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Theorem uvcfval 19273
Description: Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypotheses
Ref Expression
uvcfval.u  |-  U  =  ( R unitVec  I )
uvcfval.o  |-  .1.  =  ( 1r `  R )
uvcfval.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
uvcfval  |-  ( ( R  e.  V  /\  I  e.  W )  ->  U  =  ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  )
) ) )
Distinct variable groups:    .1. , j,
k    R, j, k    j, I, k    .0. , j, k
Allowed substitution hints:    U( j, k)    V( j, k)    W( j, k)

Proof of Theorem uvcfval
Dummy variables  i 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uvcfval.u . 2  |-  U  =  ( R unitVec  I )
2 elex 3096 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
3 elex 3096 . . 3  |-  ( I  e.  W  ->  I  e.  _V )
4 df-uvc 19272 . . . . 5  |- unitVec  =  ( r  e.  _V , 
i  e.  _V  |->  ( j  e.  i  |->  ( k  e.  i  |->  if ( k  =  j ,  ( 1r `  r ) ,  ( 0g `  r ) ) ) ) )
54a1i 11 . . . 4  |-  ( ( R  e.  _V  /\  I  e.  _V )  -> unitVec 
=  ( r  e. 
_V ,  i  e. 
_V  |->  ( j  e.  i  |->  ( k  e.  i  |->  if ( k  =  j ,  ( 1r `  r ) ,  ( 0g `  r ) ) ) ) ) )
6 simpr 462 . . . . . 6  |-  ( ( r  =  R  /\  i  =  I )  ->  i  =  I )
7 fveq2 5881 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
8 uvcfval.o . . . . . . . . . 10  |-  .1.  =  ( 1r `  R )
97, 8syl6eqr 2488 . . . . . . . . 9  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
10 fveq2 5881 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
11 uvcfval.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  R )
1210, 11syl6eqr 2488 . . . . . . . . 9  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
139, 12ifeq12d 3935 . . . . . . . 8  |-  ( r  =  R  ->  if ( k  =  j ,  ( 1r `  r ) ,  ( 0g `  r ) )  =  if ( k  =  j ,  .1.  ,  .0.  )
)
1413adantr 466 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  I )  ->  if ( k  =  j ,  ( 1r
`  r ) ,  ( 0g `  r
) )  =  if ( k  =  j ,  .1.  ,  .0.  ) )
156, 14mpteq12dv 4504 . . . . . 6  |-  ( ( r  =  R  /\  i  =  I )  ->  ( k  e.  i 
|->  if ( k  =  j ,  ( 1r
`  r ) ,  ( 0g `  r
) ) )  =  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) )
166, 15mpteq12dv 4504 . . . . 5  |-  ( ( r  =  R  /\  i  =  I )  ->  ( j  e.  i 
|->  ( k  e.  i 
|->  if ( k  =  j ,  ( 1r
`  r ) ,  ( 0g `  r
) ) ) )  =  ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) ) )
1716adantl 467 . . . 4  |-  ( ( ( R  e.  _V  /\  I  e.  _V )  /\  ( r  =  R  /\  i  =  I ) )  ->  (
j  e.  i  |->  ( k  e.  i  |->  if ( k  =  j ,  ( 1r `  r ) ,  ( 0g `  r ) ) ) )  =  ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) ) )
18 simpl 458 . . . 4  |-  ( ( R  e.  _V  /\  I  e.  _V )  ->  R  e.  _V )
19 simpr 462 . . . 4  |-  ( ( R  e.  _V  /\  I  e.  _V )  ->  I  e.  _V )
20 mptexg 6150 . . . . 5  |-  ( I  e.  _V  ->  (
j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) )  e. 
_V )
2120adantl 467 . . . 4  |-  ( ( R  e.  _V  /\  I  e.  _V )  ->  ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) )  e. 
_V )
225, 17, 18, 19, 21ovmpt2d 6438 . . 3  |-  ( ( R  e.  _V  /\  I  e.  _V )  ->  ( R unitVec  I )  =  ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) ) )
232, 3, 22syl2an 479 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( R unitVec  I )  =  ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) ) )
241, 23syl5eq 2482 1  |-  ( ( R  e.  V  /\  I  e.  W )  ->  U  =  ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087   ifcif 3915    |-> cmpt 4484   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   0gc0g 15297   1rcur 17670   unitVec cuvc 19271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-uvc 19272
This theorem is referenced by:  uvcval  19274  uvcff  19280
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