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Theorem issdrg 27505
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 27504 . . . . 5  |- SubDRing  =  ( w  e.  DivRing  |->  { s  e.  (SubRing `  w
)  |  ( ws  s )  e.  DivRing } )
21dmmptss 5169 . . . 4  |-  dom SubDRing  C_  DivRing
3 elfvdm 5554 . . . 4  |-  ( S  e.  (SubDRing `  R
)  ->  R  e.  dom SubDRing )
42, 3sseldi 3178 . . 3  |-  ( S  e.  (SubDRing `  R
)  ->  R  e.  DivRing )
5 fveq2 5525 . . . . . . 7  |-  ( w  =  R  ->  (SubRing `  w )  =  (SubRing `  R ) )
6 oveq1 5865 . . . . . . . 8  |-  ( w  =  R  ->  (
ws  s )  =  ( Rs  s ) )
76eleq1d 2349 . . . . . . 7  |-  ( w  =  R  ->  (
( ws  s )  e.  DivRing  <->  ( Rs  s )  e.  DivRing ) )
85, 7rabeqbidv 2783 . . . . . 6  |-  ( w  =  R  ->  { s  e.  (SubRing `  w
)  |  ( ws  s )  e.  DivRing }  =  { s  e.  (SubRing `  R )  |  ( Rs  s )  e.  DivRing } )
9 fvex 5539 . . . . . . 7  |-  (SubRing `  R
)  e.  _V
109rabex 4165 . . . . . 6  |-  { s  e.  (SubRing `  R
)  |  ( Rs  s )  e.  DivRing }  e.  _V
118, 1, 10fvmpt 5602 . . . . 5  |-  ( R  e.  DivRing  ->  (SubDRing `  R )  =  { s  e.  (SubRing `  R )  |  ( Rs  s )  e.  DivRing } )
1211eleq2d 2350 . . . 4  |-  ( R  e.  DivRing  ->  ( S  e.  (SubDRing `  R )  <->  S  e.  { s  e.  (SubRing `  R )  |  ( Rs  s )  e.  DivRing } ) )
13 oveq2 5866 . . . . . 6  |-  ( s  =  S  ->  ( Rs  s )  =  ( Rs  S ) )
1413eleq1d 2349 . . . . 5  |-  ( s  =  S  ->  (
( Rs  s )  e.  DivRing  <->  ( Rs  S )  e.  DivRing ) )
1514elrab 2923 . . . 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 1623    e. wcel 1684   {crab 2547   dom cdm 4689   ` cfv 5255  (class class class)co 5858   ↾s cress 13149   DivRingcdr 15512  SubRingcsubrg 15541  SubDRingcsdrg 27503
This theorem is referenced by:  issdrg2  27506
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-sdrg 27504
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