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Theorem uvcval 19110
Description: Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
Hypotheses
Ref Expression
uvcfval.u  |-  U  =  ( R unitVec  I )
uvcfval.o  |-  .1.  =  ( 1r `  R )
uvcfval.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
uvcval  |-  ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I )  ->  ( U `  J
)  =  ( k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) ) )
Distinct variable groups:    .1. , k    R, k    k, I    .0. , k    k, J
Allowed substitution hints:    U( k)    V( k)    W( k)

Proof of Theorem uvcval
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 uvcfval.u . . . . 5  |-  U  =  ( R unitVec  I )
2 uvcfval.o . . . . 5  |-  .1.  =  ( 1r `  R )
3 uvcfval.z . . . . 5  |-  .0.  =  ( 0g `  R )
41, 2, 3uvcfval 19109 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  U  =  ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  )
) ) )
54fveq1d 5850 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( U `  J
)  =  ( ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) ) `  J ) )
653adant3 1017 . 2  |-  ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I )  ->  ( U `  J
)  =  ( ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) ) `  J ) )
7 simp3 999 . . 3  |-  ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I )  ->  J  e.  I )
8 mptexg 6122 . . . 4  |-  ( I  e.  W  ->  (
k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) )  e.  _V )
983ad2ant2 1019 . . 3  |-  ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I )  ->  ( k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) )  e.  _V )
10 eqeq2 2417 . . . . . 6  |-  ( j  =  J  ->  (
k  =  j  <->  k  =  J ) )
1110ifbid 3906 . . . . 5  |-  ( j  =  J  ->  if ( k  =  j ,  .1.  ,  .0.  )  =  if (
k  =  J ,  .1.  ,  .0.  ) )
1211mpteq2dv 4481 . . . 4  |-  ( j  =  J  ->  (
k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) )  =  ( k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) ) )
13 eqid 2402 . . . 4  |-  ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  )
) )  =  ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) )
1412, 13fvmptg 5929 . . 3  |-  ( ( J  e.  I  /\  ( k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) )  e.  _V )  ->  ( ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  )
) ) `  J
)  =  ( k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) ) )
157, 9, 14syl2anc 659 . 2  |-  ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I )  ->  ( ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) ) `
 J )  =  ( k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) ) )
166, 15eqtrd 2443 1  |-  ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I )  ->  ( U `  J
)  =  ( k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   _Vcvv 3058   ifcif 3884    |-> cmpt 4452   ` cfv 5568  (class class class)co 6277   0gc0g 15052   1rcur 17471   unitVec cuvc 19107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-uvc 19108
This theorem is referenced by:  uvcvval  19111
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