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Theorem eigposi 26578
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 6303 . . . . . . . 8  |-  ( ( T `  A )  =  ( B  .h  A )  ->  ( A  .ih  ( T `  A ) )  =  ( A  .ih  ( B  .h  A )
) )
21eleq1d 2536 . . . . . . 7  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( A  .ih  ( T `  A )
)  e.  RR  <->  ( A  .ih  ( B  .h  A
) )  e.  RR ) )
3 oveq1 6302 . . . . . . . . 9  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( T `  A
)  .ih  A )  =  ( ( B  .h  A )  .ih  A ) )
41, 3eqeq12d 2489 . . . . . . . 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 25754 . . . . . . . . 9  |-  ( B  .h  A )  e. 
~H
8 hire 25834 . . . . . . . . 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 672 . . . . . . . 8  |-  ( ( A  .ih  ( B  .h  A ) )  e.  RR  <->  ( A  .ih  ( B  .h  A
) )  =  ( ( B  .h  A
)  .ih  A )
)
104, 9syl6rbbr 264 . . . . . . 7  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( A  .ih  ( B  .h  A )
)  e.  RR  <->  ( A  .ih  ( T `  A
) )  =  ( ( T `  A
)  .ih  A )
) )
112, 10bitrd 253 . . . . . 6  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( A  .ih  ( T `  A )
)  e.  RR  <->  ( A  .ih  ( T `  A
) )  =  ( ( T `  A
)  .ih  A )
) )
1211adantr 465 . . . . 5  |-  ( ( ( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h )  -> 
( ( A  .ih  ( T `  A ) )  e.  RR  <->  ( A  .ih  ( T `  A
) )  =  ( ( T `  A
)  .ih  A )
) )
135, 6eigrei 26576 . . . . 5  |-  ( ( ( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h )  -> 
( ( A  .ih  ( T `  A ) )  =  ( ( T `  A ) 
.ih  A )  <->  B  e.  RR ) )
1412, 13bitrd 253 . . . 4  |-  ( ( ( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h )  -> 
( ( A  .ih  ( T `  A ) )  e.  RR  <->  B  e.  RR ) )
1514biimpac 486 . . 3  |-  ( ( ( A  .ih  ( T `  A )
)  e.  RR  /\  ( ( T `  A )  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  B  e.  RR )
1615adantlr 714 . 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 25825 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( A  .ih  A ) )
185, 17mpan 670 . . . 4  |-  ( A  =/=  0h  ->  0  <  ( A  .ih  A
) )
1918ad2antll 728 . . 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 754 . . . 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 727 . . . . 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 25826 . . . . . . 7  |-  ( ( B  e.  CC  /\  A  e.  ~H  /\  A  e.  ~H )  ->  ( A  .ih  ( B  .h  A ) )  =  ( ( * `  B )  x.  ( A  .ih  A ) ) )
236, 5, 5, 22mp3an 1324 . . . . . 6  |-  ( A 
.ih  ( B  .h  A ) )  =  ( ( * `  B )  x.  ( A  .ih  A ) )
2416cjred 13039 . . . . . . 7  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  ( * `  B )  =  B )
2524oveq1d 6310 . . . . . 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 2520 . . . . 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 2508 . . . 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 4477 . . 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 25835 . . . . 5  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
305, 29ax-mp 5 . . . 4  |-  ( A 
.ih  A )  e.  RR
31 prodge02 10402 . . . 4  |-  ( ( ( B  e.  RR  /\  ( A  .ih  A
)  e.  RR )  /\  ( 0  < 
( A  .ih  A
)  /\  0  <_  ( B  x.  ( A 
.ih  A ) ) ) )  ->  0  <_  B )
3230, 31mpanl2 681 . . 3  |-  ( ( B  e.  RR  /\  ( 0  <  ( A  .ih  A )  /\  0  <_  ( B  x.  ( A  .ih  A ) ) ) )  -> 
0  <_  B )
3316, 19, 28, 32syl12anc 1226 . 2  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  0  <_  B
)
3416, 33jca 532 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504    x. cmul 9509    < clt 9640    <_ cle 9641   *ccj 12909   ~Hchil 25659    .h csm 25661    .ih csp 25662   0hc0v 25664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-hfvmul 25745  ax-hfi 25819  ax-his1 25822  ax-his3 25824  ax-his4 25825
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-2 10606  df-cj 12912  df-re 12913  df-im 12914
This theorem is referenced by: (None)
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