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Theorem eigposi 22412
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 5828 . . . . . . . 8  |-  ( ( T `  A )  =  ( B  .h  A )  ->  ( A  .ih  ( T `  A ) )  =  ( A  .ih  ( B  .h  A )
) )
21eleq1d 2350 . . . . . . 7  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( A  .ih  ( T `  A )
)  e.  RR  <->  ( A  .ih  ( B  .h  A
) )  e.  RR ) )
3 oveq1 5827 . . . . . . . . 9  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( T `  A
)  .ih  A )  =  ( ( B  .h  A )  .ih  A ) )
41, 3eqeq12d 2298 . . . . . . . 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 21590 . . . . . . . . 9  |-  ( B  .h  A )  e. 
~H
8 hire 21669 . . . . . . . . 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 653 . . . . . . . 8  |-  ( ( A  .ih  ( B  .h  A ) )  e.  RR  <->  ( A  .ih  ( B  .h  A
) )  =  ( ( B  .h  A
)  .ih  A )
)
104, 9syl6rbbr 255 . . . . . . 7  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( A  .ih  ( B  .h  A )
)  e.  RR  <->  ( A  .ih  ( T `  A
) )  =  ( ( T `  A
)  .ih  A )
) )
112, 10bitrd 244 . . . . . 6  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( A  .ih  ( T `  A )
)  e.  RR  <->  ( A  .ih  ( T `  A
) )  =  ( ( T `  A
)  .ih  A )
) )
1211adantr 451 . . . . 5  |-  ( ( ( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h )  -> 
( ( A  .ih  ( T `  A ) )  e.  RR  <->  ( A  .ih  ( T `  A
) )  =  ( ( T `  A
)  .ih  A )
) )
135, 6eigrei 22410 . . . . 5  |-  ( ( ( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h )  -> 
( ( A  .ih  ( T `  A ) )  =  ( ( T `  A ) 
.ih  A )  <->  B  e.  RR ) )
1412, 13bitrd 244 . . . 4  |-  ( ( ( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h )  -> 
( ( A  .ih  ( T `  A ) )  e.  RR  <->  B  e.  RR ) )
1514biimpac 472 . . 3  |-  ( ( ( A  .ih  ( T `  A )
)  e.  RR  /\  ( ( T `  A )  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  B  e.  RR )
1615adantlr 695 . 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 21660 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( A  .ih  A ) )
185, 17mpan 651 . . . 4  |-  ( A  =/=  0h  ->  0  <  ( A  .ih  A
) )
1918ad2antll 709 . . 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 731 . . . 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 708 . . . . 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 21661 . . . . . . 7  |-  ( ( B  e.  CC  /\  A  e.  ~H  /\  A  e.  ~H )  ->  ( A  .ih  ( B  .h  A ) )  =  ( ( * `  B )  x.  ( A  .ih  A ) ) )
236, 5, 5, 22mp3an 1277 . . . . . 6  |-  ( A 
.ih  ( B  .h  A ) )  =  ( ( * `  B )  x.  ( A  .ih  A ) )
2416cjred 11707 . . . . . . 7  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  ( * `  B )  =  B )
2524oveq1d 5835 . . . . . 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 2328 . . . . 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 2316 . . . 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 4048 . . 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 21670 . . . . 5  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
305, 29ax-mp 8 . . . 4  |-  ( A 
.ih  A )  e.  RR
31 prodge02 9600 . . . 4  |-  ( ( ( B  e.  RR  /\  ( A  .ih  A
)  e.  RR )  /\  ( 0  < 
( A  .ih  A
)  /\  0  <_  ( B  x.  ( A 
.ih  A ) ) ) )  ->  0  <_  B )
3230, 31mpanl2 662 . . 3  |-  ( ( B  e.  RR  /\  ( 0  <  ( A  .ih  A )  /\  0  <_  ( B  x.  ( A  .ih  A ) ) ) )  -> 
0  <_  B )
3316, 19, 28, 32syl12anc 1180 . 2  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  0  <_  B
)
3416, 33jca 518 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 176    /\ wa 358    = wceq 1623    e. wcel 1685    =/= wne 2447   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   CCcc 8731   RRcr 8732   0cc0 8733    x. cmul 8738    < clt 8863    <_ cle 8864   *ccj 11577   ~Hchil 21495    .h csm 21497    .ih csp 21498   0hc0v 21500
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-13 1687  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-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-hfvmul 21581  ax-hfi 21654  ax-his1 21657  ax-his3 21659  ax-his4 21660
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-2 9800  df-cj 11580  df-re 11581  df-im 11582
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