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Theorem uvcval 18345
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 18344 . . . 4  |-  ( ( R  e.  V  /\  I  e.  W )  ->  U  =  ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  )
) ) )
54fveq1d 5804 . . 3  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( U `  J
)  =  ( ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) ) `  J ) )
653adant3 1008 . 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 990 . . 3  |-  ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I )  ->  J  e.  I )
8 mptexg 6059 . . . 4  |-  ( I  e.  W  ->  (
k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) )  e.  _V )
983ad2ant2 1010 . . 3  |-  ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I )  ->  ( k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) )  e.  _V )
10 eqeq2 2469 . . . . . 6  |-  ( j  =  J  ->  (
k  =  j  <->  k  =  J ) )
1110ifbid 3922 . . . . 5  |-  ( j  =  J  ->  if ( k  =  j ,  .1.  ,  .0.  )  =  if (
k  =  J ,  .1.  ,  .0.  ) )
1211mpteq2dv 4490 . . . 4  |-  ( j  =  J  ->  (
k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) )  =  ( k  e.  I  |->  if ( k  =  J ,  .1.  ,  .0.  ) ) )
13 eqid 2454 . . . 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 5884 . . 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 661 . 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 2495 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3078   ifcif 3902    |-> cmpt 4461   ` cfv 5529  (class class class)co 6203   0gc0g 14501   1rcur 16735   unitVec cuvc 18342
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-uvc 18343
This theorem is referenced by:  uvcvval  18346
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