HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  lnfnaddi Unicode version

Theorem lnfnaddi 22623
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 8795 . . 3  |-  1  e.  CC
2 lnfnl.1 . . . 4  |-  T  e. 
LinFn
32lnfnli 22620 . . 3  |-  ( ( 1  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( (
1  .h  A )  +h  B ) )  =  ( ( 1  x.  ( T `  A ) )  +  ( T `  B
) ) )
41, 3mp3an1 1264 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  (
( 1  .h  A
)  +h  B ) )  =  ( ( 1  x.  ( T `
 A ) )  +  ( T `  B ) ) )
5 ax-hvmulid 21586 . . . . 5  |-  ( A  e.  ~H  ->  (
1  .h  A )  =  A )
65oveq1d 5873 . . . 4  |-  ( A  e.  ~H  ->  (
( 1  .h  A
)  +h  B )  =  ( A  +h  B ) )
76fveq2d 5529 . . 3  |-  ( A  e.  ~H  ->  ( T `  ( (
1  .h  A )  +h  B ) )  =  ( T `  ( A  +h  B
) ) )
87adantr 451 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  (
( 1  .h  A
)  +h  B ) )  =  ( T `
 ( A  +h  B ) ) )
92lnfnfi 22621 . . . . . 6  |-  T : ~H
--> CC
109ffvelrni 5664 . . . . 5  |-  ( A  e.  ~H  ->  ( T `  A )  e.  CC )
1110mulid2d 8853 . . . 4  |-  ( A  e.  ~H  ->  (
1  x.  ( T `
 A ) )  =  ( T `  A ) )
1211adantr 451 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( 1  x.  ( T `  A )
)  =  ( T `
 A ) )
1312oveq1d 5873 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( 1  x.  ( T `  A
) )  +  ( T `  B ) )  =  ( ( T `  A )  +  ( T `  B ) ) )
144, 8, 133eqtr3d 2323 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  B ) )  =  ( ( T `
 A )  +  ( T `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    x. cmul 8742   ~Hchil 21499    +h cva 21500    .h csm 21501   LinFnclf 21534
This theorem is referenced by:  lnfnaddmuli  22625  nlelshi  22640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-mulcl 8799  ax-mulcom 8801  ax-mulass 8803  ax-distr 8804  ax-1rid 8807  ax-cnre 8810  ax-hilex 21579  ax-hvmulid 21586
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-lnfn 22428
  Copyright terms: Public domain W3C validator