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Theorem sraval 16139
Description: Lemma for srabase 16141 through sravsca 16145. (Contributed by Mario Carneiro, 27-Nov-2014.)
Assertion
Ref Expression
sraval  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  (
( subringAlg  `  W ) `  S )  =  ( ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)

Proof of Theorem sraval
Dummy variables  s  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2881 . . . 4  |-  ( W  e.  V  ->  W  e.  _V )
21adantr 451 . . 3  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  W  e.  _V )
3 fveq2 5632 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
43pweqd 3719 . . . . 5  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P ( Base `  W
) )
5 id 19 . . . . . . 7  |-  ( w  =  W  ->  w  =  W )
6 oveq1 5988 . . . . . . . 8  |-  ( w  =  W  ->  (
ws  s )  =  ( Ws  s ) )
76opeq2d 3905 . . . . . . 7  |-  ( w  =  W  ->  <. (Scalar ` 
ndx ) ,  ( ws  s ) >.  =  <. (Scalar `  ndx ) ,  ( Ws  s ) >. )
85, 7oveq12d 5999 . . . . . 6  |-  ( w  =  W  ->  (
w sSet  <. (Scalar `  ndx ) ,  ( ws  s
) >. )  =  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  s
) >. ) )
9 fveq2 5632 . . . . . . 7  |-  ( w  =  W  ->  ( .r `  w )  =  ( .r `  W
) )
109opeq2d 3905 . . . . . 6  |-  ( w  =  W  ->  <. ( .s `  ndx ) ,  ( .r `  w
) >.  =  <. ( .s `  ndx ) ,  ( .r `  W
) >. )
118, 10oveq12d 5999 . . . . 5  |-  ( w  =  W  ->  (
( w sSet  <. (Scalar ` 
ndx ) ,  ( ws  s ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  w ) >. )  =  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  s ) >.
) sSet  <. ( .s `  ndx ) ,  ( .r
`  W ) >.
) )
124, 11mpteq12dv 4200 . . . 4  |-  ( w  =  W  ->  (
s  e.  ~P ( Base `  w )  |->  ( ( w sSet  <. (Scalar ` 
ndx ) ,  ( ws  s ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  w ) >. )
)  =  ( s  e.  ~P ( Base `  W )  |->  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  s
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) ) )
13 df-sra 16135 . . . 4  |- subringAlg  =  ( w  e.  _V  |->  ( s  e.  ~P ( Base `  w )  |->  ( ( w sSet  <. (Scalar ` 
ndx ) ,  ( ws  s ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  w ) >. )
) )
14 fvex 5646 . . . . . 6  |-  ( Base `  W )  e.  _V
1514pwex 4295 . . . . 5  |-  ~P ( Base `  W )  e. 
_V
1615mptex 5866 . . . 4  |-  ( s  e.  ~P ( Base `  W )  |->  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  s
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) )  e. 
_V
1712, 13, 16fvmpt 5709 . . 3  |-  ( W  e.  _V  ->  ( subringAlg  `  W )  =  ( s  e.  ~P ( Base `  W )  |->  ( ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  s ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
) )
182, 17syl 15 . 2  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  ( subringAlg  `  W )  =  ( s  e.  ~P ( Base `  W )  |->  ( ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  s ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
) )
19 simpr 447 . . . . . 6  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  ->  s  =  S )
2019oveq2d 5997 . . . . 5  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  ->  ( Ws  s )  =  ( Ws  S ) )
2120opeq2d 3905 . . . 4  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  -> 
<. (Scalar `  ndx ) ,  ( Ws  s ) >.  =  <. (Scalar `  ndx ) ,  ( Ws  S
) >. )
2221oveq2d 5997 . . 3  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  ->  ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  s ) >. )  =  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. )
)
2322oveq1d 5996 . 2  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  ->  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  s ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )  =  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)
24 simpr 447 . . 3  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  S  C_  ( Base `  W
) )
2514elpw2 4277 . . 3  |-  ( S  e.  ~P ( Base `  W )  <->  S  C_  ( Base `  W ) )
2624, 25sylibr 203 . 2  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  S  e.  ~P ( Base `  W
) )
27 ovex 6006 . . 3  |-  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. )  e.  _V
2827a1i 10 . 2  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  (
( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. )  e.  _V )
2918, 23, 26, 28fvmptd 5713 1  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  (
( subringAlg  `  W ) `  S )  =  ( ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   _Vcvv 2873    C_ wss 3238   ~Pcpw 3714   <.cop 3732    e. cmpt 4179   ` cfv 5358  (class class class)co 5981   ndxcnx 13353   sSet csts 13354   Basecbs 13356   ↾s cress 13357   .rcmulr 13417  Scalarcsca 13419   .scvsca 13420   subringAlg csra 16131
This theorem is referenced by:  sralem  16140  srasca  16144  sravsca  16145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-sra 16135
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