Theorem issdrg 20825

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 20824	. . . 4 ⊢ SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing})
2	1	mptrcl 6980	. . 3 ⊢ (𝑆 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing)
3		fveq2 6862	. . . . . . 7 ⊢ (𝑤 = 𝑅 → (SubRing‘𝑤) = (SubRing‘𝑅))
4		oveq1 7398	. . . . . . . 8 ⊢ (𝑤 = 𝑅 → (𝑤 ↾s 𝑠) = (𝑅 ↾s 𝑠))
5	4	eleq1d 2846	. . . . . . 7 ⊢ (𝑤 = 𝑅 → ((𝑤 ↾s 𝑠) ∈ DivRing ↔ (𝑅 ↾s 𝑠) ∈ DivRing))
6	3, 5	rabeqbidv 3431	. . . . . 6 ⊢ (𝑤 = 𝑅 → {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing} = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing})
7		fvex 6875	. . . . . . 7 ⊢ (SubRing‘𝑅) ∈ V
8	7	rabex 5292	. . . . . 6 ⊢ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing} ∈ V
9	6, 1, 8	fvmpt 6970	. . . . 5 ⊢ (𝑅 ∈ DivRing → (SubDRing‘𝑅) = {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing})
10	9	eleq2d 2847	. . . 4 ⊢ (𝑅 ∈ DivRing → (𝑆 ∈ (SubDRing‘𝑅) ↔ 𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing}))
11		oveq2 7399	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑅 ↾s 𝑠) = (𝑅 ↾s 𝑆))
12	11	eleq1d 2846	. . . . 5 ⊢ (𝑠 = 𝑆 → ((𝑅 ↾s 𝑠) ∈ DivRing ↔ (𝑅 ↾s 𝑆) ∈ DivRing))
13	12	elrab 3649	. . . 4 ⊢ (𝑆 ∈ {𝑠 ∈ (SubRing‘𝑅) ∣ (𝑅 ↾s 𝑠) ∈ DivRing} ↔ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing))
14	10, 13	bitrdi 289	. . 3 ⊢ (𝑅 ∈ DivRing → (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing)))
15	2, 14	biadanii 831	. 2 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing)))
16		3anass 1105	. 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing) ↔ (𝑅 ∈ DivRing ∧ (𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing)))
17	15, 16	bitr4i 280	1 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  {crab 3413  ‘cfv 6516  (class class class)co 7391   ↾_s cress 17257  SubRingcsubrg 20606  DivRingcdr 20766  SubDRingcsdrg 20823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fv 6524  df-ov 7394  df-sdrg 20824

This theorem is referenced by:  sdrgrcl  20826  sdrgdrng  20827  sdrgsubrg  20828  sdrgid  20829  sdrgss  20830  issdrg2  20832  fldsdrgfld  20835  sdrgint  20841  primefld  20842  primefld0cl  20843  primefld1cl  20844  subsdrg  33446  sdrgdvcl  33447  sdrginvcl  33448  primefldchr  33449  fldgensdrg  33462  fldgenssp  33466  primefldgen1  33469  1fldgenq  33470  fldextsdrg  33912  fldextrspunlem2  33935  fldextrspundgdvdslem  33938  fldextrspundgdvds  33939  irngnzply1lem  33948  irngnzply1  33949  ply1annig1p  33962  minplycl  33964  ply1annprmidl  33965  algextdeglem1  33975  algextdeglem2  33976  algextdeglem3  33977  algextdeglem4  33978  algextdeglem5  33979  constrextdg2  34007  constrext2chnlem  34008  constrcon  34032  2sqr3minply  34038  cos9thpiminply  34046