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Theorem sraval 17372
Description: Lemma for srabase 17374 through sravsca 17378. (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 3080 . . . 4  |-  ( W  e.  V  ->  W  e.  _V )
21adantr 465 . . 3  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  W  e.  _V )
3 fveq2 5792 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
43pweqd 3966 . . . . 5  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P ( Base `  W
) )
5 id 22 . . . . . . . 8  |-  ( w  =  W  ->  w  =  W )
6 oveq1 6200 . . . . . . . . 9  |-  ( w  =  W  ->  (
ws  s )  =  ( Ws  s ) )
76opeq2d 4167 . . . . . . . 8  |-  ( w  =  W  ->  <. (Scalar ` 
ndx ) ,  ( ws  s ) >.  =  <. (Scalar `  ndx ) ,  ( Ws  s ) >. )
85, 7oveq12d 6211 . . . . . . 7  |-  ( w  =  W  ->  (
w sSet  <. (Scalar `  ndx ) ,  ( ws  s
) >. )  =  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  s
) >. ) )
9 fveq2 5792 . . . . . . . 8  |-  ( w  =  W  ->  ( .r `  w )  =  ( .r `  W
) )
109opeq2d 4167 . . . . . . 7  |-  ( w  =  W  ->  <. ( .s `  ndx ) ,  ( .r `  w
) >.  =  <. ( .s `  ndx ) ,  ( .r `  W
) >. )
118, 10oveq12d 6211 . . . . . 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 4167 . . . . . 6  |-  ( w  =  W  ->  <. ( .i `  ndx ) ,  ( .r `  w
) >.  =  <. ( .i `  ndx ) ,  ( .r `  W
) >. )
1311, 12oveq12d 6211 . . . . 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 4471 . . . 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 17368 . . . 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 5802 . . . . . 6  |-  ( Base `  W )  e.  _V
1716pwex 4576 . . . . 5  |-  ~P ( Base `  W )  e. 
_V
1817mptex 6050 . . . 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 5876 . . 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 461 . . . . . . 7  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  ->  s  =  S )
2221oveq2d 6209 . . . . . 6  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  ->  ( Ws  s )  =  ( Ws  S ) )
2322opeq2d 4167 . . . . 5  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  -> 
<. (Scalar `  ndx ) ,  ( Ws  s ) >.  =  <. (Scalar `  ndx ) ,  ( Ws  S
) >. )
2423oveq2d 6209 . . . 4  |-  ( ( ( W  e.  V  /\  S  C_  ( Base `  W ) )  /\  s  =  S )  ->  ( W sSet  <. (Scalar ` 
ndx ) ,  ( Ws  s ) >. )  =  ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. )
)
2524oveq1d 6208 . . 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 6208 . 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 461 . . 3  |-  ( ( W  e.  V  /\  S  C_  ( Base `  W
) )  ->  S  C_  ( Base `  W
) )
2816elpw2 4557 . . 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 6218 . . 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 5881 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 369    = wceq 1370    e. wcel 1758   _Vcvv 3071    C_ wss 3429   ~Pcpw 3961   <.cop 3984    |-> cmpt 4451   ` cfv 5519  (class class class)co 6193   ndxcnx 14282   sSet csts 14283   Basecbs 14285   ↾s cress 14286   .rcmulr 14350  Scalarcsca 14352   .scvsca 14353   .icip 14354  subringAlg csra 17364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-sra 17368
This theorem is referenced by:  sralem  17373  srasca  17377  sravsca  17378  sraip  17379  rlmval2  17390
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