Theorem issdrg 37767

Description: Property of a division subring. (Contributed by Stefan O'Rear, 3-Oct-2015.)

Assertion

Ref	Expression
issdrg	|- ( S e. (SubDRing ` R ) <-> ( R e. DivRing /\ S e. (SubRing ` R ) /\ ( R ↾s S ) e. 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 37766	. . . . 5 |- SubDRing = ( w e. DivRing |-> { s e. (SubRing ` w ) | ( w ↾s s ) e. DivRing } )
2	1	dmmptss 5631	. . . 4 |- dom SubDRing C_ DivRing
3		elfvdm 6220	. . . 4 |- ( S e. (SubDRing ` R ) -> R e. dom SubDRing )
4	2, 3	sseldi 3601	. . 3 |- ( S e. (SubDRing ` R ) -> R e. DivRing )
5		fveq2 6191	. . . . . . 7 |- ( w = R -> (SubRing ` w ) = (SubRing ` R ) )
6		oveq1 6657	. . . . . . . 8 |- ( w = R -> ( w ↾s s ) = ( R ↾s s ) )
7	6	eleq1d 2686	. . . . . . 7 |- ( w = R -> ( ( w ↾s s ) e. DivRing <-> ( R ↾s s ) e. DivRing ) )
8	5, 7	rabeqbidv 3195	. . . . . 6 |- ( w = R -> { s e. (SubRing ` w ) | ( w ↾s s ) e. DivRing } = { s e. (SubRing ` R ) | ( R ↾s s ) e. DivRing } )
9		fvex 6201	. . . . . . 7 |- (SubRing ` R ) e. _V
10	9	rabex 4813	. . . . . 6 |- { s e. (SubRing ` R ) | ( R ↾s s ) e. DivRing } e. _V
11	8, 1, 10	fvmpt 6282	. . . . 5 |- ( R e. DivRing -> (SubDRing ` R ) = { s e. (SubRing ` R ) | ( R ↾s s ) e. DivRing } )
12	11	eleq2d 2687	. . . 4 |- ( R e. DivRing -> ( S e. (SubDRing ` R ) <-> S e. { s e. (SubRing ` R ) | ( R ↾s s ) e. DivRing } ) )
13		oveq2 6658	. . . . . 6 |- ( s = S -> ( R ↾s s ) = ( R ↾s S ) )
14	13	eleq1d 2686	. . . . 5 |- ( s = S -> ( ( R ↾s s ) e. DivRing <-> ( R ↾s S ) e. DivRing ) )
15	14	elrab 3363	. . . 4 |- ( S e. { s e. (SubRing ` R ) | ( R ↾s s ) e. DivRing } <-> ( S e. (SubRing ` R ) /\ ( R ↾s S ) e. DivRing ) )
16	12, 15	syl6bb 276	. . 3 |- ( R e. DivRing -> ( S e. (SubDRing ` R ) <-> ( S e. (SubRing ` R ) /\ ( R ↾s S ) e. DivRing ) ) )
17	4, 16	biadan2 674	. 2 |- ( S e. (SubDRing ` R ) <-> ( R e. DivRing /\ ( S e. (SubRing ` R ) /\ ( R ↾s S ) e. DivRing ) ) )
18		3anass 1042	. 2 |- ( ( R e. DivRing /\ S e. (SubRing ` R ) /\ ( R ↾s S ) e. DivRing ) <-> ( R e. DivRing /\ ( S e. (SubRing ` R ) /\ ( R ↾s S ) e. DivRing ) ) )
19	17, 18	bitr4i 267	1 |- ( S e. (SubDRing ` R ) <-> ( R e. DivRing /\ S e. (SubRing ` R ) /\ ( R ↾s S ) e. DivRing ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   dom cdm 5114   ` cfv 5888  (class class class)co 6650   ↾_s cress 15858   DivRingcdr 18747  SubRingcsubrg 18776  SubDRingcsdrg 37765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-sdrg 37766

This theorem is referenced by:  issdrg2  37768