Theorem leopg 28981

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 6658	. . . 4 |- ( t = T -> ( u -op t ) = ( u -op T ) )
2	1	eleq1d 2686	. . 3 |- ( t = T -> ( ( u -op t ) e. HrmOp <-> ( u -op T ) e. HrmOp ) )
3	1	fveq1d 6193	. . . . . 6 |- ( t = T -> ( ( u -op t ) ` x ) = ( ( u -op T ) ` x ) )
4	3	oveq1d 6665	. . . . 5 |- ( t = T -> ( ( ( u -op t ) ` x ) .ih x ) = ( ( ( u -op T ) ` x ) .ih x ) )
5	4	breq2d 4665	. . . 4 |- ( t = T -> ( 0 <_ ( ( ( u -op t ) ` x ) .ih x ) <-> 0 <_ ( ( ( u -op T ) ` x ) .ih x ) ) )
6	5	ralbidv 2986	. . 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 747	. 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 6657	. . . 4 |- ( u = U -> ( u -op T ) = ( U -op T ) )
9	8	eleq1d 2686	. . 3 |- ( u = U -> ( ( u -op T ) e. HrmOp <-> ( U -op T ) e. HrmOp ) )
10	8	fveq1d 6193	. . . . . 6 |- ( u = U -> ( ( u -op T ) ` x ) = ( ( U -op T ) ` x ) )
11	10	oveq1d 6665	. . . . 5 |- ( u = U -> ( ( ( u -op T ) ` x ) .ih x ) = ( ( ( U -op T ) ` x ) .ih x ) )
12	11	breq2d 4665	. . . 4 |- ( u = U -> ( 0 <_ ( ( ( u -op T ) ` x ) .ih x ) <-> 0 <_ ( ( ( U -op T ) ` x ) .ih x ) ) )
13	12	ralbidv 2986	. . 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 747	. 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 28711	. 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 4994	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 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   <_ cle 10075   ~Hchil 27776   .ih csp 27779   -op chod 27797   HrmOpcho 27807   <_op cleo 27815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896  df-ov 6653  df-leop 28711

This theorem is referenced by:  leop  28982  leoprf2  28986