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

Theorem issubrg 17300
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 17298 . . . 4  |- SubRing  =  ( r  e.  Ring  |->  { s  e.  ~P ( Base `  r )  |  ( ( rs  s )  e. 
Ring  /\  ( 1r `  r )  e.  s ) } )
21dmmptss 5509 . . 3  |-  dom SubRing  C_  Ring
3 elfvdm 5898 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  dom SubRing )
42, 3sseldi 3507 . 2  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
5 simpll 753 . 2  |-  ( ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  B  /\  .1.  e.  A
) )  ->  R  e.  Ring )
6 fveq2 5872 . . . . . . . 8  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
7 issubrg.b . . . . . . . 8  |-  B  =  ( Base `  R
)
86, 7syl6eqr 2526 . . . . . . 7  |-  ( r  =  R  ->  ( Base `  r )  =  B )
98pweqd 4021 . . . . . 6  |-  ( r  =  R  ->  ~P ( Base `  r )  =  ~P B )
10 oveq1 6302 . . . . . . . 8  |-  ( r  =  R  ->  (
rs  s )  =  ( Rs  s ) )
1110eleq1d 2536 . . . . . . 7  |-  ( r  =  R  ->  (
( rs  s )  e. 
Ring 
<->  ( Rs  s )  e. 
Ring ) )
12 fveq2 5872 . . . . . . . . 9  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
13 issubrg.i . . . . . . . . 9  |-  .1.  =  ( 1r `  R )
1412, 13syl6eqr 2526 . . . . . . . 8  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
1514eleq1d 2536 . . . . . . 7  |-  ( r  =  R  ->  (
( 1r `  r
)  e.  s  <->  .1.  e.  s ) )
1611, 15anbi12d 710 . . . . . 6  |-  ( r  =  R  ->  (
( ( rs  s )  e.  Ring  /\  ( 1r `  r )  e.  s )  <->  ( ( Rs  s )  e.  Ring  /\  .1.  e.  s ) ) )
179, 16rabeqbidv 3113 . . . . 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 5882 . . . . . . . 8  |-  ( Base `  R )  e.  _V
197, 18eqeltri 2551 . . . . . . 7  |-  B  e. 
_V
2019pwex 4636 . . . . . 6  |-  ~P B  e.  _V
2120rabex 4604 . . . . 5  |-  { s  e.  ~P B  | 
( ( Rs  s )  e.  Ring  /\  .1.  e.  s ) }  e.  _V
2217, 1, 21fvmpt 5957 . . . 4  |-  ( R  e.  Ring  ->  (SubRing `  R
)  =  { s  e.  ~P B  | 
( ( Rs  s )  e.  Ring  /\  .1.  e.  s ) } )
2322eleq2d 2537 . . 3  |-  ( R  e.  Ring  ->  ( A  e.  (SubRing `  R
)  <->  A  e.  { s  e.  ~P B  | 
( ( Rs  s )  e.  Ring  /\  .1.  e.  s ) } ) )
24 oveq2 6303 . . . . . . . 8  |-  ( s  =  A  ->  ( Rs  s )  =  ( Rs  A ) )
2524eleq1d 2536 . . . . . . 7  |-  ( s  =  A  ->  (
( Rs  s )  e. 
Ring 
<->  ( Rs  A )  e.  Ring ) )
26 eleq2 2540 . . . . . . 7  |-  ( s  =  A  ->  (  .1.  e.  s  <->  .1.  e.  A ) )
2725, 26anbi12d 710 . . . . . 6  |-  ( s  =  A  ->  (
( ( Rs  s )  e.  Ring  /\  .1.  e.  s )  <->  ( ( Rs  A )  e.  Ring  /\  .1.  e.  A ) ) )
2827elrab 3266 . . . . 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 4617 . . . . . 6  |-  ( A  e.  ~P B  <->  A  C_  B
)
3029anbi1i 695 . . . . 5  |-  ( ( A  e.  ~P B  /\  ( ( Rs  A )  e.  Ring  /\  .1.  e.  A ) )  <->  ( A  C_  B  /\  ( ( Rs  A )  e.  Ring  /\  .1.  e.  A ) ) )
31 an12 795 . . . . 5  |-  ( ( A  C_  B  /\  ( ( Rs  A )  e.  Ring  /\  .1.  e.  A ) )  <->  ( ( Rs  A )  e.  Ring  /\  ( A  C_  B  /\  .1.  e.  A ) ) )
3228, 30, 313bitri 271 . . . 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 504 . . . . 5  |-  ( R  e.  Ring  ->  ( ( Rs  A )  e.  Ring  <->  ( R  e.  Ring  /\  ( Rs  A )  e.  Ring ) ) )
3433anbi1d 704 . . . 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 257 . . 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 253 . 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 353 1  |-  ( A  e.  (SubRing `  R
)  <->  ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  B  /\  .1.  e.  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2821   _Vcvv 3118    C_ wss 3481   ~Pcpw 4016   dom cdm 5005   ` cfv 5594  (class class class)co 6295   Basecbs 14507   ↾s cress 14508   1rcur 17025   Ringcrg 17070  SubRingcsubrg 17296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-subrg 17298
This theorem is referenced by:  subrgss  17301  subrgid  17302  subrgring  17303  subrgrcl  17305  subrg1cl  17308  issubrg2  17320  subsubrg  17326  subrgpropd  17334  issubassa  17843  subrgpsr  17944  cphsubrglem  21492
  Copyright terms: Public domain W3C validator