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Theorem ipffval 16834
Description: The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.)
Hypotheses
Ref Expression
ipffval.1  |-  V  =  ( Base `  W
)
ipffval.2  |-  .,  =  ( .i `  W )
ipffval.3  |-  .x.  =  ( .i f `  W
)
Assertion
Ref Expression
ipffval  |-  .x.  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) )
Distinct variable groups:    x, y,  .,    x, V, y    x, W, y
Allowed substitution hints:    .x. ( x, y)

Proof of Theorem ipffval
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 ipffval.3 . 2  |-  .x.  =  ( .i f `  W
)
2 fveq2 5687 . . . . . 6  |-  ( g  =  W  ->  ( Base `  g )  =  ( Base `  W
) )
3 ipffval.1 . . . . . 6  |-  V  =  ( Base `  W
)
42, 3syl6eqr 2454 . . . . 5  |-  ( g  =  W  ->  ( Base `  g )  =  V )
5 fveq2 5687 . . . . . . 7  |-  ( g  =  W  ->  ( .i `  g )  =  ( .i `  W
) )
6 ipffval.2 . . . . . . 7  |-  .,  =  ( .i `  W )
75, 6syl6eqr 2454 . . . . . 6  |-  ( g  =  W  ->  ( .i `  g )  = 
.,  )
87oveqd 6057 . . . . 5  |-  ( g  =  W  ->  (
x ( .i `  g ) y )  =  ( x  .,  y ) )
94, 4, 8mpt2eq123dv 6095 . . . 4  |-  ( g  =  W  ->  (
x  e.  ( Base `  g ) ,  y  e.  ( Base `  g
)  |->  ( x ( .i `  g ) y ) )  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) ) )
10 df-ipf 16813 . . . 4  |-  .i f  =  ( g  e. 
_V  |->  ( x  e.  ( Base `  g
) ,  y  e.  ( Base `  g
)  |->  ( x ( .i `  g ) y ) ) )
11 df-ov 6043 . . . . . . . 8  |-  ( x 
.,  y )  =  (  .,  `  <. x ,  y >. )
12 fvrn0 5712 . . . . . . . 8  |-  (  .,  ` 
<. x ,  y >.
)  e.  ( ran  .,  u.  { (/) } )
1311, 12eqeltri 2474 . . . . . . 7  |-  ( x 
.,  y )  e.  ( ran  .,  u.  {
(/) } )
1413rgen2w 2734 . . . . . 6  |-  A. x  e.  V  A. y  e.  V  ( x  .,  y )  e.  ( ran  .,  u.  { (/) } )
15 eqid 2404 . . . . . . 7  |-  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) )  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) )
1615fmpt2 6377 . . . . . 6  |-  ( A. x  e.  V  A. y  e.  V  (
x  .,  y )  e.  ( ran  .,  u.  {
(/) } )  <->  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) ) : ( V  X.  V
) --> ( ran  .,  u.  {
(/) } ) )
1714, 16mpbi 200 . . . . 5  |-  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) ) : ( V  X.  V ) --> ( ran  .,  u.  { (/) } )
18 fvex 5701 . . . . . . 7  |-  ( Base `  W )  e.  _V
193, 18eqeltri 2474 . . . . . 6  |-  V  e. 
_V
2019, 19xpex 4949 . . . . 5  |-  ( V  X.  V )  e. 
_V
21 fvex 5701 . . . . . . . 8  |-  ( .i
`  W )  e. 
_V
226, 21eqeltri 2474 . . . . . . 7  |-  .,  e.  _V
2322rnex 5092 . . . . . 6  |-  ran  .,  e.  _V
24 p0ex 4346 . . . . . 6  |-  { (/) }  e.  _V
2523, 24unex 4666 . . . . 5  |-  ( ran  .,  u.  { (/) } )  e.  _V
26 fex2 5562 . . . . 5  |-  ( ( ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) ) : ( V  X.  V ) --> ( ran  .,  u.  {
(/) } )  /\  ( V  X.  V )  e. 
_V  /\  ( ran  .,  u.  { (/) } )  e.  _V )  -> 
( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) )  e.  _V )
2717, 20, 25, 26mp3an 1279 . . . 4  |-  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) )  e.  _V
289, 10, 27fvmpt 5765 . . 3  |-  ( W  e.  _V  ->  ( .i f `  W )  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) ) )
29 fvprc 5681 . . . . 5  |-  ( -.  W  e.  _V  ->  ( .i f `  W
)  =  (/) )
30 mpt20 6386 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) )  =  (/)
3129, 30syl6eqr 2454 . . . 4  |-  ( -.  W  e.  _V  ->  ( .i f `  W
)  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) ) )
32 fvprc 5681 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
333, 32syl5eq 2448 . . . . 5  |-  ( -.  W  e.  _V  ->  V  =  (/) )
34 mpt2eq12 6093 . . . . 5  |-  ( ( V  =  (/)  /\  V  =  (/) )  ->  (
x  e.  V , 
y  e.  V  |->  ( x  .,  y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) ) )
3533, 33, 34syl2anc 643 . . . 4  |-  ( -.  W  e.  _V  ->  ( x  e.  V , 
y  e.  V  |->  ( x  .,  y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) ) )
3631, 35eqtr4d 2439 . . 3  |-  ( -.  W  e.  _V  ->  ( .i f `  W
)  =  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) ) )
3728, 36pm2.61i 158 . 2  |-  ( .i f `  W )  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) )
381, 37eqtri 2424 1  |-  .x.  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    u. cun 3278   (/)c0 3588   {csn 3774   <.cop 3777    X. cxp 4835   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   Basecbs 13424   .icip 13489   .i fcipf 16811
This theorem is referenced by:  ipfval  16835  ipfeq  16836  ipffn  16837  phlipf  16838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-ipf 16813
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