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Theorem resvval 27970
Description: Value of structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
Hypotheses
Ref Expression
resvsca.r  |-  R  =  ( Wv  A )
resvsca.f  |-  F  =  (Scalar `  W )
resvsca.b  |-  B  =  ( Base `  F
)
Assertion
Ref Expression
resvval  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  R  =  if ( B  C_  A ,  W ,  ( W sSet  <.
(Scalar `  ndx ) ,  ( Fs  A ) >. )
) )

Proof of Theorem resvval
Dummy variables  x  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resvsca.r . 2  |-  R  =  ( Wv  A )
2 elex 3118 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 elex 3118 . . 3  |-  ( A  e.  Y  ->  A  e.  _V )
4 ovex 6324 . . . . . 6  |-  ( W sSet  <. (Scalar `  ndx ) ,  ( Fs  A ) >. )  e.  _V
5 ifcl 3986 . . . . . 6  |-  ( ( W  e.  _V  /\  ( W sSet  <. (Scalar `  ndx ) ,  ( Fs  A ) >. )  e.  _V )  ->  if ( B 
C_  A ,  W ,  ( W sSet  <. (Scalar `  ndx ) ,  ( Fs  A ) >. )
)  e.  _V )
64, 5mpan2 671 . . . . 5  |-  ( W  e.  _V  ->  if ( B  C_  A ,  W ,  ( W sSet  <.
(Scalar `  ndx ) ,  ( Fs  A ) >. )
)  e.  _V )
76adantr 465 . . . 4  |-  ( ( W  e.  _V  /\  A  e.  _V )  ->  if ( B  C_  A ,  W , 
( W sSet  <. (Scalar `  ndx ) ,  ( Fs  A ) >. ) )  e. 
_V )
8 simpl 457 . . . . . . . . . . 11  |-  ( ( w  =  W  /\  x  =  A )  ->  w  =  W )
98fveq2d 5876 . . . . . . . . . 10  |-  ( ( w  =  W  /\  x  =  A )  ->  (Scalar `  w )  =  (Scalar `  W )
)
10 resvsca.f . . . . . . . . . 10  |-  F  =  (Scalar `  W )
119, 10syl6eqr 2516 . . . . . . . . 9  |-  ( ( w  =  W  /\  x  =  A )  ->  (Scalar `  w )  =  F )
1211fveq2d 5876 . . . . . . . 8  |-  ( ( w  =  W  /\  x  =  A )  ->  ( Base `  (Scalar `  w ) )  =  ( Base `  F
) )
13 resvsca.b . . . . . . . 8  |-  B  =  ( Base `  F
)
1412, 13syl6eqr 2516 . . . . . . 7  |-  ( ( w  =  W  /\  x  =  A )  ->  ( Base `  (Scalar `  w ) )  =  B )
15 simpr 461 . . . . . . 7  |-  ( ( w  =  W  /\  x  =  A )  ->  x  =  A )
1614, 15sseq12d 3528 . . . . . 6  |-  ( ( w  =  W  /\  x  =  A )  ->  ( ( Base `  (Scalar `  w ) )  C_  x 
<->  B  C_  A )
)
1711, 15oveq12d 6314 . . . . . . . 8  |-  ( ( w  =  W  /\  x  =  A )  ->  ( (Scalar `  w
)s  x )  =  ( Fs  A ) )
1817opeq2d 4226 . . . . . . 7  |-  ( ( w  =  W  /\  x  =  A )  -> 
<. (Scalar `  ndx ) ,  ( (Scalar `  w
)s  x ) >.  =  <. (Scalar `  ndx ) ,  ( Fs  A ) >. )
198, 18oveq12d 6314 . . . . . 6  |-  ( ( w  =  W  /\  x  =  A )  ->  ( w sSet  <. (Scalar ` 
ndx ) ,  ( (Scalar `  w )s  x
) >. )  =  ( W sSet  <. (Scalar `  ndx ) ,  ( Fs  A
) >. ) )
2016, 8, 19ifbieq12d 3971 . . . . 5  |-  ( ( w  =  W  /\  x  =  A )  ->  if ( ( Base `  (Scalar `  w )
)  C_  x ,  w ,  ( w sSet  <.
(Scalar `  ndx ) ,  ( (Scalar `  w
)s  x ) >. )
)  =  if ( B  C_  A ,  W ,  ( W sSet  <.
(Scalar `  ndx ) ,  ( Fs  A ) >. )
) )
21 df-resv 27968 . . . . 5  |-v  =  ( w  e.  _V ,  x  e. 
_V  |->  if ( (
Base `  (Scalar `  w
) )  C_  x ,  w ,  ( w sSet  <. (Scalar `  ndx ) ,  ( (Scalar `  w
)s  x ) >. )
) )
2220, 21ovmpt2ga 6431 . . . 4  |-  ( ( W  e.  _V  /\  A  e.  _V  /\  if ( B  C_  A ,  W ,  ( W sSet  <.
(Scalar `  ndx ) ,  ( Fs  A ) >. )
)  e.  _V )  ->  ( Wv  A )  =  if ( B  C_  A ,  W ,  ( W sSet  <. (Scalar `  ndx ) ,  ( Fs  A ) >. )
) )
237, 22mpd3an3 1325 . . 3  |-  ( ( W  e.  _V  /\  A  e.  _V )  ->  ( Wv  A )  =  if ( B  C_  A ,  W ,  ( W sSet  <. (Scalar `  ndx ) ,  ( Fs  A ) >. )
) )
242, 3, 23syl2an 477 . 2  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  ( Wv  A )  =  if ( B  C_  A ,  W ,  ( W sSet  <. (Scalar `  ndx ) ,  ( Fs  A ) >. )
) )
251, 24syl5eq 2510 1  |-  ( ( W  e.  X  /\  A  e.  Y )  ->  R  =  if ( B  C_  A ,  W ,  ( W sSet  <.
(Scalar `  ndx ) ,  ( Fs  A ) >. )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109    C_ wss 3471   ifcif 3944   <.cop 4038   ` cfv 5594  (class class class)co 6296   ndxcnx 14640   sSet csts 14641   Basecbs 14643   ↾s cress 14644  Scalarcsca 14714   ↾v cresv 27967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-resv 27968
This theorem is referenced by:  resvid2  27971  resvval2  27972
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