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Theorem ellnop 26568
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 5870 . . . . . 6  |-  ( t  =  T  ->  (
t `  ( (
x  .h  y )  +h  z ) )  =  ( T `  ( ( x  .h  y )  +h  z
) ) )
2 fveq1 5870 . . . . . . . 8  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
32oveq2d 6310 . . . . . . 7  |-  ( t  =  T  ->  (
x  .h  ( t `
 y ) )  =  ( x  .h  ( T `  y
) ) )
4 fveq1 5870 . . . . . . 7  |-  ( t  =  T  ->  (
t `  z )  =  ( T `  z ) )
53, 4oveq12d 6312 . . . . . 6  |-  ( t  =  T  ->  (
( x  .h  (
t `  y )
)  +h  ( t `
 z ) )  =  ( ( x  .h  ( T `  y ) )  +h  ( T `  z
) ) )
61, 5eqeq12d 2489 . . . . 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 2906 . . . 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 2911 . . 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 26551 . . 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 3268 . 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 25707 . . . 4  |-  ~H  e.  _V
1211, 11elmap 7457 . . 3  |-  ( T  e.  ( ~H  ^m  ~H )  <->  T : ~H --> ~H )
1312anbi1i 695 . 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 249 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   -->wf 5589   ` cfv 5593  (class class class)co 6294    ^m cmap 7430   CCcc 9500   ~Hchil 25627    +h cva 25628    .h csm 25629   LinOpclo 25655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-hilex 25707
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-map 7432  df-lnop 26551
This theorem is referenced by:  lnopf  26569  lnopl  26624  unoplin  26630  hmoplin  26652  lnopmi  26710  lnophsi  26711  lnopcoi  26713  cnlnadjlem6  26782  adjlnop  26796
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