299  primefldgen1  33302  1fldgenq  33303  irngnzply1lem  33704  irngnzply1  33705  ply1annig1p  33711  minplycl  33713  ply1annprmidl  33714  algextdeglem1  33722  algextdeglem2  33723  algextdeglem3  33724  algextdeglem4  33725  algextdeglem5  33726  2sqr3minply  33752
  Copyright terms: Public domain W3C validator