Theorem leopg 23578

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.

Step	Hyp	Ref	Expression
1		oveq2 6048	. . . 4 |- ( t = T -> ( u -op t ) = ( u -op T ) )
2	1	eleq1d 2470	. . 3 |- ( t = T -> ( ( u -op t ) e. HrmOp <-> ( u -op T ) e. HrmOp ) )
3	1	fveq1d 5689	. . . . . 6 |- ( t = T -> ( ( u -op t ) ` x ) = ( ( u -op T ) ` x ) )
4	3	oveq1d 6055	. . . . 5 |- ( t = T -> ( ( ( u -op t ) ` x ) .ih x ) = ( ( ( u -op T ) ` x ) .ih x ) )
5	4	breq2d 4184	. . . 4 |- ( t = T -> ( 0 <_ ( ( ( u -op t ) ` x ) .ih x ) <-> 0 <_ ( ( ( u -op T ) ` x ) .ih x ) ) )
6	5	ralbidv 2686	. . 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 ) ) )
7	2, 6	anbi12d 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 6047	. . . 4 |- ( u = U -> ( u -op T ) = ( U -op T ) )
9	8	eleq1d 2470	. . 3 |- ( u = U -> ( ( u -op T ) e. HrmOp <-> ( U -op T ) e. HrmOp ) )
10	8	fveq1d 5689	. . . . . 6 |- ( u = U -> ( ( u -op T ) ` x ) = ( ( U -op T ) ` x ) )
11	10	oveq1d 6055	. . . . 5 |- ( u = U -> ( ( ( u -op T ) ` x ) .ih x ) = ( ( ( U -op T ) ` x ) .ih x ) )
12	11	breq2d 4184	. . . 4 |- ( u = U -> ( 0 <_ ( ( ( u -op T ) ` x ) .ih x ) <-> 0 <_ ( ( ( U -op T ) ` x ) .ih x ) ) )
13	12	ralbidv 2686	. . 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 ) ) )
14	9, 13	anbi12d 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 23308	. 2 |- <_op = { <. t , u >. | ( ( u -op t ) e. HrmOp /\ A. x e. ~H 0 <_ ( ( ( u -op t ) ` x ) .ih x ) ) }
16	7, 14, 15	brabg 4434	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 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   A.wral 2666   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   0cc0 8946   <_ cle 9077   ~Hchil 22375   .ih csp 22378   -op chod 22396   HrmOpcho 22406   <_op cleo 22414

This theorem is referenced by:  leop  23579  leoprf2  23583

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-iota 5377  df-fv 5421  df-ov 6043  df-leop 23308