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Theorem hfsmval 25147
Description: Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
Assertion
Ref Expression
hfsmval  |-  ( ( S : ~H --> CC  /\  T : ~H --> CC )  ->  ( S  +fn  T )  =  ( x  e.  ~H  |->  ( ( S `  x )  +  ( T `  x ) ) ) )
Distinct variable groups:    x, S    x, T

Proof of Theorem hfsmval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnex 9368 . . 3  |-  CC  e.  _V
2 ax-hilex 24406 . . 3  |-  ~H  e.  _V
31, 2elmap 7246 . 2  |-  ( S  e.  ( CC  ^m  ~H )  <->  S : ~H --> CC )
41, 2elmap 7246 . 2  |-  ( T  e.  ( CC  ^m  ~H )  <->  T : ~H --> CC )
5 fveq1 5695 . . . . 5  |-  ( f  =  S  ->  (
f `  x )  =  ( S `  x ) )
65oveq1d 6111 . . . 4  |-  ( f  =  S  ->  (
( f `  x
)  +  ( g `
 x ) )  =  ( ( S `
 x )  +  ( g `  x
) ) )
76mpteq2dv 4384 . . 3  |-  ( f  =  S  ->  (
x  e.  ~H  |->  ( ( f `  x
)  +  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( ( S `  x )  +  ( g `  x ) ) ) )
8 fveq1 5695 . . . . 5  |-  ( g  =  T  ->  (
g `  x )  =  ( T `  x ) )
98oveq2d 6112 . . . 4  |-  ( g  =  T  ->  (
( S `  x
)  +  ( g `
 x ) )  =  ( ( S `
 x )  +  ( T `  x
) ) )
109mpteq2dv 4384 . . 3  |-  ( g  =  T  ->  (
x  e.  ~H  |->  ( ( S `  x
)  +  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( ( S `  x )  +  ( T `  x ) ) ) )
11 df-hfsum 25142 . . 3  |-  +fn  =  ( f  e.  ( CC  ^m  ~H ) ,  g  e.  ( CC  ^m  ~H )  |->  ( x  e.  ~H  |->  ( ( f `  x
)  +  ( g `
 x ) ) ) )
122mptex 5953 . . 3  |-  ( x  e.  ~H  |->  ( ( S `  x )  +  ( T `  x ) ) )  e.  _V
137, 10, 11, 12ovmpt2 6231 . 2  |-  ( ( S  e.  ( CC 
^m  ~H )  /\  T  e.  ( CC  ^m  ~H ) )  ->  ( S  +fn  T )  =  ( x  e.  ~H  |->  ( ( S `  x )  +  ( T `  x ) ) ) )
143, 4, 13syl2anbr 480 1  |-  ( ( S : ~H --> CC  /\  T : ~H --> CC )  ->  ( S  +fn  T )  =  ( x  e.  ~H  |->  ( ( S `  x )  +  ( T `  x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    e. cmpt 4355   -->wf 5419   ` cfv 5423  (class class class)co 6096    ^m cmap 7219   CCcc 9285    + caddc 9290   ~Hchil 24326    +fn chfs 24348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-hilex 24406
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-map 7221  df-hfsum 25142
This theorem is referenced by:  hfsval  25152
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