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Theorem eigrei 23329
Description: A necessary and sufficient condition (that holds when  T is a Hermitian operator) for an eigenvalue  B to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 21-Jan-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
eigre.1  |-  A  e. 
~H
eigre.2  |-  B  e.  CC
Assertion
Ref Expression
eigrei  |-  ( ( ( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h )  -> 
( ( A  .ih  ( T `  A ) )  =  ( ( T `  A ) 
.ih  A )  <->  B  e.  RR ) )

Proof of Theorem eigrei
StepHypRef Expression
1 oveq2 6081 . . . . 5  |-  ( ( T `  A )  =  ( B  .h  A )  ->  ( A  .ih  ( T `  A ) )  =  ( A  .ih  ( B  .h  A )
) )
2 eigre.2 . . . . . 6  |-  B  e.  CC
3 eigre.1 . . . . . 6  |-  A  e. 
~H
4 his5 22580 . . . . . 6  |-  ( ( B  e.  CC  /\  A  e.  ~H  /\  A  e.  ~H )  ->  ( A  .ih  ( B  .h  A ) )  =  ( ( * `  B )  x.  ( A  .ih  A ) ) )
52, 3, 3, 4mp3an 1279 . . . . 5  |-  ( A 
.ih  ( B  .h  A ) )  =  ( ( * `  B )  x.  ( A  .ih  A ) )
61, 5syl6eq 2483 . . . 4  |-  ( ( T `  A )  =  ( B  .h  A )  ->  ( A  .ih  ( T `  A ) )  =  ( ( * `  B )  x.  ( A  .ih  A ) ) )
7 oveq1 6080 . . . . 5  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( T `  A
)  .ih  A )  =  ( ( B  .h  A )  .ih  A ) )
8 ax-his3 22578 . . . . . 6  |-  ( ( B  e.  CC  /\  A  e.  ~H  /\  A  e.  ~H )  ->  (
( B  .h  A
)  .ih  A )  =  ( B  x.  ( A  .ih  A ) ) )
92, 3, 3, 8mp3an 1279 . . . . 5  |-  ( ( B  .h  A ) 
.ih  A )  =  ( B  x.  ( A  .ih  A ) )
107, 9syl6eq 2483 . . . 4  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( T `  A
)  .ih  A )  =  ( B  x.  ( A  .ih  A ) ) )
116, 10eqeq12d 2449 . . 3  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( A  .ih  ( T `  A )
)  =  ( ( T `  A ) 
.ih  A )  <->  ( (
* `  B )  x.  ( A  .ih  A
) )  =  ( B  x.  ( A 
.ih  A ) ) ) )
123, 3hicli 22575 . . . 4  |-  ( A 
.ih  A )  e.  CC
13 ax-his4 22579 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( A  .ih  A ) )
143, 13mpan 652 . . . . 5  |-  ( A  =/=  0h  ->  0  <  ( A  .ih  A
) )
1514gt0ne0d 9583 . . . 4  |-  ( A  =/=  0h  ->  ( A  .ih  A )  =/=  0 )
162cjcli 11966 . . . . 5  |-  ( * `
 B )  e.  CC
17 mulcan2 9652 . . . . 5  |-  ( ( ( * `  B
)  e.  CC  /\  B  e.  CC  /\  (
( A  .ih  A
)  e.  CC  /\  ( A  .ih  A )  =/=  0 ) )  ->  ( ( ( * `  B )  x.  ( A  .ih  A ) )  =  ( B  x.  ( A 
.ih  A ) )  <-> 
( * `  B
)  =  B ) )
1816, 2, 17mp3an12 1269 . . . 4  |-  ( ( ( A  .ih  A
)  e.  CC  /\  ( A  .ih  A )  =/=  0 )  -> 
( ( ( * `
 B )  x.  ( A  .ih  A
) )  =  ( B  x.  ( A 
.ih  A ) )  <-> 
( * `  B
)  =  B ) )
1912, 15, 18sylancr 645 . . 3  |-  ( A  =/=  0h  ->  (
( ( * `  B )  x.  ( A  .ih  A ) )  =  ( B  x.  ( A  .ih  A ) )  <->  ( * `  B )  =  B ) )
2011, 19sylan9bb 681 . 2  |-  ( ( ( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h )  -> 
( ( A  .ih  ( T `  A ) )  =  ( ( T `  A ) 
.ih  A )  <->  ( * `  B )  =  B ) )
212cjrebi 11971 . 2  |-  ( B  e.  RR  <->  ( * `  B )  =  B )
2220, 21syl6bbr 255 1  |-  ( ( ( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h )  -> 
( ( A  .ih  ( T `  A ) )  =  ( ( T `  A ) 
.ih  A )  <->  B  e.  RR ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982    x. cmul 8987    < clt 9112   *ccj 11893   ~Hchil 22414    .h csm 22416    .ih csp 22417   0hc0v 22419
This theorem is referenced by:  eigre  23330  eigposi  23331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-hfvmul 22500  ax-hfi 22573  ax-his1 22576  ax-his3 22578  ax-his4 22579
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-2 10050  df-cj 11896  df-re 11897  df-im 11898
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