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Theorem issubrg 16865
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 16863 . . . 4  |- SubRing  =  ( r  e.  Ring  |->  { s  e.  ~P ( Base `  r )  |  ( ( rs  s )  e. 
Ring  /\  ( 1r `  r )  e.  s ) } )
21dmmptss 5334 . . 3  |-  dom SubRing  C_  Ring
3 elfvdm 5716 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  dom SubRing )
42, 3sseldi 3354 . 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 5691 . . . . . . . 8  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
7 issubrg.b . . . . . . . 8  |-  B  =  ( Base `  R
)
86, 7syl6eqr 2493 . . . . . . 7  |-  ( r  =  R  ->  ( Base `  r )  =  B )
98pweqd 3865 . . . . . 6  |-  ( r  =  R  ->  ~P ( Base `  r )  =  ~P B )
10 oveq1 6098 . . . . . . . 8  |-  ( r  =  R  ->  (
rs  s )  =  ( Rs  s ) )
1110eleq1d 2509 . . . . . . 7  |-  ( r  =  R  ->  (
( rs  s )  e. 
Ring 
<->  ( Rs  s )  e. 
Ring ) )
12 fveq2 5691 . . . . . . . . 9  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
13 issubrg.i . . . . . . . . 9  |-  .1.  =  ( 1r `  R )
1412, 13syl6eqr 2493 . . . . . . . 8  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
1514eleq1d 2509 . . . . . . 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 2967 . . . . 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 5701 . . . . . . . 8  |-  ( Base `  R )  e.  _V
197, 18eqeltri 2513 . . . . . . 7  |-  B  e. 
_V
2019pwex 4475 . . . . . 6  |-  ~P B  e.  _V
2120rabex 4443 . . . . 5  |-  { s  e.  ~P B  | 
( ( Rs  s )  e.  Ring  /\  .1.  e.  s ) }  e.  _V
2217, 1, 21fvmpt 5774 . . . 4  |-  ( R  e.  Ring  ->  (SubRing `  R
)  =  { s  e.  ~P B  | 
( ( Rs  s )  e.  Ring  /\  .1.  e.  s ) } )
2322eleq2d 2510 . . 3  |-  ( R  e.  Ring  ->  ( A  e.  (SubRing `  R
)  <->  A  e.  { s  e.  ~P B  | 
( ( Rs  s )  e.  Ring  /\  .1.  e.  s ) } ) )
24 oveq2 6099 . . . . . . . 8  |-  ( s  =  A  ->  ( Rs  s )  =  ( Rs  A ) )
2524eleq1d 2509 . . . . . . 7  |-  ( s  =  A  ->  (
( Rs  s )  e. 
Ring 
<->  ( Rs  A )  e.  Ring ) )
26 eleq2 2504 . . . . . . 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 3117 . . . . 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 4456 . . . . . 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 1369    e. wcel 1756   {crab 2719   _Vcvv 2972    C_ wss 3328   ~Pcpw 3860   dom cdm 4840   ` cfv 5418  (class class class)co 6091   Basecbs 14174   ↾s cress 14175   1rcur 16603   Ringcrg 16645  SubRingcsubrg 16861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-subrg 16863
This theorem is referenced by:  subrgss  16866  subrgid  16867  subrgrng  16868  subrgrcl  16870  subrg1cl  16873  issubrg2  16885  subsubrg  16891  subrgpropd  16899  issubassa  17395  subrgpsr  17491  cphsubrglem  20696
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