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Theorem lnopaddi 23474
Description: Additive property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005.) (New usage is discouraged.)
Hypothesis
Ref Expression
lnopl.1  |-  T  e. 
LinOp
Assertion
Ref Expression
lnopaddi  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  B ) )  =  ( ( T `
 A )  +h  ( T `  B
) ) )

Proof of Theorem lnopaddi
StepHypRef Expression
1 ax-1cn 9048 . . 3  |-  1  e.  CC
2 lnopl.1 . . . 4  |-  T  e. 
LinOp
32lnopli 23471 . . 3  |-  ( ( 1  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( (
1  .h  A )  +h  B ) )  =  ( ( 1  .h  ( T `  A ) )  +h  ( T `  B
) ) )
41, 3mp3an1 1266 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  (
( 1  .h  A
)  +h  B ) )  =  ( ( 1  .h  ( T `
 A ) )  +h  ( T `  B ) ) )
5 ax-hvmulid 22509 . . . . 5  |-  ( A  e.  ~H  ->  (
1  .h  A )  =  A )
65oveq1d 6096 . . . 4  |-  ( A  e.  ~H  ->  (
( 1  .h  A
)  +h  B )  =  ( A  +h  B ) )
76fveq2d 5732 . . 3  |-  ( A  e.  ~H  ->  ( T `  ( (
1  .h  A )  +h  B ) )  =  ( T `  ( A  +h  B
) ) )
87adantr 452 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  (
( 1  .h  A
)  +h  B ) )  =  ( T `
 ( A  +h  B ) ) )
92lnopfi 23472 . . . . . 6  |-  T : ~H
--> ~H
109ffvelrni 5869 . . . . 5  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
11 ax-hvmulid 22509 . . . . 5  |-  ( ( T `  A )  e.  ~H  ->  (
1  .h  ( T `
 A ) )  =  ( T `  A ) )
1210, 11syl 16 . . . 4  |-  ( A  e.  ~H  ->  (
1  .h  ( T `
 A ) )  =  ( T `  A ) )
1312adantr 452 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( 1  .h  ( T `  A )
)  =  ( T `
 A ) )
1413oveq1d 6096 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( 1  .h  ( T `  A
) )  +h  ( T `  B )
)  =  ( ( T `  A )  +h  ( T `  B ) ) )
154, 8, 143eqtr3d 2476 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  ( A  +h  B ) )  =  ( ( T `
 A )  +h  ( T `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   CCcc 8988   1c1 8991   ~Hchil 22422    +h cva 22423    .h csm 22424   LinOpclo 22450
This theorem is referenced by:  lnopaddmuli  23476  lnophsi  23504  lnopeq0lem1  23508  lnophmlem2  23520  imaelshi  23561  cnlnadjlem2  23571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-1cn 9048  ax-hilex 22502  ax-hvmulid 22509
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-lnop 23344
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