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Theorem leopg 26714
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 6290 . . . 4  |-  ( t  =  T  ->  (
u  -op  t )  =  ( u  -op  T ) )
21eleq1d 2536 . . 3  |-  ( t  =  T  ->  (
( u  -op  t
)  e.  HrmOp  <->  ( u  -op  T )  e.  HrmOp ) )
31fveq1d 5866 . . . . . 6  |-  ( t  =  T  ->  (
( u  -op  t
) `  x )  =  ( ( u  -op  T ) `  x ) )
43oveq1d 6297 . . . . 5  |-  ( t  =  T  ->  (
( ( u  -op  t ) `  x
)  .ih  x )  =  ( ( ( u  -op  T ) `
 x )  .ih  x ) )
54breq2d 4459 . . . 4  |-  ( t  =  T  ->  (
0  <_  ( (
( u  -op  t
) `  x )  .ih  x )  <->  0  <_  ( ( ( u  -op  T ) `  x ) 
.ih  x ) ) )
65ralbidv 2903 . . 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 710 . 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 6289 . . . 4  |-  ( u  =  U  ->  (
u  -op  T )  =  ( U  -op  T ) )
98eleq1d 2536 . . 3  |-  ( u  =  U  ->  (
( u  -op  T
)  e.  HrmOp  <->  ( U  -op  T )  e.  HrmOp ) )
108fveq1d 5866 . . . . . 6  |-  ( u  =  U  ->  (
( u  -op  T
) `  x )  =  ( ( U  -op  T ) `  x ) )
1110oveq1d 6297 . . . . 5  |-  ( u  =  U  ->  (
( ( u  -op  T ) `  x ) 
.ih  x )  =  ( ( ( U  -op  T ) `  x )  .ih  x
) )
1211breq2d 4459 . . . 4  |-  ( u  =  U  ->  (
0  <_  ( (
( u  -op  T
) `  x )  .ih  x )  <->  0  <_  ( ( ( U  -op  T ) `  x ) 
.ih  x ) ) )
1312ralbidv 2903 . . 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 710 . 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 26444 . 2  |-  <_op  =  { <. t ,  u >.  |  ( ( u  -op  t )  e. 
HrmOp  /\  A. x  e. 
~H  0  <_  (
( ( u  -op  t ) `  x
)  .ih  x )
) }
167, 14, 15brabg 4766 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   0cc0 9488    <_ cle 9625   ~Hchil 25509    .ih csp 25512    -op chod 25530   HrmOpcho 25540    <_op cleo 25548
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-iota 5549  df-fv 5594  df-ov 6285  df-leop 26444
This theorem is referenced by:  leop  26715  leoprf2  26719
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