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Theorem uvcfval 18622
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 3122 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
3 elex 3122 . . 3  |-  ( I  e.  W  ->  I  e.  _V )
4 df-uvc 18621 . . . . 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 461 . . . . . 6  |-  ( ( r  =  R  /\  i  =  I )  ->  i  =  I )
7 fveq2 5866 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
8 uvcfval.o . . . . . . . . . 10  |-  .1.  =  ( 1r `  R )
97, 8syl6eqr 2526 . . . . . . . . 9  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
10 fveq2 5866 . . . . . . . . . 10  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
11 uvcfval.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  R )
1210, 11syl6eqr 2526 . . . . . . . . 9  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
139, 12ifeq12d 3959 . . . . . . . 8  |-  ( r  =  R  ->  if ( k  =  j ,  ( 1r `  r ) ,  ( 0g `  r ) )  =  if ( k  =  j ,  .1.  ,  .0.  )
)
1413adantr 465 . . . . . . 7  |-  ( ( r  =  R  /\  i  =  I )  ->  if ( k  =  j ,  ( 1r
`  r ) ,  ( 0g `  r
) )  =  if ( k  =  j ,  .1.  ,  .0.  ) )
156, 14mpteq12dv 4525 . . . . . 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 4525 . . . . 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 466 . . . 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 457 . . . 4  |-  ( ( R  e.  _V  /\  I  e.  _V )  ->  R  e.  _V )
19 simpr 461 . . . 4  |-  ( ( R  e.  _V  /\  I  e.  _V )  ->  I  e.  _V )
20 mptexg 6131 . . . . 5  |-  ( I  e.  _V  ->  (
j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) )  e. 
_V )
2120adantl 466 . . . 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 6415 . . 3  |-  ( ( R  e.  _V  /\  I  e.  _V )  ->  ( R unitVec  I )  =  ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) ) )
232, 3, 22syl2an 477 . 2  |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( R unitVec  I )  =  ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) ) )
241, 23syl5eq 2520 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 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   ifcif 3939    |-> cmpt 4505   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   0gc0g 14698   1rcur 16967   unitVec cuvc 18620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-uvc 18621
This theorem is referenced by:  uvcval  18623  uvcff  18629
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