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Theorem ipfval 17978
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.  =  ( .if `  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 6099 . 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.  =  ( .if `  W
)
52, 3, 4ipffval 17977 . 2  |-  .x.  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) )
6 ovex 6115 . 2  |-  ( X 
.,  Y )  e. 
_V
71, 5, 6ovmpt2a 6220 1  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  ( X  .x.  Y
)  =  ( X 
.,  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   ` cfv 5415  (class class class)co 6090   Basecbs 14170   .icip 14239   .ifcipf 17954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-ipf 17956
This theorem is referenced by:  ipcn  20658  cnmpt1ip  20659  cnmpt2ip  20660
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