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Theorem lnopaddi 22567
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 8811 . . 3  |-  1  e.  CC
2 lnopl.1 . . . 4  |-  T  e. 
LinOp
32lnopli 22564 . . 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 1264 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( T `  (
( 1  .h  A
)  +h  B ) )  =  ( ( 1  .h  ( T `
 A ) )  +h  ( T `  B ) ) )
5 ax-hvmulid 21602 . . . . 5  |-  ( A  e.  ~H  ->  (
1  .h  A )  =  A )
65oveq1d 5889 . . . 4  |-  ( A  e.  ~H  ->  (
( 1  .h  A
)  +h  B )  =  ( A  +h  B ) )
76fveq2d 5545 . . 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 ) ) )
92lnopfi 22565 . . . . . 6  |-  T : ~H
--> ~H
109ffvelrni 5680 . . . . 5  |-  ( A  e.  ~H  ->  ( T `  A )  e.  ~H )
11 ax-hvmulid 21602 . . . . 5  |-  ( ( T `  A )  e.  ~H  ->  (
1  .h  ( T `
 A ) )  =  ( T `  A ) )
1210, 11syl 15 . . . 4  |-  ( A  e.  ~H  ->  (
1  .h  ( T `
 A ) )  =  ( T `  A ) )
1312adantr 451 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( 1  .h  ( T `  A )
)  =  ( T `
 A ) )
1413oveq1d 5889 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( 1  .h  ( T `  A
) )  +h  ( T `  B )
)  =  ( ( T `  A )  +h  ( T `  B ) ) )
154, 8, 143eqtr3d 2336 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 358    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754   ~Hchil 21515    +h cva 21516    .h csm 21517   LinOpclo 21543
This theorem is referenced by:  lnopaddmuli  22569  lnophsi  22597  lnopeq0lem1  22601  lnophmlem2  22613  imaelshi  22654  cnlnadjlem2  22664
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-1cn 8811  ax-hilex 21595  ax-hvmulid 21602
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-lnop 22437
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