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Theorem eigposi 22432
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 5882 . . . . . . . 8  |-  ( ( T `  A )  =  ( B  .h  A )  ->  ( A  .ih  ( T `  A ) )  =  ( A  .ih  ( B  .h  A )
) )
21eleq1d 2362 . . . . . . 7  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( A  .ih  ( T `  A )
)  e.  RR  <->  ( A  .ih  ( B  .h  A
) )  e.  RR ) )
3 oveq1 5881 . . . . . . . . 9  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( T `  A
)  .ih  A )  =  ( ( B  .h  A )  .ih  A ) )
41, 3eqeq12d 2310 . . . . . . . 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 21610 . . . . . . . . 9  |-  ( B  .h  A )  e. 
~H
8 hire 21689 . . . . . . . . 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 22430 . . . . 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 21680 . . . . 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 21681 . . . . . . 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 11727 . . . . . . 7  |-  ( ( ( ( A  .ih  ( T `  A ) )  e.  RR  /\  0  <_  ( A  .ih  ( T `  A ) ) )  /\  (
( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h ) )  ->  ( * `  B )  =  B )
2524oveq1d 5889 . . . . . 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 2340 . . . . 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 2328 . . . 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 4063 . . 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 21690 . . . . 5  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
305, 29ax-mp 8 . . . 4  |-  ( A 
.ih  A )  e.  RR
31 prodge02 9620 . . . 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    x. cmul 8758    < clt 8883    <_ cle 8884   *ccj 11597   ~Hchil 21515    .h csm 21517    .ih csp 21518   0hc0v 21520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-hfvmul 21601  ax-hfi 21674  ax-his1 21677  ax-his3 21679  ax-his4 21680
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-cj 11600  df-re 11601  df-im 11602
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