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Theorem hfsmval 26330
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 9569 . . 3  |-  CC  e.  _V
2 ax-hilex 25589 . . 3  |-  ~H  e.  _V
31, 2elmap 7444 . 2  |-  ( S  e.  ( CC  ^m  ~H )  <->  S : ~H --> CC )
41, 2elmap 7444 . 2  |-  ( T  e.  ( CC  ^m  ~H )  <->  T : ~H --> CC )
5 fveq1 5863 . . . . 5  |-  ( f  =  S  ->  (
f `  x )  =  ( S `  x ) )
65oveq1d 6297 . . . 4  |-  ( f  =  S  ->  (
( f `  x
)  +  ( g `
 x ) )  =  ( ( S `
 x )  +  ( g `  x
) ) )
76mpteq2dv 4534 . . 3  |-  ( f  =  S  ->  (
x  e.  ~H  |->  ( ( f `  x
)  +  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( ( S `  x )  +  ( g `  x ) ) ) )
8 fveq1 5863 . . . . 5  |-  ( g  =  T  ->  (
g `  x )  =  ( T `  x ) )
98oveq2d 6298 . . . 4  |-  ( g  =  T  ->  (
( S `  x
)  +  ( g `
 x ) )  =  ( ( S `
 x )  +  ( T `  x
) ) )
109mpteq2dv 4534 . . 3  |-  ( g  =  T  ->  (
x  e.  ~H  |->  ( ( S `  x
)  +  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( ( S `  x )  +  ( T `  x ) ) ) )
11 df-hfsum 26325 . . 3  |-  +fn  =  ( f  e.  ( CC  ^m  ~H ) ,  g  e.  ( CC  ^m  ~H )  |->  ( x  e.  ~H  |->  ( ( f `  x
)  +  ( g `
 x ) ) ) )
122mptex 6129 . . 3  |-  ( x  e.  ~H  |->  ( ( S `  x )  +  ( T `  x ) ) )  e.  _V
137, 10, 11, 12ovmpt2 6420 . 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 1379    e. wcel 1767    |-> cmpt 4505   -->wf 5582   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   CCcc 9486    + caddc 9491   ~Hchil 25509    +fn chfs 25531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-hilex 25589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-hfsum 26325
This theorem is referenced by:  hfsval  26335
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