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Theorem lnopl 25463
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 25407 . . . . . 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 464 . . . . 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 6200 . . . . . . . . 9  |-  ( x  =  A  ->  (
x  .h  y )  =  ( A  .h  y ) )
43oveq1d 6208 . . . . . . . 8  |-  ( x  =  A  ->  (
( x  .h  y
)  +h  z )  =  ( ( A  .h  y )  +h  z ) )
54fveq2d 5796 . . . . . . 7  |-  ( x  =  A  ->  ( T `  ( (
x  .h  y )  +h  z ) )  =  ( T `  ( ( A  .h  y )  +h  z
) ) )
6 oveq1 6200 . . . . . . . 8  |-  ( x  =  A  ->  (
x  .h  ( T `
 y ) )  =  ( A  .h  ( T `  y ) ) )
76oveq1d 6208 . . . . . . 7  |-  ( x  =  A  ->  (
( x  .h  ( T `  y )
)  +h  ( T `
 z ) )  =  ( ( A  .h  ( T `  y ) )  +h  ( T `  z
) ) )
85, 7eqeq12d 2473 . . . . . 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 6201 . . . . . . . . 9  |-  ( y  =  B  ->  ( A  .h  y )  =  ( A  .h  B ) )
109oveq1d 6208 . . . . . . . 8  |-  ( y  =  B  ->  (
( A  .h  y
)  +h  z )  =  ( ( A  .h  B )  +h  z ) )
1110fveq2d 5796 . . . . . . 7  |-  ( y  =  B  ->  ( T `  ( ( A  .h  y )  +h  z ) )  =  ( T `  (
( A  .h  B
)  +h  z ) ) )
12 fveq2 5792 . . . . . . . . 9  |-  ( y  =  B  ->  ( T `  y )  =  ( T `  B ) )
1312oveq2d 6209 . . . . . . . 8  |-  ( y  =  B  ->  ( A  .h  ( T `  y ) )  =  ( A  .h  ( T `  B )
) )
1413oveq1d 6208 . . . . . . 7  |-  ( y  =  B  ->  (
( A  .h  ( T `  y )
)  +h  ( T `
 z ) )  =  ( ( A  .h  ( T `  B ) )  +h  ( T `  z
) ) )
1511, 14eqeq12d 2473 . . . . . 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 6201 . . . . . . . 8  |-  ( z  =  C  ->  (
( A  .h  B
)  +h  z )  =  ( ( A  .h  B )  +h  C ) )
1716fveq2d 5796 . . . . . . 7  |-  ( z  =  C  ->  ( T `  ( ( A  .h  B )  +h  z ) )  =  ( T `  (
( A  .h  B
)  +h  C ) ) )
18 fveq2 5792 . . . . . . . 8  |-  ( z  =  C  ->  ( T `  z )  =  ( T `  C ) )
1918oveq2d 6209 . . . . . . 7  |-  ( z  =  C  ->  (
( A  .h  ( T `  B )
)  +h  ( T `
 z ) )  =  ( ( A  .h  ( T `  B ) )  +h  ( T `  C
) ) )
2017, 19eqeq12d 2473 . . . . . 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 3182 . . . . 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 32 . . . 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 1189 . . 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 430 . 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 648 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795   -->wf 5515   ` cfv 5519  (class class class)co 6193   CCcc 9384   ~Hchil 24466    +h cva 24467    .h csm 24468   LinOpclo 24494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-hilex 24546
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-map 7319  df-lnop 25390
This theorem is referenced by:  lnop0  25515  lnopmul  25516  lnopli  25517
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