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Theorem ellnop 22546
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 5604 . . . . . 6  |-  ( t  =  T  ->  (
t `  ( (
x  .h  y )  +h  z ) )  =  ( T `  ( ( x  .h  y )  +h  z
) ) )
2 fveq1 5604 . . . . . . . 8  |-  ( t  =  T  ->  (
t `  y )  =  ( T `  y ) )
32oveq2d 5958 . . . . . . 7  |-  ( t  =  T  ->  (
x  .h  ( t `
 y ) )  =  ( x  .h  ( T `  y
) ) )
4 fveq1 5604 . . . . . . 7  |-  ( t  =  T  ->  (
t `  z )  =  ( T `  z ) )
53, 4oveq12d 5960 . . . . . 6  |-  ( t  =  T  ->  (
( x  .h  (
t `  y )
)  +h  ( t `
 z ) )  =  ( ( x  .h  ( T `  y ) )  +h  ( T `  z
) ) )
61, 5eqeq12d 2372 . . . . 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 2639 . . . 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 2661 . . 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 22529 . . 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 3001 . 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 21687 . . . 4  |-  ~H  e.  _V
1211, 11elmap 6881 . . 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 1642    e. wcel 1710   A.wral 2619   -->wf 5330   ` cfv 5334  (class class class)co 5942    ^m cmap 6857   CCcc 8822   ~Hchil 21607    +h cva 21608    .h csm 21609   LinOpclo 21635
This theorem is referenced by:  lnopf  22547  lnopl  22602  unoplin  22608  hmoplin  22630  lnopmi  22688  lnophsi  22689  lnopcoi  22691  cnlnadjlem6  22760  adjlnop  22774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-hilex 21687
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-map 6859  df-lnop 22529
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