Theorem leopg 22698

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 5828	. . . 4 |- ( t = T -> ( u -op t ) = ( u -op T ) )
2	1	eleq1d 2350	. . 3 |- ( t = T -> ( ( u -op t ) e. HrmOp <-> ( u -op T ) e. HrmOp ) )
3	1	fveq1d 5488	. . . . . 6 |- ( t = T -> ( ( u -op t ) ` x ) = ( ( u -op T ) ` x ) )
4	3	oveq1d 5835	. . . . 5 |- ( t = T -> ( ( ( u -op t ) ` x ) .ih x ) = ( ( ( u -op T ) ` x ) .ih x ) )
5	4	breq2d 4036	. . . 4 |- ( t = T -> ( 0 <_ ( ( ( u -op t ) ` x ) .ih x ) <-> 0 <_ ( ( ( u -op T ) ` x ) .ih x ) ) )
6	5	ralbidv 2564	. . 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 691	. 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 5827	. . . 4 |- ( u = U -> ( u -op T ) = ( U -op T ) )
9	8	eleq1d 2350	. . 3 |- ( u = U -> ( ( u -op T ) e. HrmOp <-> ( U -op T ) e. HrmOp ) )
10	8	fveq1d 5488	. . . . . 6 |- ( u = U -> ( ( u -op T ) ` x ) = ( ( U -op T ) ` x ) )
11	10	oveq1d 5835	. . . . 5 |- ( u = U -> ( ( ( u -op T ) ` x ) .ih x ) = ( ( ( U -op T ) ` x ) .ih x ) )
12	11	breq2d 4036	. . . 4 |- ( u = U -> ( 0 <_ ( ( ( u -op T ) ` x ) .ih x ) <-> 0 <_ ( ( ( U -op T ) ` x ) .ih x ) ) )
13	12	ralbidv 2564	. . 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 691	. 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 22428	. 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 4283	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 176   /\ wa 358   = wceq 1623   e. wcel 1685   A.wral 2544   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   0cc0 8733   <_ cle 8864   ~Hchil 21495   .ih csp 21498   -op chod 21516   HrmOpcho 21526   <_op cleo 21534

This theorem is referenced by:  leop  22699  leoprf2  22703

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fv 5229  df-ov 5823  df-leop 22428