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Theorem ipfeq 16873
Description: If the inner product operation is already a function, the functionalization of it is equal to the original operation. (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
ipfeq  |-  (  .,  Fn  ( V  X.  V
)  ->  .x.  =  .,  )

Proof of Theorem ipfeq
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnov 6170 . . 3  |-  (  .,  Fn  ( V  X.  V
)  <->  .,  =  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) ) )
21biimpi 187 . 2  |-  (  .,  Fn  ( V  X.  V
)  ->  .,  =  ( x  e.  V , 
y  e.  V  |->  ( x  .,  y ) ) )
3 ipffval.1 . . 3  |-  V  =  ( Base `  W
)
4 ipffval.2 . . 3  |-  .,  =  ( .i `  W )
5 ipffval.3 . . 3  |-  .x.  =  ( .i f `  W
)
63, 4, 5ipffval 16871 . 2  |-  .x.  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) )
72, 6syl6reqr 2486 1  |-  (  .,  Fn  ( V  X.  V
)  ->  .x.  =  .,  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    X. cxp 4868    Fn wfn 5441   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   Basecbs 13461   .icip 13526   .i fcipf 16848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-ipf 16850
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