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Theorem issdrg 27608
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 )  /\  ( Rs  S )  e.  DivRing ) )

Proof of Theorem issdrg
Dummy variables  w  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sdrg 27607 . . . . 5  |- SubDRing  =  ( w  e.  DivRing  |->  { s  e.  (SubRing `  w
)  |  ( ws  s )  e.  DivRing } )
21dmmptss 5185 . . . 4  |-  dom SubDRing  C_  DivRing
3 elfvdm 5570 . . . 4  |-  ( S  e.  (SubDRing `  R
)  ->  R  e.  dom SubDRing )
42, 3sseldi 3191 . . 3  |-  ( S  e.  (SubDRing `  R
)  ->  R  e.  DivRing )
5 fveq2 5541 . . . . . . 7  |-  ( w  =  R  ->  (SubRing `  w )  =  (SubRing `  R ) )
6 oveq1 5881 . . . . . . . 8  |-  ( w  =  R  ->  (
ws  s )  =  ( Rs  s ) )
76eleq1d 2362 . . . . . . 7  |-  ( w  =  R  ->  (
( ws  s )  e.  DivRing  <->  ( Rs  s )  e.  DivRing ) )
85, 7rabeqbidv 2796 . . . . . 6  |-  ( w  =  R  ->  { s  e.  (SubRing `  w
)  |  ( ws  s )  e.  DivRing }  =  { s  e.  (SubRing `  R )  |  ( Rs  s )  e.  DivRing } )
9 fvex 5555 . . . . . . 7  |-  (SubRing `  R
)  e.  _V
109rabex 4181 . . . . . 6  |-  { s  e.  (SubRing `  R
)  |  ( Rs  s )  e.  DivRing }  e.  _V
118, 1, 10fvmpt 5618 . . . . 5  |-  ( R  e.  DivRing  ->  (SubDRing `  R )  =  { s  e.  (SubRing `  R )  |  ( Rs  s )  e.  DivRing } )
1211eleq2d 2363 . . . 4  |-  ( R  e.  DivRing  ->  ( S  e.  (SubDRing `  R )  <->  S  e.  { s  e.  (SubRing `  R )  |  ( Rs  s )  e.  DivRing } ) )
13 oveq2 5882 . . . . . 6  |-  ( s  =  S  ->  ( Rs  s )  =  ( Rs  S ) )
1413eleq1d 2362 . . . . 5  |-  ( s  =  S  ->  (
( Rs  s )  e.  DivRing  <->  ( Rs  S )  e.  DivRing ) )
1514elrab 2936 . . . 4  |-  ( S  e.  { s  e.  (SubRing `  R )  |  ( Rs  s )  e.  DivRing }  <->  ( S  e.  (SubRing `  R )  /\  ( Rs  S )  e.  DivRing ) )
1612, 15syl6bb 252 . . 3  |-  ( R  e.  DivRing  ->  ( S  e.  (SubDRing `  R )  <->  ( S  e.  (SubRing `  R
)  /\  ( Rs  S
)  e.  DivRing ) ) )
174, 16biadan2 623 . 2  |-  ( S  e.  (SubDRing `  R
)  <->  ( R  e.  DivRing 
/\  ( S  e.  (SubRing `  R )  /\  ( Rs  S )  e.  DivRing ) ) )
18 3anass 938 . 2  |-  ( ( R  e.  DivRing  /\  S  e.  (SubRing `  R )  /\  ( Rs  S )  e.  DivRing )  <-> 
( R  e.  DivRing  /\  ( S  e.  (SubRing `  R )  /\  ( Rs  S )  e.  DivRing ) ) )
1917, 18bitr4i 243 1  |-  ( S  e.  (SubDRing `  R
)  <->  ( R  e.  DivRing 
/\  S  e.  (SubRing `  R )  /\  ( Rs  S )  e.  DivRing ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {crab 2560   dom cdm 4705   ` cfv 5271  (class class class)co 5874   ↾s cress 13165   DivRingcdr 15528  SubRingcsubrg 15557  SubDRingcsdrg 27606
This theorem is referenced by:  issdrg2  27609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-sdrg 27607
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