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Theorem ellnop 22493
Description: Property defining a linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ellnop  |-  ( 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
) ) ) )
Distinct variable group:    x, y, z, T

Proof of Theorem ellnop
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 fveq1 5562 . . . . . 6  |-  ( t  =  T  ->  (
t `  ( (
x  .h  y )  +h  z ) )  =  ( T `  ( ( x  .h  y )  +h  z
) ) )
2 fveq1 5562 . . . . . . . 8  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
32oveq2d 5916 . . . . . . 7  |-  ( t  =  T  ->  (
x  .h  ( t `
 y ) )  =  ( x  .h  ( T `  y
) ) )
4 fveq1 5562 . . . . . . 7  |-  ( t  =  T  ->  (
t `  z )  =  ( T `  z ) )
53, 4oveq12d 5918 . . . . . 6  |-  ( t  =  T  ->  (
( x  .h  (
t `  y )
)  +h  ( t `
 z ) )  =  ( ( x  .h  ( T `  y ) )  +h  ( T `  z
) ) )
61, 5eqeq12d 2330 . . . . 5  |-  ( t  =  T  ->  (
( t `  (
( x  .h  y
)  +h  z ) )  =  ( ( x  .h  ( t `
 y ) )  +h  ( t `  z ) )  <->  ( T `  ( ( x  .h  y )  +h  z
) )  =  ( ( x  .h  ( T `  y )
)  +h  ( T `
 z ) ) ) )
76ralbidv 2597 . . . 4  |-  ( t  =  T  ->  ( A. z  e.  ~H  ( t `  (
( x  .h  y
)  +h  z ) )  =  ( ( x  .h  ( t `
 y ) )  +h  ( t `  z ) )  <->  A. z  e.  ~H  ( T `  ( ( x  .h  y )  +h  z
) )  =  ( ( x  .h  ( T `  y )
)  +h  ( T `
 z ) ) ) )
872ralbidv 2619 . . 3  |-  ( t  =  T  ->  ( 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
) )  <->  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
) ) ) )
9 df-lnop 22476 . . 3  |-  LinOp  =  {
t  e.  ( ~H 
^m  ~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 ) ) }
108, 9elrab2 2959 . 2  |-  ( T  e.  LinOp 
<->  ( T  e.  ( ~H  ^m  ~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
) ) ) )
11 ax-hilex 21634 . . . 4  |-  ~H  e.  _V
1211, 11elmap 6839 . . 3  |-  ( T  e.  ( ~H  ^m  ~H )  <->  T : ~H --> ~H )
1312anbi1i 676 . 2  |-  ( ( T  e.  ( ~H 
^m  ~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 : ~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
) ) ) )
1410, 13bitri 240 1  |-  ( 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
) ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577   -->wf 5288   ` cfv 5292  (class class class)co 5900    ^m cmap 6815   CCcc 8780   ~Hchil 21554    +h cva 21555    .h csm 21556   LinOpclo 21582
This theorem is referenced by:  lnopf  22494  lnopl  22549  unoplin  22555  hmoplin  22577  lnopmi  22635  lnophsi  22636  lnopcoi  22638  cnlnadjlem6  22707  adjlnop  22721
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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