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Theorem ipfval 16659
Description: The inner product operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
ipffval.1  |-  V  =  ( Base `  W
)
ipffval.2  |-  .,  =  ( .i `  W )
ipffval.3  |-  .x.  =  ( .i f `  W
)
Assertion
Ref Expression
ipfval  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  ( X  .x.  Y
)  =  ( X 
.,  Y ) )

Proof of Theorem ipfval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 5954 . 2  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x  .,  y
)  =  ( X 
.,  Y ) )
2 ipffval.1 . . 3  |-  V  =  ( Base `  W
)
3 ipffval.2 . . 3  |-  .,  =  ( .i `  W )
4 ipffval.3 . . 3  |-  .x.  =  ( .i f `  W
)
52, 3, 4ipffval 16658 . 2  |-  .x.  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) )
6 ovex 5970 . 2  |-  ( X 
.,  Y )  e. 
_V
71, 5, 6ovmpt2a 6065 1  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  ( X  .x.  Y
)  =  ( X 
.,  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   ` cfv 5337  (class class class)co 5945   Basecbs 13245   .icip 13310   .i fcipf 16635
This theorem is referenced by:  ipcn  18777  cnmpt1ip  18778  cnmpt2ip  18779
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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