MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  issubrg Unicode version

Theorem issubrg 15561
Description: The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Hypotheses
Ref Expression
issubrg.b  |-  B  =  ( Base `  R
)
issubrg.i  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
issubrg  |-  ( A  e.  (SubRing `  R
)  <->  ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  B  /\  .1.  e.  A
) ) )

Proof of Theorem issubrg
Dummy variables  s 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-subrg 15559 . . . 4  |- SubRing  =  ( r  e.  Ring  |->  { s  e.  ~P ( Base `  r )  |  ( ( rs  s )  e. 
Ring  /\  ( 1r `  r )  e.  s ) } )
21dmmptss 5185 . . 3  |-  dom SubRing  C_  Ring
3 elfvdm 5570 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  dom SubRing )
42, 3sseldi 3191 . 2  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
5 simpll 730 . 2  |-  ( ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  B  /\  .1.  e.  A
) )  ->  R  e.  Ring )
6 fveq2 5541 . . . . . . . 8  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
7 issubrg.b . . . . . . . 8  |-  B  =  ( Base `  R
)
86, 7syl6eqr 2346 . . . . . . 7  |-  ( r  =  R  ->  ( Base `  r )  =  B )
98pweqd 3643 . . . . . 6  |-  ( r  =  R  ->  ~P ( Base `  r )  =  ~P B )
10 oveq1 5881 . . . . . . . 8  |-  ( r  =  R  ->  (
rs  s )  =  ( Rs  s ) )
1110eleq1d 2362 . . . . . . 7  |-  ( r  =  R  ->  (
( rs  s )  e. 
Ring 
<->  ( Rs  s )  e. 
Ring ) )
12 fveq2 5541 . . . . . . . . 9  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
13 issubrg.i . . . . . . . . 9  |-  .1.  =  ( 1r `  R )
1412, 13syl6eqr 2346 . . . . . . . 8  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
1514eleq1d 2362 . . . . . . 7  |-  ( r  =  R  ->  (
( 1r `  r
)  e.  s  <->  .1.  e.  s ) )
1611, 15anbi12d 691 . . . . . 6  |-  ( r  =  R  ->  (
( ( rs  s )  e.  Ring  /\  ( 1r `  r )  e.  s )  <->  ( ( Rs  s )  e.  Ring  /\  .1.  e.  s ) ) )
179, 16rabeqbidv 2796 . . . . 5  |-  ( r  =  R  ->  { s  e.  ~P ( Base `  r )  |  ( ( rs  s )  e. 
Ring  /\  ( 1r `  r )  e.  s ) }  =  {
s  e.  ~P B  |  ( ( Rs  s )  e.  Ring  /\  .1.  e.  s ) } )
18 fvex 5555 . . . . . . . 8  |-  ( Base `  R )  e.  _V
197, 18eqeltri 2366 . . . . . . 7  |-  B  e. 
_V
2019pwex 4209 . . . . . 6  |-  ~P B  e.  _V
2120rabex 4181 . . . . 5  |-  { s  e.  ~P B  | 
( ( Rs  s )  e.  Ring  /\  .1.  e.  s ) }  e.  _V
2217, 1, 21fvmpt 5618 . . . 4  |-  ( R  e.  Ring  ->  (SubRing `  R
)  =  { s  e.  ~P B  | 
( ( Rs  s )  e.  Ring  /\  .1.  e.  s ) } )
2322eleq2d 2363 . . 3  |-  ( R  e.  Ring  ->  ( A  e.  (SubRing `  R
)  <->  A  e.  { s  e.  ~P B  | 
( ( Rs  s )  e.  Ring  /\  .1.  e.  s ) } ) )
24 oveq2 5882 . . . . . . . 8  |-  ( s  =  A  ->  ( Rs  s )  =  ( Rs  A ) )
2524eleq1d 2362 . . . . . . 7  |-  ( s  =  A  ->  (
( Rs  s )  e. 
Ring 
<->  ( Rs  A )  e.  Ring ) )
26 eleq2 2357 . . . . . . 7  |-  ( s  =  A  ->  (  .1.  e.  s  <->  .1.  e.  A ) )
2725, 26anbi12d 691 . . . . . 6  |-  ( s  =  A  ->  (
( ( Rs  s )  e.  Ring  /\  .1.  e.  s )  <->  ( ( Rs  A )  e.  Ring  /\  .1.  e.  A ) ) )
2827elrab 2936 . . . . 5  |-  ( A  e.  { s  e. 
~P B  |  ( ( Rs  s )  e. 
Ring  /\  .1.  e.  s ) }  <->  ( A  e.  ~P B  /\  (
( Rs  A )  e.  Ring  /\  .1.  e.  A ) ) )
2919elpw2 4191 . . . . . 6  |-  ( A  e.  ~P B  <->  A  C_  B
)
3029anbi1i 676 . . . . 5  |-  ( ( A  e.  ~P B  /\  ( ( Rs  A )  e.  Ring  /\  .1.  e.  A ) )  <->  ( A  C_  B  /\  ( ( Rs  A )  e.  Ring  /\  .1.  e.  A ) ) )
31 an12 772 . . . . 5  |-  ( ( A  C_  B  /\  ( ( Rs  A )  e.  Ring  /\  .1.  e.  A ) )  <->  ( ( Rs  A )  e.  Ring  /\  ( A  C_  B  /\  .1.  e.  A ) ) )
3228, 30, 313bitri 262 . . . 4  |-  ( A  e.  { s  e. 
~P B  |  ( ( Rs  s )  e. 
Ring  /\  .1.  e.  s ) }  <->  ( ( Rs  A )  e.  Ring  /\  ( A  C_  B  /\  .1.  e.  A ) ) )
33 ibar 490 . . . . 5  |-  ( R  e.  Ring  ->  ( ( Rs  A )  e.  Ring  <->  ( R  e.  Ring  /\  ( Rs  A )  e.  Ring ) ) )
3433anbi1d 685 . . . 4  |-  ( R  e.  Ring  ->  ( ( ( Rs  A )  e.  Ring  /\  ( A  C_  B  /\  .1.  e.  A ) )  <->  ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  B  /\  .1.  e.  A
) ) ) )
3532, 34syl5bb 248 . . 3  |-  ( R  e.  Ring  ->  ( A  e.  { s  e. 
~P B  |  ( ( Rs  s )  e. 
Ring  /\  .1.  e.  s ) }  <->  ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  B  /\  .1.  e.  A
) ) ) )
3623, 35bitrd 244 . 2  |-  ( R  e.  Ring  ->  ( A  e.  (SubRing `  R
)  <->  ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  B  /\  .1.  e.  A
) ) ) )
374, 5, 36pm5.21nii 342 1  |-  ( A  e.  (SubRing `  R
)  <->  ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  B  /\  .1.  e.  A
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   dom cdm 4705   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   Ringcrg 15353   1rcur 15355  SubRingcsubrg 15557
This theorem is referenced by:  subrgss  15562  subrgid  15563  subrgrng  15564  subrgrcl  15566  subrg1cl  15569  issubrg2  15581  subsubrg  15587  subrgpropd  15595  issubassa  16080  subrgpsr  16179  cphsubrglem  18629
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-pw 3640  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-subrg 15559
  Copyright terms: Public domain W3C validator