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Theorem ipffval 16658
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 5608 . . . . . 6  |-  ( g  =  W  ->  ( Base `  g )  =  ( Base `  W
) )
3 ipffval.1 . . . . . 6  |-  V  =  ( Base `  W
)
42, 3syl6eqr 2408 . . . . 5  |-  ( g  =  W  ->  ( Base `  g )  =  V )
5 fveq2 5608 . . . . . . 7  |-  ( g  =  W  ->  ( .i `  g )  =  ( .i `  W
) )
6 ipffval.2 . . . . . . 7  |-  .,  =  ( .i `  W )
75, 6syl6eqr 2408 . . . . . 6  |-  ( g  =  W  ->  ( .i `  g )  = 
.,  )
87oveqd 5962 . . . . 5  |-  ( g  =  W  ->  (
x ( .i `  g ) y )  =  ( x  .,  y ) )
94, 4, 8mpt2eq123dv 5997 . . . 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 16637 . . . 4  |-  .i f  =  ( g  e. 
_V  |->  ( x  e.  ( Base `  g
) ,  y  e.  ( Base `  g
)  |->  ( x ( .i `  g ) y ) ) )
11 df-ov 5948 . . . . . . . 8  |-  ( x 
.,  y )  =  (  .,  `  <. x ,  y >. )
12 fvrn0 5633 . . . . . . . 8  |-  (  .,  ` 
<. x ,  y >.
)  e.  ( ran  .,  u.  { (/) } )
1311, 12eqeltri 2428 . . . . . . 7  |-  ( x 
.,  y )  e.  ( ran  .,  u.  {
(/) } )
1413rgen2w 2687 . . . . . 6  |-  A. x  e.  V  A. y  e.  V  ( x  .,  y )  e.  ( ran  .,  u.  { (/) } )
15 eqid 2358 . . . . . . 7  |-  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) )  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) )
1615fmpt2 6278 . . . . . 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 199 . . . . 5  |-  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) ) : ( V  X.  V ) --> ( ran  .,  u.  { (/) } )
18 fvex 5622 . . . . . . 7  |-  ( Base `  W )  e.  _V
193, 18eqeltri 2428 . . . . . 6  |-  V  e. 
_V
2019, 19xpex 4883 . . . . 5  |-  ( V  X.  V )  e. 
_V
21 fvex 5622 . . . . . . . 8  |-  ( .i
`  W )  e. 
_V
226, 21eqeltri 2428 . . . . . . 7  |-  .,  e.  _V
2322rnex 5024 . . . . . 6  |-  ran  .,  e.  _V
24 p0ex 4278 . . . . . 6  |-  { (/) }  e.  _V
2523, 24unex 4600 . . . . 5  |-  ( ran  .,  u.  { (/) } )  e.  _V
26 fex2 5484 . . . . 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 1277 . . . 4  |-  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) )  e.  _V
289, 10, 27fvmpt 5685 . . 3  |-  ( W  e.  _V  ->  ( .i f `  W )  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) ) )
29 fvprc 5602 . . . . 5  |-  ( -.  W  e.  _V  ->  ( .i f `  W
)  =  (/) )
30 mpt20 6286 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) )  =  (/)
3129, 30syl6eqr 2408 . . . 4  |-  ( -.  W  e.  _V  ->  ( .i f `  W
)  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) ) )
32 fvprc 5602 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
333, 32syl5eq 2402 . . . . 5  |-  ( -.  W  e.  _V  ->  V  =  (/) )
34 mpt2eq12 5995 . . . . 5  |-  ( ( V  =  (/)  /\  V  =  (/) )  ->  (
x  e.  V , 
y  e.  V  |->  ( x  .,  y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) ) )
3533, 33, 34syl2anc 642 . . . 4  |-  ( -.  W  e.  _V  ->  ( x  e.  V , 
y  e.  V  |->  ( x  .,  y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) ) )
3631, 35eqtr4d 2393 . . 3  |-  ( -.  W  e.  _V  ->  ( .i f `  W
)  =  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) ) )
3728, 36pm2.61i 156 . 2  |-  ( .i f `  W )  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) )
381, 37eqtri 2378 1  |-  .x.  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864    u. cun 3226   (/)c0 3531   {csn 3716   <.cop 3719    X. cxp 4769   ran crn 4772   -->wf 5333   ` cfv 5337  (class class class)co 5945    e. cmpt2 5947   Basecbs 13245   .icip 13310   .i fcipf 16635
This theorem is referenced by:  ipfval  16659  ipfeq  16660  ipffn  16661  phlipf  16662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-ipf 16637
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