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Theorem ipffval 18981
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.  =  ( .if `  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.  =  ( .if `  W
)
2 fveq2 5849 . . . . . 6  |-  ( g  =  W  ->  ( Base `  g )  =  ( Base `  W
) )
3 ipffval.1 . . . . . 6  |-  V  =  ( Base `  W
)
42, 3syl6eqr 2461 . . . . 5  |-  ( g  =  W  ->  ( Base `  g )  =  V )
5 fveq2 5849 . . . . . . 7  |-  ( g  =  W  ->  ( .i `  g )  =  ( .i `  W
) )
6 ipffval.2 . . . . . . 7  |-  .,  =  ( .i `  W )
75, 6syl6eqr 2461 . . . . . 6  |-  ( g  =  W  ->  ( .i `  g )  = 
.,  )
87oveqd 6295 . . . . 5  |-  ( g  =  W  ->  (
x ( .i `  g ) y )  =  ( x  .,  y ) )
94, 4, 8mpt2eq123dv 6340 . . . 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 18960 . . . 4  |-  .if 
=  ( g  e. 
_V  |->  ( x  e.  ( Base `  g
) ,  y  e.  ( Base `  g
)  |->  ( x ( .i `  g ) y ) ) )
11 df-ov 6281 . . . . . . . 8  |-  ( x 
.,  y )  =  (  .,  `  <. x ,  y >. )
12 fvrn0 5871 . . . . . . . 8  |-  (  .,  ` 
<. x ,  y >.
)  e.  ( ran  .,  u.  { (/) } )
1311, 12eqeltri 2486 . . . . . . 7  |-  ( x 
.,  y )  e.  ( ran  .,  u.  {
(/) } )
1413rgen2w 2766 . . . . . 6  |-  A. x  e.  V  A. y  e.  V  ( x  .,  y )  e.  ( ran  .,  u.  { (/) } )
15 eqid 2402 . . . . . . 7  |-  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) )  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) )
1615fmpt2 6851 . . . . . 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 208 . . . . 5  |-  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) ) : ( V  X.  V ) --> ( ran  .,  u.  { (/) } )
18 fvex 5859 . . . . . . 7  |-  ( Base `  W )  e.  _V
193, 18eqeltri 2486 . . . . . 6  |-  V  e. 
_V
2019, 19xpex 6586 . . . . 5  |-  ( V  X.  V )  e. 
_V
21 fvex 5859 . . . . . . . 8  |-  ( .i
`  W )  e. 
_V
226, 21eqeltri 2486 . . . . . . 7  |-  .,  e.  _V
2322rnex 6718 . . . . . 6  |-  ran  .,  e.  _V
24 p0ex 4581 . . . . . 6  |-  { (/) }  e.  _V
2523, 24unex 6580 . . . . 5  |-  ( ran  .,  u.  { (/) } )  e.  _V
26 fex2 6739 . . . . 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 1326 . . . 4  |-  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) )  e.  _V
289, 10, 27fvmpt 5932 . . 3  |-  ( W  e.  _V  ->  ( .if `  W )  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) ) )
29 fvprc 5843 . . . . 5  |-  ( -.  W  e.  _V  ->  ( .if `  W
)  =  (/) )
30 mpt20 6348 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) )  =  (/)
3129, 30syl6eqr 2461 . . . 4  |-  ( -.  W  e.  _V  ->  ( .if `  W
)  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) ) )
32 fvprc 5843 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
333, 32syl5eq 2455 . . . . 5  |-  ( -.  W  e.  _V  ->  V  =  (/) )
34 mpt2eq12 6338 . . . . 5  |-  ( ( V  =  (/)  /\  V  =  (/) )  ->  (
x  e.  V , 
y  e.  V  |->  ( x  .,  y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) ) )
3533, 33, 34syl2anc 659 . . . 4  |-  ( -.  W  e.  _V  ->  ( x  e.  V , 
y  e.  V  |->  ( x  .,  y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) ) )
3631, 35eqtr4d 2446 . . 3  |-  ( -.  W  e.  _V  ->  ( .if `  W
)  =  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) ) )
3728, 36pm2.61i 164 . 2  |-  ( .if `  W )  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) )
381, 37eqtri 2431 1  |-  .x.  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1405    e. wcel 1842   A.wral 2754   _Vcvv 3059    u. cun 3412   (/)c0 3738   {csn 3972   <.cop 3978    X. cxp 4821   ran crn 4824   -->wf 5565   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   Basecbs 14841   .icip 14914   .ifcipf 18958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-ipf 18960
This theorem is referenced by:  ipfval  18982  ipfeq  18983  ipffn  18984  phlipf  18985
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