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Theorem eigposi 23292
Description: A sufficient condition (first conjunct pair, that holds when  T is a positive operator) for an eigenvalue  B (second conjunct pair) to be nonnegative. Remark (ii) in [Hughes] p. 137. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
eigpos.1  |-  A  e. 
~H
eigpos.2  |-  B  e.  CC
Assertion
Ref Expression
eigposi  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  ( B  e.  RR  /\  0  <_  B ) )

Proof of Theorem eigposi
StepHypRef Expression
1 oveq2 6048 . . . . . . . 8  |-  ( ( T `  A )  =  ( B  .h  A )  ->  ( A  .ih  ( T `  A ) )  =  ( A  .ih  ( B  .h  A )
) )
21eleq1d 2470 . . . . . . 7  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( A  .ih  ( T `  A )
)  e.  RR  <->  ( A  .ih  ( B  .h  A
) )  e.  RR ) )
3 oveq1 6047 . . . . . . . . 9  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( T `  A
)  .ih  A )  =  ( ( B  .h  A )  .ih  A ) )
41, 3eqeq12d 2418 . . . . . . . 8  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( A  .ih  ( T `  A )
)  =  ( ( T `  A ) 
.ih  A )  <->  ( A  .ih  ( B  .h  A
) )  =  ( ( B  .h  A
)  .ih  A )
) )
5 eigpos.1 . . . . . . . . 9  |-  A  e. 
~H
6 eigpos.2 . . . . . . . . . 10  |-  B  e.  CC
76, 5hvmulcli 22470 . . . . . . . . 9  |-  ( B  .h  A )  e. 
~H
8 hire 22549 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  ( B  .h  A
)  e.  ~H )  ->  ( ( A  .ih  ( B  .h  A
) )  e.  RR  <->  ( A  .ih  ( B  .h  A ) )  =  ( ( B  .h  A )  .ih  A ) ) )
95, 7, 8mp2an 654 . . . . . . . 8  |-  ( ( A  .ih  ( B  .h  A ) )  e.  RR  <->  ( A  .ih  ( B  .h  A
) )  =  ( ( B  .h  A
)  .ih  A )
)
104, 9syl6rbbr 256 . . . . . . 7  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( A  .ih  ( B  .h  A )
)  e.  RR  <->  ( A  .ih  ( T `  A
) )  =  ( ( T `  A
)  .ih  A )
) )
112, 10bitrd 245 . . . . . 6  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( A  .ih  ( T `  A )
)  e.  RR  <->  ( A  .ih  ( T `  A
) )  =  ( ( T `  A
)  .ih  A )
) )
1211adantr 452 . . . . 5  |-  ( ( ( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h )  -> 
( ( A  .ih  ( T `  A ) )  e.  RR  <->  ( A  .ih  ( T `  A
) )  =  ( ( T `  A
)  .ih  A )
) )
135, 6eigrei 23290 . . . . 5  |-  ( ( ( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h )  -> 
( ( A  .ih  ( T `  A ) )  =  ( ( T `  A ) 
.ih  A )  <->  B  e.  RR ) )
1412, 13bitrd 245 . . . 4  |-  ( ( ( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h )  -> 
( ( A  .ih  ( T `  A ) )  e.  RR  <->  B  e.  RR ) )
1514biimpac 473 . . 3  |-  ( ( ( A  .ih  ( T `  A )
)  e.  RR  /\  ( ( T `  A )  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  B  e.  RR )
1615adantlr 696 . 2  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  B  e.  RR )
17 ax-his4 22540 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( A  .ih  A ) )
185, 17mpan 652 . . . 4  |-  ( A  =/=  0h  ->  0  <  ( A  .ih  A
) )
1918ad2antll 710 . . 3  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  0  <  ( A  .ih  A ) )
20 simplr 732 . . . 4  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  0  <_  ( A  .ih  ( T `  A ) ) )
211ad2antrl 709 . . . . 5  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  ( A  .ih  ( T `  A ) )  =  ( A 
.ih  ( B  .h  A ) ) )
22 his5 22541 . . . . . . 7  |-  ( ( B  e.  CC  /\  A  e.  ~H  /\  A  e.  ~H )  ->  ( A  .ih  ( B  .h  A ) )  =  ( ( * `  B )  x.  ( A  .ih  A ) ) )
236, 5, 5, 22mp3an 1279 . . . . . 6  |-  ( A 
.ih  ( B  .h  A ) )  =  ( ( * `  B )  x.  ( A  .ih  A ) )
2416cjred 11986 . . . . . . 7  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  ( * `  B )  =  B )
2524oveq1d 6055 . . . . . 6  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  ( ( * `
 B )  x.  ( A  .ih  A
) )  =  ( B  x.  ( A 
.ih  A ) ) )
2623, 25syl5eq 2448 . . . . 5  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  ( A  .ih  ( B  .h  A
) )  =  ( B  x.  ( A 
.ih  A ) ) )
2721, 26eqtrd 2436 . . . 4  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  ( A  .ih  ( T `  A ) )  =  ( B  x.  ( A  .ih  A ) ) )
2820, 27breqtrd 4196 . . 3  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  0  <_  ( B  x.  ( A  .ih  A ) ) )
29 hiidrcl 22550 . . . . 5  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
305, 29ax-mp 8 . . . 4  |-  ( A 
.ih  A )  e.  RR
31 prodge02 9814 . . . 4  |-  ( ( ( B  e.  RR  /\  ( A  .ih  A
)  e.  RR )  /\  ( 0  < 
( A  .ih  A
)  /\  0  <_  ( B  x.  ( A 
.ih  A ) ) ) )  ->  0  <_  B )
3230, 31mpanl2 663 . . 3  |-  ( ( B  e.  RR  /\  ( 0  <  ( A  .ih  A )  /\  0  <_  ( B  x.  ( A  .ih  A ) ) ) )  -> 
0  <_  B )
3316, 19, 28, 32syl12anc 1182 . 2  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  0  <_  B
)
3416, 33jca 519 1  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  ( B  e.  RR  /\  0  <_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946    x. cmul 8951    < clt 9076    <_ cle 9077   *ccj 11856   ~Hchil 22375    .h csm 22377    .ih csp 22378   0hc0v 22380
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-13 1723  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-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-hfvmul 22461  ax-hfi 22534  ax-his1 22537  ax-his3 22539  ax-his4 22540
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-cj 11859  df-re 11860  df-im 11861
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