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Theorem ipffval 18445
Description: The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.)
Hypotheses
Ref Expression
ipffval.1  |-  V  =  ( Base `  W
)
ipffval.2  |-  .,  =  ( .i `  W )
ipffval.3  |-  .x.  =  ( .if `  W
)
Assertion
Ref Expression
ipffval  |-  .x.  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) )
Distinct variable groups:    x, y,  .,    x, V, y    x, W, y
Allowed substitution hints:    .x. ( x, y)

Proof of Theorem ipffval
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 ipffval.3 . 2  |-  .x.  =  ( .if `  W
)
2 fveq2 5859 . . . . . 6  |-  ( g  =  W  ->  ( Base `  g )  =  ( Base `  W
) )
3 ipffval.1 . . . . . 6  |-  V  =  ( Base `  W
)
42, 3syl6eqr 2521 . . . . 5  |-  ( g  =  W  ->  ( Base `  g )  =  V )
5 fveq2 5859 . . . . . . 7  |-  ( g  =  W  ->  ( .i `  g )  =  ( .i `  W
) )
6 ipffval.2 . . . . . . 7  |-  .,  =  ( .i `  W )
75, 6syl6eqr 2521 . . . . . 6  |-  ( g  =  W  ->  ( .i `  g )  = 
.,  )
87oveqd 6294 . . . . 5  |-  ( g  =  W  ->  (
x ( .i `  g ) y )  =  ( x  .,  y ) )
94, 4, 8mpt2eq123dv 6336 . . . 4  |-  ( g  =  W  ->  (
x  e.  ( Base `  g ) ,  y  e.  ( Base `  g
)  |->  ( x ( .i `  g ) y ) )  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) ) )
10 df-ipf 18424 . . . 4  |-  .if 
=  ( g  e. 
_V  |->  ( x  e.  ( Base `  g
) ,  y  e.  ( Base `  g
)  |->  ( x ( .i `  g ) y ) ) )
11 df-ov 6280 . . . . . . . 8  |-  ( x 
.,  y )  =  (  .,  `  <. x ,  y >. )
12 fvrn0 5881 . . . . . . . 8  |-  (  .,  ` 
<. x ,  y >.
)  e.  ( ran  .,  u.  { (/) } )
1311, 12eqeltri 2546 . . . . . . 7  |-  ( x 
.,  y )  e.  ( ran  .,  u.  {
(/) } )
1413rgen2w 2821 . . . . . 6  |-  A. x  e.  V  A. y  e.  V  ( x  .,  y )  e.  ( ran  .,  u.  { (/) } )
15 eqid 2462 . . . . . . 7  |-  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) )  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) )
1615fmpt2 6843 . . . . . 6  |-  ( A. x  e.  V  A. y  e.  V  (
x  .,  y )  e.  ( ran  .,  u.  {
(/) } )  <->  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) ) : ( V  X.  V
) --> ( ran  .,  u.  {
(/) } ) )
1714, 16mpbi 208 . . . . 5  |-  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) ) : ( V  X.  V ) --> ( ran  .,  u.  { (/) } )
18 fvex 5869 . . . . . . 7  |-  ( Base `  W )  e.  _V
193, 18eqeltri 2546 . . . . . 6  |-  V  e. 
_V
2019, 19xpex 6706 . . . . 5  |-  ( V  X.  V )  e. 
_V
21 fvex 5869 . . . . . . . 8  |-  ( .i
`  W )  e. 
_V
226, 21eqeltri 2546 . . . . . . 7  |-  .,  e.  _V
2322rnex 6710 . . . . . 6  |-  ran  .,  e.  _V
24 p0ex 4629 . . . . . 6  |-  { (/) }  e.  _V
2523, 24unex 6575 . . . . 5  |-  ( ran  .,  u.  { (/) } )  e.  _V
26 fex2 6731 . . . . 5  |-  ( ( ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) ) : ( V  X.  V ) --> ( ran  .,  u.  {
(/) } )  /\  ( V  X.  V )  e. 
_V  /\  ( ran  .,  u.  { (/) } )  e.  _V )  -> 
( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) )  e.  _V )
2717, 20, 25, 26mp3an 1319 . . . 4  |-  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) )  e.  _V
289, 10, 27fvmpt 5943 . . 3  |-  ( W  e.  _V  ->  ( .if `  W )  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) ) )
29 fvprc 5853 . . . . 5  |-  ( -.  W  e.  _V  ->  ( .if `  W
)  =  (/) )
30 mpt20 6344 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) )  =  (/)
3129, 30syl6eqr 2521 . . . 4  |-  ( -.  W  e.  _V  ->  ( .if `  W
)  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) ) )
32 fvprc 5853 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
333, 32syl5eq 2515 . . . . 5  |-  ( -.  W  e.  _V  ->  V  =  (/) )
34 mpt2eq12 6334 . . . . 5  |-  ( ( V  =  (/)  /\  V  =  (/) )  ->  (
x  e.  V , 
y  e.  V  |->  ( x  .,  y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) ) )
3533, 33, 34syl2anc 661 . . . 4  |-  ( -.  W  e.  _V  ->  ( x  e.  V , 
y  e.  V  |->  ( x  .,  y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x  .,  y ) ) )
3631, 35eqtr4d 2506 . . 3  |-  ( -.  W  e.  _V  ->  ( .if `  W
)  =  ( x  e.  V ,  y  e.  V  |->  ( x 
.,  y ) ) )
3728, 36pm2.61i 164 . 2  |-  ( .if `  W )  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) )
381, 37eqtri 2491 1  |-  .x.  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1374    e. wcel 1762   A.wral 2809   _Vcvv 3108    u. cun 3469   (/)c0 3780   {csn 4022   <.cop 4028    X. cxp 4992   ran crn 4995   -->wf 5577   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   Basecbs 14481   .icip 14551   .ifcipf 18422
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-ipf 18424
This theorem is referenced by:  ipfval  18446  ipfeq  18447  ipffn  18448  phlipf  18449
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