MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ipfeq Structured version   Unicode version

Theorem ipfeq 18875
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.  =  ( .if `  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 6347 . . 3  |-  (  .,  Fn  ( V  X.  V
)  <->  .,  =  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) ) )
21biimpi 194 . 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.  =  ( .if `  W
)
63, 4, 5ipffval 18873 . 2  |-  .x.  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) )
72, 6syl6reqr 2462 1  |-  (  .,  Fn  ( V  X.  V
)  ->  .x.  =  .,  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    X. cxp 4940    Fn wfn 5520   ` cfv 5525  (class class class)co 6234    |-> cmpt2 6236   Basecbs 14733   .icip 14806   .ifcipf 18850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739  df-ipf 18852
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator