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Theorem sraval 16211
Description: Lemma for srabase 16213 through sravsca 16217. (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 2932 . . . 4  |-  ( W  e.  V  ->  W  e.  _V )
21adantr 452 . . 3  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  W  e.  _V )
3 fveq2 5695 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
43pweqd 3772 . . . . 5  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P ( Base `  W
) )
5 id 20 . . . . . . 7  |-  ( w  =  W  ->  w  =  W )
6 oveq1 6055 . . . . . . . 8  |-  ( w  =  W  ->  (
ws  s )  =  ( Ws  s ) )
76opeq2d 3959 . . . . . . 7  |-  ( w  =  W  ->  <. (Scalar ` 
ndx ) ,  ( ws  s ) >.  =  <. (Scalar `  ndx ) ,  ( Ws  s ) >. )
85, 7oveq12d 6066 . . . . . 6  |-  ( w  =  W  ->  (
w sSet  <. (Scalar `  ndx ) ,  ( ws  s
) >. )  =  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  s
) >. ) )
9 fveq2 5695 . . . . . . 7  |-  ( w  =  W  ->  ( .r `  w )  =  ( .r `  W
) )
109opeq2d 3959 . . . . . 6  |-  ( w  =  W  ->  <. ( .s `  ndx ) ,  ( .r `  w
) >.  =  <. ( .s `  ndx ) ,  ( .r `  W
) >. )
118, 10oveq12d 6066 . . . . 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 4255 . . . 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 16207 . . . 4  |- subringAlg  =  ( w  e.  _V  |->  ( s  e.  ~P ( Base `  w )  |->  ( ( w sSet  <. (Scalar ` 
ndx ) ,  ( ws  s ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  w ) >. )
) )
14 fvex 5709 . . . . . 6  |-  ( Base `  W )  e.  _V
1514pwex 4350 . . . . 5  |-  ~P ( Base `  W )  e. 
_V
1615mptex 5933 . . . 4  |-  ( s  e.  ~P ( Base `  W )  |->  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  s
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) )  e. 
_V
1712, 13, 16fvmpt 5773 . . 3  |-  ( W  e.  _V  ->  ( subringAlg  `  W )  =  ( s  e.  ~P ( Base `  W )  |->  ( ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  s ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. )
) )
182, 17syl 16 . 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 448 . . . . . 6  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  ->  s  =  S )
2019oveq2d 6064 . . . . 5  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  ->  ( Ws  s )  =  ( Ws  S ) )
2120opeq2d 3959 . . . 4  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  -> 
<. (Scalar `  ndx ) ,  ( Ws  s ) >.  =  <. (Scalar `  ndx ) ,  ( Ws  S
) >. )
2221oveq2d 6064 . . 3  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  ->  ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  s ) >. )  =  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. )
)
2322oveq1d 6063 . 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 448 . . 3  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  S  C_  ( Base `  W
) )
2514elpw2 4332 . . 3  |-  ( S  e.  ~P ( Base `  W )  <->  S  C_  ( Base `  W ) )
2624, 25sylibr 204 . 2  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  S  e.  ~P ( Base `  W
) )
27 ovex 6073 . . 3  |-  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. )  e.  _V
2827a1i 11 . 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 5777 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 359    = wceq 1649    e. wcel 1721   _Vcvv 2924    C_ wss 3288   ~Pcpw 3767   <.cop 3785    e. cmpt 4234   ` cfv 5421  (class class class)co 6048   ndxcnx 13429   sSet csts 13430   Basecbs 13432   ↾s cress 13433   .rcmulr 13493  Scalarcsca 13495   .scvsca 13496   subringAlg csra 16203
This theorem is referenced by:  sralem  16212  srasca  16216  sravsca  16217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-sra 16207
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