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Theorem uvcvval 18991
Description: Value of a unit vector coordinate 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
uvcvval  |-  ( ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I
)  /\  K  e.  I )  ->  (
( U `  J
) `  K )  =  if ( K  =  J ,  .1.  ,  .0.  ) )

Proof of Theorem uvcvval
Dummy variable  k 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, 3uvcval 18990 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I )  ->  ( U `  J
)  =  ( k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) ) )
54fveq1d 5850 . . 3  |-  ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I )  ->  ( ( U `  J ) `  K
)  =  ( ( k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) ) `  K
) )
65adantr 463 . 2  |-  ( ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I
)  /\  K  e.  I )  ->  (
( U `  J
) `  K )  =  ( ( k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) ) `
 K ) )
7 simpr 459 . . 3  |-  ( ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I
)  /\  K  e.  I )  ->  K  e.  I )
8 fvex 5858 . . . . 5  |-  ( 1r
`  R )  e. 
_V
92, 8eqeltri 2538 . . . 4  |-  .1.  e.  _V
10 fvex 5858 . . . . 5  |-  ( 0g
`  R )  e. 
_V
113, 10eqeltri 2538 . . . 4  |-  .0.  e.  _V
129, 11ifex 3997 . . 3  |-  if ( K  =  J ,  .1.  ,  .0.  )  e. 
_V
13 eqeq1 2458 . . . . 5  |-  ( k  =  K  ->  (
k  =  J  <->  K  =  J ) )
1413ifbid 3951 . . . 4  |-  ( k  =  K  ->  if ( k  =  J ,  .1.  ,  .0.  )  =  if ( K  =  J ,  .1.  ,  .0.  ) )
15 eqid 2454 . . . 4  |-  ( k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) )  =  ( k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) )
1614, 15fvmptg 5929 . . 3  |-  ( ( K  e.  I  /\  if ( K  =  J ,  .1.  ,  .0.  )  e.  _V )  ->  ( ( k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) ) `  K )  =  if ( K  =  J ,  .1.  ,  .0.  ) )
177, 12, 16sylancl 660 . 2  |-  ( ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I
)  /\  K  e.  I )  ->  (
( k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) ) `  K
)  =  if ( K  =  J ,  .1.  ,  .0.  ) )
186, 17eqtrd 2495 1  |-  ( ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I
)  /\  K  e.  I )  ->  (
( U `  J
) `  K )  =  if ( K  =  J ,  .1.  ,  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106   ifcif 3929    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   0gc0g 14932   1rcur 17351   unitVec cuvc 18987
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-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-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-oprab 6274  df-mpt2 6275  df-uvc 18988
This theorem is referenced by:  uvcvvcl  18992  uvcvvcl2  18993  uvcvv1  18994  uvcvv0  18995
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