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Theorem sraval 15929
Description: Lemma for srabase 15931 through sravsca 15935. (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 2796 . . . 4  |-  ( W  e.  V  ->  W  e.  _V )
21adantr 451 . . 3  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  W  e.  _V )
3 fveq2 5525 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
43pweqd 3630 . . . . 5  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P ( Base `  W
) )
5 id 19 . . . . . . 7  |-  ( w  =  W  ->  w  =  W )
6 oveq1 5865 . . . . . . . 8  |-  ( w  =  W  ->  (
ws  s )  =  ( Ws  s ) )
76opeq2d 3803 . . . . . . 7  |-  ( w  =  W  ->  <. (Scalar ` 
ndx ) ,  ( ws  s ) >.  =  <. (Scalar `  ndx ) ,  ( Ws  s ) >. )
85, 7oveq12d 5876 . . . . . 6  |-  ( w  =  W  ->  (
w sSet  <. (Scalar `  ndx ) ,  ( ws  s
) >. )  =  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  s
) >. ) )
9 fveq2 5525 . . . . . . 7  |-  ( w  =  W  ->  ( .r `  w )  =  ( .r `  W
) )
109opeq2d 3803 . . . . . 6  |-  ( w  =  W  ->  <. ( .s `  ndx ) ,  ( .r `  w
) >.  =  <. ( .s `  ndx ) ,  ( .r `  W
) >. )
118, 10oveq12d 5876 . . . . 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 4098 . . . 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 15925 . . . 4  |- subringAlg  =  ( w  e.  _V  |->  ( s  e.  ~P ( Base `  w )  |->  ( ( w sSet  <. (Scalar ` 
ndx ) ,  ( ws  s ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  w ) >. )
) )
14 fvex 5539 . . . . . 6  |-  ( Base `  W )  e.  _V
1514pwex 4193 . . . . 5  |-  ~P ( Base `  W )  e. 
_V
1615mptex 5746 . . . 4  |-  ( s  e.  ~P ( Base `  W )  |->  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  s
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) )  e. 
_V
1712, 13, 16fvmpt 5602 . . 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 5874 . . . . 5  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  ->  ( Ws  s )  =  ( Ws  S ) )
2120opeq2d 3803 . . . 4  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  -> 
<. (Scalar `  ndx ) ,  ( Ws  s ) >.  =  <. (Scalar `  ndx ) ,  ( Ws  S
) >. )
2221oveq2d 5874 . . 3  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  ->  ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  s ) >. )  =  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. )
)
2322oveq1d 5873 . 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 4175 . . 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 5883 . . 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 5606 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 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   <.cop 3643    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   ndxcnx 13145   sSet csts 13146   Basecbs 13148   ↾s cress 13149   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   subringAlg csra 15921
This theorem is referenced by:  sralem  15930  srasca  15934  sravsca  15935
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-sra 15925
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