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Theorem leopg 23625
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 6089 . . . 4  |-  ( t  =  T  ->  (
u  -op  t )  =  ( u  -op  T ) )
21eleq1d 2502 . . 3  |-  ( t  =  T  ->  (
( u  -op  t
)  e.  HrmOp  <->  ( u  -op  T )  e.  HrmOp ) )
31fveq1d 5730 . . . . . 6  |-  ( t  =  T  ->  (
( u  -op  t
) `  x )  =  ( ( u  -op  T ) `  x ) )
43oveq1d 6096 . . . . 5  |-  ( t  =  T  ->  (
( ( u  -op  t ) `  x
)  .ih  x )  =  ( ( ( u  -op  T ) `
 x )  .ih  x ) )
54breq2d 4224 . . . 4  |-  ( t  =  T  ->  (
0  <_  ( (
( u  -op  t
) `  x )  .ih  x )  <->  0  <_  ( ( ( u  -op  T ) `  x ) 
.ih  x ) ) )
65ralbidv 2725 . . 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 692 . 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 6088 . . . 4  |-  ( u  =  U  ->  (
u  -op  T )  =  ( U  -op  T ) )
98eleq1d 2502 . . 3  |-  ( u  =  U  ->  (
( u  -op  T
)  e.  HrmOp  <->  ( U  -op  T )  e.  HrmOp ) )
108fveq1d 5730 . . . . . 6  |-  ( u  =  U  ->  (
( u  -op  T
) `  x )  =  ( ( U  -op  T ) `  x ) )
1110oveq1d 6096 . . . . 5  |-  ( u  =  U  ->  (
( ( u  -op  T ) `  x ) 
.ih  x )  =  ( ( ( U  -op  T ) `  x )  .ih  x
) )
1211breq2d 4224 . . . 4  |-  ( u  =  U  ->  (
0  <_  ( (
( u  -op  T
) `  x )  .ih  x )  <->  0  <_  ( ( ( U  -op  T ) `  x ) 
.ih  x ) ) )
1312ralbidv 2725 . . 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 692 . 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 23355 . 2  |-  <_op  =  { <. t ,  u >.  |  ( ( u  -op  t )  e. 
HrmOp  /\  A. x  e. 
~H  0  <_  (
( ( u  -op  t ) `  x
)  .ih  x )
) }
167, 14, 15brabg 4474 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   0cc0 8990    <_ cle 9121   ~Hchil 22422    .ih csp 22425    -op chod 22443   HrmOpcho 22453    <_op cleo 22461
This theorem is referenced by:  leop  23626  leoprf2  23630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-iota 5418  df-fv 5462  df-ov 6084  df-leop 23355
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