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Theorem lnfnaddi 27696
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 9597 . . 3  |-  1  e.  CC
2 lnfnl.1 . . . 4  |-  T  e. 
LinFn
32lnfnli 27693 . . 3  |-  ( ( 1  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( (
1  .h  A )  +h  B ) )  =  ( ( 1  x.  ( T `  A ) )  +  ( T `  B
) ) )
41, 3mp3an1 1351 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  (
( 1  .h  A
)  +h  B ) )  =  ( ( 1  x.  ( T `
 A ) )  +  ( T `  B ) ) )
5 ax-hvmulid 26659 . . . . 5  |-  ( A  e.  ~H  ->  (
1  .h  A )  =  A )
65oveq1d 6305 . . . 4  |-  ( A  e.  ~H  ->  (
( 1  .h  A
)  +h  B )  =  ( A  +h  B ) )
76fveq2d 5869 . . 3  |-  ( A  e.  ~H  ->  ( T `  ( (
1  .h  A )  +h  B ) )  =  ( T `  ( A  +h  B
) ) )
87adantr 467 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  (
( 1  .h  A
)  +h  B ) )  =  ( T `
 ( A  +h  B ) ) )
92lnfnfi 27694 . . . . . 6  |-  T : ~H
--> CC
109ffvelrni 6021 . . . . 5  |-  ( A  e.  ~H  ->  ( T `  A )  e.  CC )
1110mulid2d 9661 . . . 4  |-  ( A  e.  ~H  ->  (
1  x.  ( T `
 A ) )  =  ( T `  A ) )
1211adantr 467 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( 1  x.  ( T `  A )
)  =  ( T `
 A ) )
1312oveq1d 6305 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( 1  x.  ( T `  A
) )  +  ( T `  B ) )  =  ( ( T `  A )  +  ( T `  B ) ) )
144, 8, 133eqtr3d 2493 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 371    = wceq 1444    e. wcel 1887   ` cfv 5582  (class class class)co 6290   CCcc 9537   1c1 9540    + caddc 9542    x. cmul 9544   ~Hchil 26572    +h cva 26573    .h csm 26574   LinFnclf 26607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-mulcl 9601  ax-mulcom 9603  ax-mulass 9605  ax-distr 9606  ax-1rid 9609  ax-cnre 9612  ax-hilex 26652  ax-hvmulid 26659
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-map 7474  df-lnfn 27501
This theorem is referenced by:  lnfnaddmuli  27698  nlelshi  27713
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