HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  leopg Unicode version

Theorem leopg 22698
Description: Ordering relation for positive operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
Assertion
Ref Expression
leopg  |-  ( ( T  e.  A  /\  U  e.  B )  ->  ( T  <_op  U  <->  ( ( U  -op  T
)  e.  HrmOp  /\  A. x  e.  ~H  0  <_  ( ( ( U  -op  T ) `  x )  .ih  x
) ) ) )
Distinct variable groups:    x, A    x, B    x, T    x, U

Proof of Theorem leopg
Dummy variables  u  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5828 . . . 4  |-  ( t  =  T  ->  (
u  -op  t )  =  ( u  -op  T ) )
21eleq1d 2350 . . 3  |-  ( t  =  T  ->  (
( u  -op  t
)  e.  HrmOp  <->  ( u  -op  T )  e.  HrmOp ) )
31fveq1d 5488 . . . . . 6  |-  ( t  =  T  ->  (
( u  -op  t
) `  x )  =  ( ( u  -op  T ) `  x ) )
43oveq1d 5835 . . . . 5  |-  ( t  =  T  ->  (
( ( u  -op  t ) `  x
)  .ih  x )  =  ( ( ( u  -op  T ) `
 x )  .ih  x ) )
54breq2d 4036 . . . 4  |-  ( t  =  T  ->  (
0  <_  ( (
( u  -op  t
) `  x )  .ih  x )  <->  0  <_  ( ( ( u  -op  T ) `  x ) 
.ih  x ) ) )
65ralbidv 2564 . . 3  |-  ( t  =  T  ->  ( A. x  e.  ~H  0  <_  ( ( ( u  -op  t ) `
 x )  .ih  x )  <->  A. x  e.  ~H  0  <_  (
( ( u  -op  T ) `  x ) 
.ih  x ) ) )
72, 6anbi12d 691 . 2  |-  ( t  =  T  ->  (
( ( u  -op  t )  e.  HrmOp  /\ 
A. x  e.  ~H  0  <_  ( ( ( u  -op  t ) `
 x )  .ih  x ) )  <->  ( (
u  -op  T )  e.  HrmOp  /\  A. x  e.  ~H  0  <_  (
( ( u  -op  T ) `  x ) 
.ih  x ) ) ) )
8 oveq1 5827 . . . 4  |-  ( u  =  U  ->  (
u  -op  T )  =  ( U  -op  T ) )
98eleq1d 2350 . . 3  |-  ( u  =  U  ->  (
( u  -op  T
)  e.  HrmOp  <->  ( U  -op  T )  e.  HrmOp ) )
108fveq1d 5488 . . . . . 6  |-  ( u  =  U  ->  (
( u  -op  T
) `  x )  =  ( ( U  -op  T ) `  x ) )
1110oveq1d 5835 . . . . 5  |-  ( u  =  U  ->  (
( ( u  -op  T ) `  x ) 
.ih  x )  =  ( ( ( U  -op  T ) `  x )  .ih  x
) )
1211breq2d 4036 . . . 4  |-  ( u  =  U  ->  (
0  <_  ( (
( u  -op  T
) `  x )  .ih  x )  <->  0  <_  ( ( ( U  -op  T ) `  x ) 
.ih  x ) ) )
1312ralbidv 2564 . . 3  |-  ( u  =  U  ->  ( A. x  e.  ~H  0  <_  ( ( ( u  -op  T ) `
 x )  .ih  x )  <->  A. x  e.  ~H  0  <_  (
( ( U  -op  T ) `  x ) 
.ih  x ) ) )
149, 13anbi12d 691 . 2  |-  ( u  =  U  ->  (
( ( u  -op  T )  e.  HrmOp  /\  A. x  e.  ~H  0  <_  ( ( ( u  -op  T ) `  x )  .ih  x
) )  <->  ( ( U  -op  T )  e. 
HrmOp  /\  A. x  e. 
~H  0  <_  (
( ( U  -op  T ) `  x ) 
.ih  x ) ) ) )
15 df-leop 22428 . 2  |-  <_op  =  { <. t ,  u >.  |  ( ( u  -op  t )  e. 
HrmOp  /\  A. x  e. 
~H  0  <_  (
( ( u  -op  t ) `  x
)  .ih  x )
) }
167, 14, 15brabg 4283 1  |-  ( ( T  e.  A  /\  U  e.  B )  ->  ( T  <_op  U  <->  ( ( U  -op  T
)  e.  HrmOp  /\  A. x  e.  ~H  0  <_  ( ( ( U  -op  T ) `  x )  .ih  x
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1685   A.wral 2544   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   0cc0 8733    <_ cle 8864   ~Hchil 21495    .ih csp 21498    -op chod 21516   HrmOpcho 21526    <_op cleo 21534
This theorem is referenced by:  leop  22699  leoprf2  22703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fv 5229  df-ov 5823  df-leop 22428
  Copyright terms: Public domain W3C validator