Theorem issdrg 20698

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) ∧ (𝑅 ↾s 𝑆) ∈ DivRing))

Proof of Theorem issdrg

Dummy variables 𝑤 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sdrg 20697	. . . 4 ⊢ SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing})
2	1	mptrcl 6933	. . 3 ⊢ (𝑆 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing)
3		fveq2 6817	. . . . . . 7 ⊢ (𝑤 = 𝑅 → (SubRing‘𝑤) = (SubRing‘𝑅))
4		oveq1 7348	. . . . . . . 8 ⊢ (𝑤 = 𝑅 → (𝑤 ↾s 𝑠) = (𝑅 ↾s 𝑠))
5	4	eleq1d 2816	. . . . . . 7 ⊢ (𝑤 = 𝑅 → ((𝑤 ↾s 𝑠) ∈ DivRing ↔ (𝑅 ↾s 𝑠) ∈ DivRing))
6	3, 5	rabeqbidv 3413	. . . . . 6 ⊢ (𝑤 = 𝑅 → {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing} = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing})
7		fvex 6830	. . . . . . 7 ⊢ (SubRing‘𝑅) ∈ V
8	7	rabex 5272	. . . . . 6 ⊢ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing} ∈ V
9	6, 1, 8	fvmpt 6924	. . . . 5 ⊢ (𝑅 ∈ DivRing → (SubDRing‘𝑅) = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing})
10	9	eleq2d 2817	. . . 4 ⊢ (𝑅 ∈ DivRing → (𝑆 ∈ (SubDRing‘𝑅) ↔ 𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing}))
11		oveq2 7349	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑅 ↾s 𝑠) = (𝑅 ↾s 𝑆))
12	11	eleq1d 2816	. . . . 5 ⊢ (𝑠 = 𝑆 → ((𝑅 ↾s 𝑠) ∈ DivRing ↔ (𝑅 ↾s 𝑆) ∈ DivRing))
13	12	elrab 3642	. . . 4 ⊢ (𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing} ↔ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing))
14	10, 13	bitrdi 287	. . 3 ⊢ (𝑅 ∈ DivRing → (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing)))
15	2, 14	biadanii 821	. 2 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing)))
16		3anass 1094	. 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing)))
17	15, 16	bitr4i 278	1 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  {crab 3395  ‘cfv 6476  (class class class)co 7341   ↾_s cress 17136  SubRingcsubrg 20479  DivRingcdr 20639  SubDRingcsdrg 20696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fv 6484  df-ov 7344  df-sdrg 20697

This theorem is referenced by:  sdrgrcl  20699  sdrgdrng  20700  sdrgsubrg  20701  sdrgid  20702  sdrgss  20703  issdrg2  20705  fldsdrgfld  20708  sdrgint  20714  primefld  20715  primefld0cl  20716  primefld1cl  20717  subsdrg  33256  sdrgdvcl  33257  sdrginvcl  33258  primefldchr  33259  fldgensdrg  33272  fldgenssp  33276  primefldgen1  33279  1fldgenq  33280  fldextsdrg  33659  fldextrspunlem2  33682  fldextrspundgdvdslem  33685  fldextrspundgdvds  33686  irngnzply1lem  33695  irngnzply1  33696  ply1annig1p  33709  minplycl  33711  ply1annprmidl  33712  algextdeglem1  33722  algextdeglem2  33723  algextdeglem3  33724  algextdeglem4  33725  algextdeglem5  33726  constrextdg2  33754  constrext2chnlem  33755  constrcon  33779  2sqr3minply  33785  cos9thpiminply  33793