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Theorem lnopl 23419
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 23363 . . . . . 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 452 . . . . 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 6090 . . . . . . . . 9  |-  ( x  =  A  ->  (
x  .h  y )  =  ( A  .h  y ) )
43oveq1d 6098 . . . . . . . 8  |-  ( x  =  A  ->  (
( x  .h  y
)  +h  z )  =  ( ( A  .h  y )  +h  z ) )
54fveq2d 5734 . . . . . . 7  |-  ( x  =  A  ->  ( T `  ( (
x  .h  y )  +h  z ) )  =  ( T `  ( ( A  .h  y )  +h  z
) ) )
6 oveq1 6090 . . . . . . . 8  |-  ( x  =  A  ->  (
x  .h  ( T `
 y ) )  =  ( A  .h  ( T `  y ) ) )
76oveq1d 6098 . . . . . . 7  |-  ( x  =  A  ->  (
( x  .h  ( T `  y )
)  +h  ( T `
 z ) )  =  ( ( A  .h  ( T `  y ) )  +h  ( T `  z
) ) )
85, 7eqeq12d 2452 . . . . . 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 6091 . . . . . . . . 9  |-  ( y  =  B  ->  ( A  .h  y )  =  ( A  .h  B ) )
109oveq1d 6098 . . . . . . . 8  |-  ( y  =  B  ->  (
( A  .h  y
)  +h  z )  =  ( ( A  .h  B )  +h  z ) )
1110fveq2d 5734 . . . . . . 7  |-  ( y  =  B  ->  ( T `  ( ( A  .h  y )  +h  z ) )  =  ( T `  (
( A  .h  B
)  +h  z ) ) )
12 fveq2 5730 . . . . . . . . 9  |-  ( y  =  B  ->  ( T `  y )  =  ( T `  B ) )
1312oveq2d 6099 . . . . . . . 8  |-  ( y  =  B  ->  ( A  .h  ( T `  y ) )  =  ( A  .h  ( T `  B )
) )
1413oveq1d 6098 . . . . . . 7  |-  ( y  =  B  ->  (
( A  .h  ( T `  y )
)  +h  ( T `
 z ) )  =  ( ( A  .h  ( T `  B ) )  +h  ( T `  z
) ) )
1511, 14eqeq12d 2452 . . . . . 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 6091 . . . . . . . 8  |-  ( z  =  C  ->  (
( A  .h  B
)  +h  z )  =  ( ( A  .h  B )  +h  C ) )
1716fveq2d 5734 . . . . . . 7  |-  ( z  =  C  ->  ( T `  ( ( A  .h  B )  +h  z ) )  =  ( T `  (
( A  .h  B
)  +h  C ) ) )
18 fveq2 5730 . . . . . . . 8  |-  ( z  =  C  ->  ( T `  z )  =  ( T `  C ) )
1918oveq2d 6099 . . . . . . 7  |-  ( z  =  C  ->  (
( A  .h  ( T `  B )
)  +h  ( T `
 z ) )  =  ( ( A  .h  ( T `  B ) )  +h  ( T `  C
) ) )
2017, 19eqeq12d 2452 . . . . . 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 3063 . . . . 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 31 . . . 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 1155 . . 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 421 . 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 631 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   -->wf 5452   ` cfv 5456  (class class class)co 6083   CCcc 8990   ~Hchil 22424    +h cva 22425    .h csm 22426   LinOpclo 22452
This theorem is referenced by:  lnop0  23471  lnopmul  23472  lnopli  23473
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-hilex 22504
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-lnop 23346
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