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Theorem resvval 26300
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 2986 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 elex 2986 . . 3  |-  ( A  e.  Y  ->  A  e.  _V )
4 ovex 6121 . . . . . 6  |-  ( W sSet  <. (Scalar `  ndx ) ,  ( Fs  A ) >. )  e.  _V
5 ifcl 3836 . . . . . 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 5700 . . . . . . . . . 10  |-  ( ( w  =  W  /\  x  =  A )  ->  (Scalar `  w )  =  (Scalar `  W )
)
10 resvsca.f . . . . . . . . . 10  |-  F  =  (Scalar `  W )
119, 10syl6eqr 2493 . . . . . . . . 9  |-  ( ( w  =  W  /\  x  =  A )  ->  (Scalar `  w )  =  F )
1211fveq2d 5700 . . . . . . . 8  |-  ( ( w  =  W  /\  x  =  A )  ->  ( Base `  (Scalar `  w ) )  =  ( Base `  F
) )
13 resvsca.b . . . . . . . 8  |-  B  =  ( Base `  F
)
1412, 13syl6eqr 2493 . . . . . . 7  |-  ( ( w  =  W  /\  x  =  A )  ->  ( Base `  (Scalar `  w ) )  =  B )
15 simpr 461 . . . . . . 7  |-  ( ( w  =  W  /\  x  =  A )  ->  x  =  A )
1614, 15sseq12d 3390 . . . . . 6  |-  ( ( w  =  W  /\  x  =  A )  ->  ( ( Base `  (Scalar `  w ) )  C_  x 
<->  B  C_  A )
)
1711, 15oveq12d 6114 . . . . . . . 8  |-  ( ( w  =  W  /\  x  =  A )  ->  ( (Scalar `  w
)s  x )  =  ( Fs  A ) )
1817opeq2d 4071 . . . . . . 7  |-  ( ( w  =  W  /\  x  =  A )  -> 
<. (Scalar `  ndx ) ,  ( (Scalar `  w
)s  x ) >.  =  <. (Scalar `  ndx ) ,  ( Fs  A ) >. )
198, 18oveq12d 6114 . . . . . 6  |-  ( ( w  =  W  /\  x  =  A )  ->  ( w sSet  <. (Scalar ` 
ndx ) ,  ( (Scalar `  w )s  x
) >. )  =  ( W sSet  <. (Scalar `  ndx ) ,  ( Fs  A
) >. ) )
2016, 8, 19ifbieq12d 3821 . . . . 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 26298 . . . . 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 6225 . . . 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 1315 . . 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 2487 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 1369    e. wcel 1756   _Vcvv 2977    C_ wss 3333   ifcif 3796   <.cop 3888   ` cfv 5423  (class class class)co 6096   ndxcnx 14176   sSet csts 14177   Basecbs 14179   ↾s cress 14180  Scalarcsca 14246   ↾v cresv 26297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-resv 26298
This theorem is referenced by:  resvid2  26301  resvval2  26302
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