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Theorem ipffval 16884
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 5731 . . . . . 6  |-  ( g  =  W  ->  ( Base `  g )  =  ( Base `  W
) )
3 ipffval.1 . . . . . 6  |-  V  =  ( Base `  W
)
42, 3syl6eqr 2488 . . . . 5  |-  ( g  =  W  ->  ( Base `  g )  =  V )
5 fveq2 5731 . . . . . . 7  |-  ( g  =  W  ->  ( .i `  g )  =  ( .i `  W
) )
6 ipffval.2 . . . . . . 7  |-  .,  =  ( .i `  W )
75, 6syl6eqr 2488 . . . . . 6  |-  ( g  =  W  ->  ( .i `  g )  = 
.,  )
87oveqd 6101 . . . . 5  |-  ( g  =  W  ->  (
x ( .i `  g ) y )  =  ( x  .,  y ) )
94, 4, 8mpt2eq123dv 6139 . . . 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 16863 . . . 4  |-  .i f  =  ( g  e. 
_V  |->  ( x  e.  ( Base `  g
) ,  y  e.  ( Base `  g
)  |->  ( x ( .i `  g ) y ) ) )
11 df-ov 6087 . . . . . . . 8  |-  ( x 
.,  y )  =  (  .,  `  <. x ,  y >. )
12 fvrn0 5756 . . . . . . . 8  |-  (  .,  ` 
<. x ,  y >.
)  e.  ( ran  .,  u.  { (/) } )
1311, 12eqeltri 2508 . . . . . . 7  |-  ( x 
.,  y )  e.  ( ran  .,  u.  {
(/) } )
1413rgen2w 2776 . . . . . 6  |-  A. x  e.  V  A. y  e.  V  ( x  .,  y )  e.  ( ran  .,  u.  { (/) } )
15 eqid 2438 . . . . . . 7  |-  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) )  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) )
1615fmpt2 6421 . . . . . 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 201 . . . . 5  |-  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) ) : ( V  X.  V ) --> ( ran  .,  u.  { (/) } )
18 fvex 5745 . . . . . . 7  |-  ( Base `  W )  e.  _V
193, 18eqeltri 2508 . . . . . 6  |-  V  e. 
_V
2019, 19xpex 4993 . . . . 5  |-  ( V  X.  V )  e. 
_V
21 fvex 5745 . . . . . . . 8  |-  ( .i
`  W )  e. 
_V
226, 21eqeltri 2508 . . . . . . 7  |-  .,  e.  _V
2322rnex 5136 . . . . . 6  |-  ran  .,  e.  _V
24 p0ex 4389 . . . . . 6  |-  { (/) }  e.  _V
2523, 24unex 4710 . . . . 5  |-  ( ran  .,  u.  { (/) } )  e.  _V
26 fex2 5606 . . . . 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 1280 . . . 4  |-  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) )  e.  _V
289, 10, 27fvmpt 5809 . . 3  |-  ( W  e.  _V  ->  ( .i f `  W )  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) ) )
29 fvprc 5725 . . . . 5  |-  ( -.  W  e.  _V  ->  ( .i f `  W
)  =  (/) )
30 mpt20 6430 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) )  =  (/)
3129, 30syl6eqr 2488 . . . 4  |-  ( -.  W  e.  _V  ->  ( .i f `  W
)  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) ) )
32 fvprc 5725 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
333, 32syl5eq 2482 . . . . 5  |-  ( -.  W  e.  _V  ->  V  =  (/) )
34 mpt2eq12 6137 . . . . 5  |-  ( ( V  =  (/)  /\  V  =  (/) )  ->  (
x  e.  V , 
y  e.  V  |->  ( x  .,  y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) ) )
3533, 33, 34syl2anc 644 . . . 4  |-  ( -.  W  e.  _V  ->  ( x  e.  V , 
y  e.  V  |->  ( x  .,  y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) ) )
3631, 35eqtr4d 2473 . . 3  |-  ( -.  W  e.  _V  ->  ( .i f `  W
)  =  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) ) )
3728, 36pm2.61i 159 . 2  |-  ( .i f `  W )  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) )
381, 37eqtri 2458 1  |-  .x.  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    u. cun 3320   (/)c0 3630   {csn 3816   <.cop 3819    X. cxp 4879   ran crn 4882   -->wf 5453   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   Basecbs 13474   .icip 13539   .i fcipf 16861
This theorem is referenced by:  ipfval  16885  ipfeq  16886  ipffn  16887  phlipf  16888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-ipf 16863
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