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Theorem leopg 27167
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 6304 . . . 4  |-  ( t  =  T  ->  (
u  -op  t )  =  ( u  -op  T ) )
21eleq1d 2526 . . 3  |-  ( t  =  T  ->  (
( u  -op  t
)  e.  HrmOp  <->  ( u  -op  T )  e.  HrmOp ) )
31fveq1d 5874 . . . . . 6  |-  ( t  =  T  ->  (
( u  -op  t
) `  x )  =  ( ( u  -op  T ) `  x ) )
43oveq1d 6311 . . . . 5  |-  ( t  =  T  ->  (
( ( u  -op  t ) `  x
)  .ih  x )  =  ( ( ( u  -op  T ) `
 x )  .ih  x ) )
54breq2d 4468 . . . 4  |-  ( t  =  T  ->  (
0  <_  ( (
( u  -op  t
) `  x )  .ih  x )  <->  0  <_  ( ( ( u  -op  T ) `  x ) 
.ih  x ) ) )
65ralbidv 2896 . . 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 6303 . . . 4  |-  ( u  =  U  ->  (
u  -op  T )  =  ( U  -op  T ) )
98eleq1d 2526 . . 3  |-  ( u  =  U  ->  (
( u  -op  T
)  e.  HrmOp  <->  ( U  -op  T )  e.  HrmOp ) )
108fveq1d 5874 . . . . . 6  |-  ( u  =  U  ->  (
( u  -op  T
) `  x )  =  ( ( U  -op  T ) `  x ) )
1110oveq1d 6311 . . . . 5  |-  ( u  =  U  ->  (
( ( u  -op  T ) `  x ) 
.ih  x )  =  ( ( ( U  -op  T ) `  x )  .ih  x
) )
1211breq2d 4468 . . . 4  |-  ( u  =  U  ->  (
0  <_  ( (
( u  -op  T
) `  x )  .ih  x )  <->  0  <_  ( ( ( U  -op  T ) `  x ) 
.ih  x ) ) )
1312ralbidv 2896 . . 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 26897 . 2  |-  <_op  =  { <. t ,  u >.  |  ( ( u  -op  t )  e. 
HrmOp  /\  A. x  e. 
~H  0  <_  (
( ( u  -op  t ) `  x
)  .ih  x )
) }
167, 14, 15brabg 4775 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 1395    e. wcel 1819   A.wral 2807   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   0cc0 9509    <_ cle 9646   ~Hchil 25962    .ih csp 25965    -op chod 25983   HrmOpcho 25993    <_op cleo 26001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-iota 5557  df-fv 5602  df-ov 6299  df-leop 26897
This theorem is referenced by:  leop  27168  leoprf2  27172
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