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Theorem issubrg 15860
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 15858 . . . 4  |- SubRing  =  ( r  e.  Ring  |->  { s  e.  ~P ( Base `  r )  |  ( ( rs  s )  e. 
Ring  /\  ( 1r `  r )  e.  s ) } )
21dmmptss 5358 . . 3  |-  dom SubRing  C_  Ring
3 elfvdm 5749 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  dom SubRing )
42, 3sseldi 3338 . 2  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
5 simpll 731 . 2  |-  ( ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  B  /\  .1.  e.  A
) )  ->  R  e.  Ring )
6 fveq2 5720 . . . . . . . 8  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
7 issubrg.b . . . . . . . 8  |-  B  =  ( Base `  R
)
86, 7syl6eqr 2485 . . . . . . 7  |-  ( r  =  R  ->  ( Base `  r )  =  B )
98pweqd 3796 . . . . . 6  |-  ( r  =  R  ->  ~P ( Base `  r )  =  ~P B )
10 oveq1 6080 . . . . . . . 8  |-  ( r  =  R  ->  (
rs  s )  =  ( Rs  s ) )
1110eleq1d 2501 . . . . . . 7  |-  ( r  =  R  ->  (
( rs  s )  e. 
Ring 
<->  ( Rs  s )  e. 
Ring ) )
12 fveq2 5720 . . . . . . . . 9  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
13 issubrg.i . . . . . . . . 9  |-  .1.  =  ( 1r `  R )
1412, 13syl6eqr 2485 . . . . . . . 8  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
1514eleq1d 2501 . . . . . . 7  |-  ( r  =  R  ->  (
( 1r `  r
)  e.  s  <->  .1.  e.  s ) )
1611, 15anbi12d 692 . . . . . 6  |-  ( r  =  R  ->  (
( ( rs  s )  e.  Ring  /\  ( 1r `  r )  e.  s )  <->  ( ( Rs  s )  e.  Ring  /\  .1.  e.  s ) ) )
179, 16rabeqbidv 2943 . . . . 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 5734 . . . . . . . 8  |-  ( Base `  R )  e.  _V
197, 18eqeltri 2505 . . . . . . 7  |-  B  e. 
_V
2019pwex 4374 . . . . . 6  |-  ~P B  e.  _V
2120rabex 4346 . . . . 5  |-  { s  e.  ~P B  | 
( ( Rs  s )  e.  Ring  /\  .1.  e.  s ) }  e.  _V
2217, 1, 21fvmpt 5798 . . . 4  |-  ( R  e.  Ring  ->  (SubRing `  R
)  =  { s  e.  ~P B  | 
( ( Rs  s )  e.  Ring  /\  .1.  e.  s ) } )
2322eleq2d 2502 . . 3  |-  ( R  e.  Ring  ->  ( A  e.  (SubRing `  R
)  <->  A  e.  { s  e.  ~P B  | 
( ( Rs  s )  e.  Ring  /\  .1.  e.  s ) } ) )
24 oveq2 6081 . . . . . . . 8  |-  ( s  =  A  ->  ( Rs  s )  =  ( Rs  A ) )
2524eleq1d 2501 . . . . . . 7  |-  ( s  =  A  ->  (
( Rs  s )  e. 
Ring 
<->  ( Rs  A )  e.  Ring ) )
26 eleq2 2496 . . . . . . 7  |-  ( s  =  A  ->  (  .1.  e.  s  <->  .1.  e.  A ) )
2725, 26anbi12d 692 . . . . . 6  |-  ( s  =  A  ->  (
( ( Rs  s )  e.  Ring  /\  .1.  e.  s )  <->  ( ( Rs  A )  e.  Ring  /\  .1.  e.  A ) ) )
2827elrab 3084 . . . . 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 4356 . . . . . 6  |-  ( A  e.  ~P B  <->  A  C_  B
)
3029anbi1i 677 . . . . 5  |-  ( ( A  e.  ~P B  /\  ( ( Rs  A )  e.  Ring  /\  .1.  e.  A ) )  <->  ( A  C_  B  /\  ( ( Rs  A )  e.  Ring  /\  .1.  e.  A ) ) )
31 an12 773 . . . . 5  |-  ( ( A  C_  B  /\  ( ( Rs  A )  e.  Ring  /\  .1.  e.  A ) )  <->  ( ( Rs  A )  e.  Ring  /\  ( A  C_  B  /\  .1.  e.  A ) ) )
3228, 30, 313bitri 263 . . . 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 491 . . . . 5  |-  ( R  e.  Ring  ->  ( ( Rs  A )  e.  Ring  <->  ( R  e.  Ring  /\  ( Rs  A )  e.  Ring ) ) )
3433anbi1d 686 . . . 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 249 . . 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 245 . 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 343 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   dom cdm 4870   ` cfv 5446  (class class class)co 6073   Basecbs 13461   ↾s cress 13462   Ringcrg 15652   1rcur 15654  SubRingcsubrg 15856
This theorem is referenced by:  subrgss  15861  subrgid  15862  subrgrng  15863  subrgrcl  15865  subrg1cl  15868  issubrg2  15880  subsubrg  15886  subrgpropd  15894  issubassa  16375  subrgpsr  16474  cphsubrglem  19132
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-pw 3793  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-subrg 15858
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