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Theorem lnopl 22494
Description: Basic linearity property of a linear Hilbert space operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
Assertion
Ref Expression
lnopl  |-  ( ( ( T  e.  LinOp  /\  A  e.  CC )  /\  ( B  e. 
~H  /\  C  e.  ~H ) )  ->  ( T `  ( ( A  .h  B )  +h  C ) )  =  ( ( A  .h  ( T `  B ) )  +h  ( T `
 C ) ) )

Proof of Theorem lnopl
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ellnop 22438 . . . . . 6  |-  ( T  e.  LinOp 
<->  ( T : ~H --> ~H  /\  A. x  e.  CC  A. y  e. 
~H  A. z  e.  ~H  ( T `  ( ( x  .h  y )  +h  z ) )  =  ( ( x  .h  ( T `  y ) )  +h  ( T `  z
) ) ) )
21simprbi 450 . . . . 5  |-  ( T  e.  LinOp  ->  A. x  e.  CC  A. y  e. 
~H  A. z  e.  ~H  ( T `  ( ( x  .h  y )  +h  z ) )  =  ( ( x  .h  ( T `  y ) )  +h  ( T `  z
) ) )
3 oveq1 5865 . . . . . . . . 9  |-  ( x  =  A  ->  (
x  .h  y )  =  ( A  .h  y ) )
43oveq1d 5873 . . . . . . . 8  |-  ( x  =  A  ->  (
( x  .h  y
)  +h  z )  =  ( ( A  .h  y )  +h  z ) )
54fveq2d 5529 . . . . . . 7  |-  ( x  =  A  ->  ( T `  ( (
x  .h  y )  +h  z ) )  =  ( T `  ( ( A  .h  y )  +h  z
) ) )
6 oveq1 5865 . . . . . . . 8  |-  ( x  =  A  ->  (
x  .h  ( T `
 y ) )  =  ( A  .h  ( T `  y ) ) )
76oveq1d 5873 . . . . . . 7  |-  ( x  =  A  ->  (
( x  .h  ( T `  y )
)  +h  ( T `
 z ) )  =  ( ( A  .h  ( T `  y ) )  +h  ( T `  z
) ) )
85, 7eqeq12d 2297 . . . . . 6  |-  ( x  =  A  ->  (
( T `  (
( x  .h  y
)  +h  z ) )  =  ( ( x  .h  ( T `
 y ) )  +h  ( T `  z ) )  <->  ( T `  ( ( A  .h  y )  +h  z
) )  =  ( ( A  .h  ( T `  y )
)  +h  ( T `
 z ) ) ) )
9 oveq2 5866 . . . . . . . . 9  |-  ( y  =  B  ->  ( A  .h  y )  =  ( A  .h  B ) )
109oveq1d 5873 . . . . . . . 8  |-  ( y  =  B  ->  (
( A  .h  y
)  +h  z )  =  ( ( A  .h  B )  +h  z ) )
1110fveq2d 5529 . . . . . . 7  |-  ( y  =  B  ->  ( T `  ( ( A  .h  y )  +h  z ) )  =  ( T `  (
( A  .h  B
)  +h  z ) ) )
12 fveq2 5525 . . . . . . . . 9  |-  ( y  =  B  ->  ( T `  y )  =  ( T `  B ) )
1312oveq2d 5874 . . . . . . . 8  |-  ( y  =  B  ->  ( A  .h  ( T `  y ) )  =  ( A  .h  ( T `  B )
) )
1413oveq1d 5873 . . . . . . 7  |-  ( y  =  B  ->  (
( A  .h  ( T `  y )
)  +h  ( T `
 z ) )  =  ( ( A  .h  ( T `  B ) )  +h  ( T `  z
) ) )
1511, 14eqeq12d 2297 . . . . . 6  |-  ( y  =  B  ->  (
( T `  (
( A  .h  y
)  +h  z ) )  =  ( ( A  .h  ( T `
 y ) )  +h  ( T `  z ) )  <->  ( T `  ( ( A  .h  B )  +h  z
) )  =  ( ( A  .h  ( T `  B )
)  +h  ( T `
 z ) ) ) )
16 oveq2 5866 . . . . . . . 8  |-  ( z  =  C  ->  (
( A  .h  B
)  +h  z )  =  ( ( A  .h  B )  +h  C ) )
1716fveq2d 5529 . . . . . . 7  |-  ( z  =  C  ->  ( T `  ( ( A  .h  B )  +h  z ) )  =  ( T `  (
( A  .h  B
)  +h  C ) ) )
18 fveq2 5525 . . . . . . . 8  |-  ( z  =  C  ->  ( T `  z )  =  ( T `  C ) )
1918oveq2d 5874 . . . . . . 7  |-  ( z  =  C  ->  (
( A  .h  ( T `  B )
)  +h  ( T `
 z ) )  =  ( ( A  .h  ( T `  B ) )  +h  ( T `  C
) ) )
2017, 19eqeq12d 2297 . . . . . 6  |-  ( z  =  C  ->  (
( T `  (
( A  .h  B
)  +h  z ) )  =  ( ( A  .h  ( T `
 B ) )  +h  ( T `  z ) )  <->  ( T `  ( ( A  .h  B )  +h  C
) )  =  ( ( A  .h  ( T `  B )
)  +h  ( T `
 C ) ) ) )
218, 15, 20rspc3v 2893 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A. x  e.  CC  A. y  e.  ~H  A. z  e.  ~H  ( T `  ( (
x  .h  y )  +h  z ) )  =  ( ( x  .h  ( T `  y ) )  +h  ( T `  z
) )  ->  ( T `  ( ( A  .h  B )  +h  C ) )  =  ( ( A  .h  ( T `  B ) )  +h  ( T `
 C ) ) ) )
222, 21syl5 28 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( T  e.  LinOp  ->  ( T `  ( ( A  .h  B )  +h  C ) )  =  ( ( A  .h  ( T `  B ) )  +h  ( T `
 C ) ) ) )
23223expb 1152 . . 3  |-  ( ( A  e.  CC  /\  ( B  e.  ~H  /\  C  e.  ~H )
)  ->  ( T  e.  LinOp  ->  ( T `  ( ( A  .h  B )  +h  C
) )  =  ( ( A  .h  ( T `  B )
)  +h  ( T `
 C ) ) ) )
2423impcom 419 . 2  |-  ( ( T  e.  LinOp  /\  ( A  e.  CC  /\  ( B  e.  ~H  /\  C  e.  ~H ) ) )  ->  ( T `  ( ( A  .h  B )  +h  C
) )  =  ( ( A  .h  ( T `  B )
)  +h  ( T `
 C ) ) )
2524anassrs 629 1  |-  ( ( ( T  e.  LinOp  /\  A  e.  CC )  /\  ( B  e. 
~H  /\  C  e.  ~H ) )  ->  ( T `  ( ( A  .h  B )  +h  C ) )  =  ( ( A  .h  ( T `  B ) )  +h  ( T `
 C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   ~Hchil 21499    +h cva 21500    .h csm 21501   LinOpclo 21527
This theorem is referenced by:  lnop0  22546  lnopmul  22547  lnopli  22548
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-hilex 21579
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-lnop 22421
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