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Theorem lnfnaddi 27255
Description: Additive property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
lnfnl.1  |-  T  e. 
LinFn
Assertion
Ref Expression
lnfnaddi  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  B ) )  =  ( ( T `
 A )  +  ( T `  B
) ) )

Proof of Theorem lnfnaddi
StepHypRef Expression
1 ax-1cn 9500 . . 3  |-  1  e.  CC
2 lnfnl.1 . . . 4  |-  T  e. 
LinFn
32lnfnli 27252 . . 3  |-  ( ( 1  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( (
1  .h  A )  +h  B ) )  =  ( ( 1  x.  ( T `  A ) )  +  ( T `  B
) ) )
41, 3mp3an1 1313 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  (
( 1  .h  A
)  +h  B ) )  =  ( ( 1  x.  ( T `
 A ) )  +  ( T `  B ) ) )
5 ax-hvmulid 26217 . . . . 5  |-  ( A  e.  ~H  ->  (
1  .h  A )  =  A )
65oveq1d 6249 . . . 4  |-  ( A  e.  ~H  ->  (
( 1  .h  A
)  +h  B )  =  ( A  +h  B ) )
76fveq2d 5809 . . 3  |-  ( A  e.  ~H  ->  ( T `  ( (
1  .h  A )  +h  B ) )  =  ( T `  ( A  +h  B
) ) )
87adantr 463 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  (
( 1  .h  A
)  +h  B ) )  =  ( T `
 ( A  +h  B ) ) )
92lnfnfi 27253 . . . . . 6  |-  T : ~H
--> CC
109ffvelrni 5964 . . . . 5  |-  ( A  e.  ~H  ->  ( T `  A )  e.  CC )
1110mulid2d 9564 . . . 4  |-  ( A  e.  ~H  ->  (
1  x.  ( T `
 A ) )  =  ( T `  A ) )
1211adantr 463 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( 1  x.  ( T `  A )
)  =  ( T `
 A ) )
1312oveq1d 6249 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( 1  x.  ( T `  A
) )  +  ( T `  B ) )  =  ( ( T `  A )  +  ( T `  B ) ) )
144, 8, 133eqtr3d 2451 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  B ) )  =  ( ( T `
 A )  +  ( T `  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   ` cfv 5525  (class class class)co 6234   CCcc 9440   1c1 9443    + caddc 9445    x. cmul 9447   ~Hchil 26130    +h cva 26131    .h csm 26132   LinFnclf 26165
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  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-mulcl 9504  ax-mulcom 9506  ax-mulass 9508  ax-distr 9509  ax-1rid 9512  ax-cnre 9515  ax-hilex 26210  ax-hvmulid 26217
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-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-br 4395  df-opab 4453  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  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-map 7379  df-lnfn 27060
This theorem is referenced by:  lnfnaddmuli  27257  nlelshi  27272
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