Theorem leopg 22663

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

Step	Hyp	Ref	Expression
1		oveq2 5800	. . . 4 |- ( t = T -> ( u -op t ) = ( u -op T ) )
2	1	eleq1d 2324	. . 3 |- ( t = T -> ( ( u -op t ) e. HrmOp <-> ( u -op T ) e. HrmOp ) )
3	1	fveq1d 5460	. . . . . 6 |- ( t = T -> ( ( u -op t ) ` x ) = ( ( u -op T ) ` x ) )
4	3	oveq1d 5807	. . . . 5 |- ( t = T -> ( ( ( u -op t ) ` x ) .ih x ) = ( ( ( u -op T ) ` x ) .ih x ) )
5	4	breq2d 4009	. . . 4 |- ( t = T -> ( 0 <_ ( ( ( u -op t ) ` x ) .ih x ) <-> 0 <_ ( ( ( u -op T ) ` x ) .ih x ) ) )
6	5	ralbidv 2538	. . 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 694	. 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 5799	. . . 4 |- ( u = U -> ( u -op T ) = ( U -op T ) )
9	8	eleq1d 2324	. . 3 |- ( u = U -> ( ( u -op T ) e. HrmOp <-> ( U -op T ) e. HrmOp ) )
10	8	fveq1d 5460	. . . . . 6 |- ( u = U -> ( ( u -op T ) ` x ) = ( ( U -op T ) ` x ) )
11	10	oveq1d 5807	. . . . 5 |- ( u = U -> ( ( ( u -op T ) ` x ) .ih x ) = ( ( ( U -op T ) ` x ) .ih x ) )
12	11	breq2d 4009	. . . 4 |- ( u = U -> ( 0 <_ ( ( ( u -op T ) ` x ) .ih x ) <-> 0 <_ ( ( ( U -op T ) ` x ) .ih x ) ) )
13	12	ralbidv 2538	. . 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 694	. 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 22393	. 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 4256	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 6   <-> wb 178   /\ wa 360   = wceq 1619   e. wcel 1621   A.wral 2518   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   0cc0 8705   <_ cle 8836   ~Hchil 21460   .ih csp 21463   -op chod 21481   HrmOpcho 21491   <_op cleo 21499

This theorem is referenced by:  leop  22664  leoprf2  22668

This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-xp 4675  df-cnv 4677  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fv 4689  df-ov 5795  df-leop 22393