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Theorem issdrg 27473
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 27472 . . . . 5  |- SubDRing  =  ( w  e.  DivRing  |->  { s  e.  (SubRing `  w
)  |  ( ws  s )  e.  DivRing } )
21dmmptss 5358 . . . 4  |-  dom SubDRing  C_  DivRing
3 elfvdm 5749 . . . 4  |-  ( S  e.  (SubDRing `  R
)  ->  R  e.  dom SubDRing )
42, 3sseldi 3338 . . 3  |-  ( S  e.  (SubDRing `  R
)  ->  R  e.  DivRing )
5 fveq2 5720 . . . . . . 7  |-  ( w  =  R  ->  (SubRing `  w )  =  (SubRing `  R ) )
6 oveq1 6080 . . . . . . . 8  |-  ( w  =  R  ->  (
ws  s )  =  ( Rs  s ) )
76eleq1d 2501 . . . . . . 7  |-  ( w  =  R  ->  (
( ws  s )  e.  DivRing  <->  ( Rs  s )  e.  DivRing ) )
85, 7rabeqbidv 2943 . . . . . 6  |-  ( w  =  R  ->  { s  e.  (SubRing `  w
)  |  ( ws  s )  e.  DivRing }  =  { s  e.  (SubRing `  R )  |  ( Rs  s )  e.  DivRing } )
9 fvex 5734 . . . . . . 7  |-  (SubRing `  R
)  e.  _V
109rabex 4346 . . . . . 6  |-  { s  e.  (SubRing `  R
)  |  ( Rs  s )  e.  DivRing }  e.  _V
118, 1, 10fvmpt 5798 . . . . 5  |-  ( R  e.  DivRing  ->  (SubDRing `  R )  =  { s  e.  (SubRing `  R )  |  ( Rs  s )  e.  DivRing } )
1211eleq2d 2502 . . . 4  |-  ( R  e.  DivRing  ->  ( S  e.  (SubDRing `  R )  <->  S  e.  { s  e.  (SubRing `  R )  |  ( Rs  s )  e.  DivRing } ) )
13 oveq2 6081 . . . . . 6  |-  ( s  =  S  ->  ( Rs  s )  =  ( Rs  S ) )
1413eleq1d 2501 . . . . 5  |-  ( s  =  S  ->  (
( Rs  s )  e.  DivRing  <->  ( Rs  S )  e.  DivRing ) )
1514elrab 3084 . . . 4  |-  ( S  e.  { s  e.  (SubRing `  R )  |  ( Rs  s )  e.  DivRing }  <->  ( S  e.  (SubRing `  R )  /\  ( Rs  S )  e.  DivRing ) )
1612, 15syl6bb 253 . . 3  |-  ( R  e.  DivRing  ->  ( S  e.  (SubDRing `  R )  <->  ( S  e.  (SubRing `  R
)  /\  ( Rs  S
)  e.  DivRing ) ) )
174, 16biadan2 624 . 2  |-  ( S  e.  (SubDRing `  R
)  <->  ( R  e.  DivRing 
/\  ( S  e.  (SubRing `  R )  /\  ( Rs  S )  e.  DivRing ) ) )
18 3anass 940 . 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 244 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2701   dom cdm 4870   ` cfv 5446  (class class class)co 6073   ↾s cress 13462   DivRingcdr 15827  SubRingcsubrg 15856  SubDRingcsdrg 27471
This theorem is referenced by:  issdrg2  27474
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-sdrg 27472
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