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Theorem leopg 22702
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 5866 . . . 4  |-  ( t  =  T  ->  (
u  -op  t )  =  ( u  -op  T ) )
21eleq1d 2349 . . 3  |-  ( t  =  T  ->  (
( u  -op  t
)  e.  HrmOp  <->  ( u  -op  T )  e.  HrmOp ) )
31fveq1d 5527 . . . . . 6  |-  ( t  =  T  ->  (
( u  -op  t
) `  x )  =  ( ( u  -op  T ) `  x ) )
43oveq1d 5873 . . . . 5  |-  ( t  =  T  ->  (
( ( u  -op  t ) `  x
)  .ih  x )  =  ( ( ( u  -op  T ) `
 x )  .ih  x ) )
54breq2d 4035 . . . 4  |-  ( t  =  T  ->  (
0  <_  ( (
( u  -op  t
) `  x )  .ih  x )  <->  0  <_  ( ( ( u  -op  T ) `  x ) 
.ih  x ) ) )
65ralbidv 2563 . . 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 5865 . . . 4  |-  ( u  =  U  ->  (
u  -op  T )  =  ( U  -op  T ) )
98eleq1d 2349 . . 3  |-  ( u  =  U  ->  (
( u  -op  T
)  e.  HrmOp  <->  ( U  -op  T )  e.  HrmOp ) )
108fveq1d 5527 . . . . . 6  |-  ( u  =  U  ->  (
( u  -op  T
) `  x )  =  ( ( U  -op  T ) `  x ) )
1110oveq1d 5873 . . . . 5  |-  ( u  =  U  ->  (
( ( u  -op  T ) `  x ) 
.ih  x )  =  ( ( ( U  -op  T ) `  x )  .ih  x
) )
1211breq2d 4035 . . . 4  |-  ( u  =  U  ->  (
0  <_  ( (
( u  -op  T
) `  x )  .ih  x )  <->  0  <_  ( ( ( U  -op  T ) `  x ) 
.ih  x ) ) )
1312ralbidv 2563 . . 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 22432 . 2  |-  <_op  =  { <. t ,  u >.  |  ( ( u  -op  t )  e. 
HrmOp  /\  A. x  e. 
~H  0  <_  (
( ( u  -op  t ) `  x
)  .ih  x )
) }
167, 14, 15brabg 4284 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 1684   A.wral 2543   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   0cc0 8737    <_ cle 8868   ~Hchil 21499    .ih csp 21502    -op chod 21520   HrmOpcho 21530    <_op cleo 21538
This theorem is referenced by:  leop  22703  leoprf2  22707
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 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-iota 5219  df-fv 5263  df-ov 5861  df-leop 22432
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