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Theorem lnopl 22549
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 22493 . . . . . 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 5907 . . . . . . . . 9  |-  ( x  =  A  ->  (
x  .h  y )  =  ( A  .h  y ) )
43oveq1d 5915 . . . . . . . 8  |-  ( x  =  A  ->  (
( x  .h  y
)  +h  z )  =  ( ( A  .h  y )  +h  z ) )
54fveq2d 5567 . . . . . . 7  |-  ( x  =  A  ->  ( T `  ( (
x  .h  y )  +h  z ) )  =  ( T `  ( ( A  .h  y )  +h  z
) ) )
6 oveq1 5907 . . . . . . . 8  |-  ( x  =  A  ->  (
x  .h  ( T `
 y ) )  =  ( A  .h  ( T `  y ) ) )
76oveq1d 5915 . . . . . . 7  |-  ( x  =  A  ->  (
( x  .h  ( T `  y )
)  +h  ( T `
 z ) )  =  ( ( A  .h  ( T `  y ) )  +h  ( T `  z
) ) )
85, 7eqeq12d 2330 . . . . . 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 5908 . . . . . . . . 9  |-  ( y  =  B  ->  ( A  .h  y )  =  ( A  .h  B ) )
109oveq1d 5915 . . . . . . . 8  |-  ( y  =  B  ->  (
( A  .h  y
)  +h  z )  =  ( ( A  .h  B )  +h  z ) )
1110fveq2d 5567 . . . . . . 7  |-  ( y  =  B  ->  ( T `  ( ( A  .h  y )  +h  z ) )  =  ( T `  (
( A  .h  B
)  +h  z ) ) )
12 fveq2 5563 . . . . . . . . 9  |-  ( y  =  B  ->  ( T `  y )  =  ( T `  B ) )
1312oveq2d 5916 . . . . . . . 8  |-  ( y  =  B  ->  ( A  .h  ( T `  y ) )  =  ( A  .h  ( T `  B )
) )
1413oveq1d 5915 . . . . . . 7  |-  ( y  =  B  ->  (
( A  .h  ( T `  y )
)  +h  ( T `
 z ) )  =  ( ( A  .h  ( T `  B ) )  +h  ( T `  z
) ) )
1511, 14eqeq12d 2330 . . . . . 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 5908 . . . . . . . 8  |-  ( z  =  C  ->  (
( A  .h  B
)  +h  z )  =  ( ( A  .h  B )  +h  C ) )
1716fveq2d 5567 . . . . . . 7  |-  ( z  =  C  ->  ( T `  ( ( A  .h  B )  +h  z ) )  =  ( T `  (
( A  .h  B
)  +h  C ) ) )
18 fveq2 5563 . . . . . . . 8  |-  ( z  =  C  ->  ( T `  z )  =  ( T `  C ) )
1918oveq2d 5916 . . . . . . 7  |-  ( z  =  C  ->  (
( A  .h  ( T `  B )
)  +h  ( T `
 z ) )  =  ( ( A  .h  ( T `  B ) )  +h  ( T `  C
) ) )
2017, 19eqeq12d 2330 . . . . . 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 2927 . . . . 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 1633    e. wcel 1701   A.wral 2577   -->wf 5288   ` cfv 5292  (class class class)co 5900   CCcc 8780   ~Hchil 21554    +h cva 21555    .h csm 21556   LinOpclo 21582
This theorem is referenced by:  lnop0  22601  lnopmul  22602  lnopli  22603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-hilex 21634
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-map 6817  df-lnop 22476
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