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Theorem sraval 18017
Description: Lemma for srabase 18019 through sravsca 18023. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.)
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 ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)

Proof of Theorem sraval
Dummy variables  s  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3115 . . . 4  |-  ( W  e.  V  ->  W  e.  _V )
21adantr 463 . . 3  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  W  e.  _V )
3 fveq2 5848 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
43pweqd 4004 . . . . 5  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P ( Base `  W
) )
5 id 22 . . . . . . . 8  |-  ( w  =  W  ->  w  =  W )
6 oveq1 6277 . . . . . . . . 9  |-  ( w  =  W  ->  (
ws  s )  =  ( Ws  s ) )
76opeq2d 4210 . . . . . . . 8  |-  ( w  =  W  ->  <. (Scalar ` 
ndx ) ,  ( ws  s ) >.  =  <. (Scalar `  ndx ) ,  ( Ws  s ) >. )
85, 7oveq12d 6288 . . . . . . 7  |-  ( w  =  W  ->  (
w sSet  <. (Scalar `  ndx ) ,  ( ws  s
) >. )  =  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  s
) >. ) )
9 fveq2 5848 . . . . . . . 8  |-  ( w  =  W  ->  ( .r `  w )  =  ( .r `  W
) )
109opeq2d 4210 . . . . . . 7  |-  ( w  =  W  ->  <. ( .s `  ndx ) ,  ( .r `  w
) >.  =  <. ( .s `  ndx ) ,  ( .r `  W
) >. )
118, 10oveq12d 6288 . . . . . 6  |-  ( w  =  W  ->  (
( w sSet  <. (Scalar ` 
ndx ) ,  ( ws  s ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  w ) >. )  =  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  s ) >.
) sSet  <. ( .s `  ndx ) ,  ( .r
`  W ) >.
) )
129opeq2d 4210 . . . . . 6  |-  ( w  =  W  ->  <. ( .i `  ndx ) ,  ( .r `  w
) >.  =  <. ( .i `  ndx ) ,  ( .r `  W
) >. )
1311, 12oveq12d 6288 . . . . 5  |-  ( w  =  W  ->  (
( ( w sSet  <. (Scalar `  ndx ) ,  ( ws  s ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  w ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  w ) >. )  =  ( ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  s
) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  W
) >. ) sSet  <. ( .i `  ndx ) ,  ( .r `  W
) >. ) )
144, 13mpteq12dv 4517 . . . 4  |-  ( w  =  W  ->  (
s  e.  ~P ( Base `  w )  |->  ( ( ( w sSet  <. (Scalar `  ndx ) ,  ( ws  s ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  w ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  w ) >. )
)  =  ( s  e.  ~P ( Base `  W )  |->  ( ( ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  s ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
) )
15 df-sra 18013 . . . 4  |- subringAlg  =  ( w  e.  _V  |->  ( s  e.  ~P ( Base `  w )  |->  ( ( ( w sSet  <. (Scalar `  ndx ) ,  ( ws  s ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  w ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  w ) >. )
) )
16 fvex 5858 . . . . . 6  |-  ( Base `  W )  e.  _V
1716pwex 4620 . . . . 5  |-  ~P ( Base `  W )  e. 
_V
1817mptex 6118 . . . 4  |-  ( s  e.  ~P ( Base `  W )  |->  ( ( ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  s ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)  e.  _V
1914, 15, 18fvmpt 5931 . . 3  |-  ( W  e.  _V  ->  (subringAlg  `  W )  =  ( s  e.  ~P ( Base `  W )  |->  ( ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  s ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
) )
202, 19syl 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 ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
) )
21 simpr 459 . . . . . . 7  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  ->  s  =  S )
2221oveq2d 6286 . . . . . 6  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  ->  ( Ws  s )  =  ( Ws  S ) )
2322opeq2d 4210 . . . . 5  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  -> 
<. (Scalar `  ndx ) ,  ( Ws  s ) >.  =  <. (Scalar `  ndx ) ,  ( Ws  S
) >. )
2423oveq2d 6286 . . . 4  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  ->  ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  s ) >. )  =  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. )
)
2524oveq1d 6285 . . 3  |-  ( ( ( 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 ) >. )
)
2625oveq1d 6285 . 2  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  ->  ( ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  s ) >.
) sSet  <. ( .s `  ndx ) ,  ( .r
`  W ) >.
) sSet  <. ( .i `  ndx ) ,  ( .r
`  W ) >.
)  =  ( ( ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
27 simpr 459 . . 3  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  S  C_  ( Base `  W
) )
2816elpw2 4601 . . 3  |-  ( S  e.  ~P ( Base `  W )  <->  S  C_  ( Base `  W ) )
2927, 28sylibr 212 . 2  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  S  e.  ~P ( Base `  W
) )
30 ovex 6298 . . 3  |-  ( ( ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )  e.  _V
3130a1i 11 . 2  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  (
( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )  e.  _V )
3220, 26, 29, 31fvmptd 5936 1  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  (
(subringAlg  `  W ) `  S )  =  ( ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <.
( .s `  ndx ) ,  ( .r `  W ) >. ) sSet  <.
( .i `  ndx ) ,  ( .r `  W ) >. )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999   <.cop 4022    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   ndxcnx 14713   sSet csts 14714   Basecbs 14716   ↾s cress 14717   .rcmulr 14785  Scalarcsca 14787   .scvsca 14788   .icip 14789  subringAlg csra 18009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-sra 18013
This theorem is referenced by:  sralem  18018  srasca  18022  sravsca  18023  sraip  18024  rlmval2  18035
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