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Theorem ipfval 16553
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 5867 . 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 16552 . 2  |-  .x.  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) )
6 ovex 5883 . 2  |-  ( X 
.,  Y )  e. 
_V
71, 5, 6ovmpt2a 5978 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .icip 13213   .i fcipf 16529
This theorem is referenced by:  ipcn  18673  cnmpt1ip  18674  cnmpt2ip  18675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-ipf 16531
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