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.

Step	Hyp	Ref	Expression
1		oveq2 6089	. . . 4 |- ( t = T -> ( u -op t ) = ( u -op T ) )
2	1	eleq1d 2502	. . 3 |- ( t = T -> ( ( u -op t ) e. HrmOp <-> ( u -op T ) e. HrmOp ) )
3	1	fveq1d 5730	. . . . . 6 |- ( t = T -> ( ( u -op t ) ` x ) = ( ( u -op T ) ` x ) )
4	3	oveq1d 6096	. . . . 5 |- ( t = T -> ( ( ( u -op t ) ` x ) .ih x ) = ( ( ( u -op T ) ` x ) .ih x ) )
5	4	breq2d 4224	. . . 4 |- ( t = T -> ( 0 <_ ( ( ( u -op t ) ` x ) .ih x ) <-> 0 <_ ( ( ( u -op T ) ` x ) .ih x ) ) )
6	5	ralbidv 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 ) ) )
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 6088	. . . 4 |- ( u = U -> ( u -op T ) = ( U -op T ) )
9	8	eleq1d 2502	. . 3 |- ( u = U -> ( ( u -op T ) e. HrmOp <-> ( U -op T ) e. HrmOp ) )
10	8	fveq1d 5730	. . . . . 6 |- ( u = U -> ( ( u -op T ) ` x ) = ( ( U -op T ) ` x ) )
11	10	oveq1d 6096	. . . . 5 |- ( u = U -> ( ( ( u -op T ) ` x ) .ih x ) = ( ( ( U -op T ) ` x ) .ih x ) )
12	11	breq2d 4224	. . . 4 |- ( u = U -> ( 0 <_ ( ( ( u -op T ) ` x ) .ih x ) <-> 0 <_ ( ( ( U -op T ) ` x ) .ih x ) ) )
13	12	ralbidv 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 ) ) )
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 23355	. 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 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 ) ) ) )

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