HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  eigre Structured version   Unicode version

Theorem eigre 23330
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, 19-Mar-2006.) (New usage is discouraged.)
Assertion
Ref Expression
eigre  |-  ( ( ( A  e.  ~H  /\  B  e.  CC )  /\  ( ( T `
 A )  =  ( B  .h  A
)  /\  A  =/=  0h ) )  ->  (
( A  .ih  ( T `  A )
)  =  ( ( T `  A ) 
.ih  A )  <->  B  e.  RR ) )

Proof of Theorem eigre
StepHypRef Expression
1 fveq2 5720 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( T `  A )  =  ( T `  if ( A  e.  ~H ,  A ,  0h )
) )
2 oveq2 6081 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( B  .h  A )  =  ( B  .h  if ( A  e.  ~H ,  A ,  0h )
) )
31, 2eqeq12d 2449 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( T `  A
)  =  ( B  .h  A )  <->  ( T `  if ( A  e. 
~H ,  A ,  0h ) )  =  ( B  .h  if ( A  e.  ~H ,  A ,  0h )
) ) )
4 neeq1 2606 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  =/=  0h  <->  if ( A  e.  ~H ,  A ,  0h )  =/=  0h ) )
53, 4anbi12d 692 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( T `  A )  =  ( B  .h  A )  /\  A  =/=  0h ) 
<->  ( ( T `  if ( A  e.  ~H ,  A ,  0h )
)  =  ( B  .h  if ( A  e.  ~H ,  A ,  0h ) )  /\  if ( A  e.  ~H ,  A ,  0h )  =/=  0h ) ) )
6 id 20 . . . . . . 7  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  A  =  if ( A  e. 
~H ,  A ,  0h ) )
76, 1oveq12d 6091 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  .ih  ( T `  A ) )  =  ( if ( A  e.  ~H ,  A ,  0h )  .ih  ( T `  if ( A  e.  ~H ,  A ,  0h ) ) ) )
81, 6oveq12d 6091 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( T `  A
)  .ih  A )  =  ( ( T `
 if ( A  e.  ~H ,  A ,  0h ) )  .ih  if ( A  e.  ~H ,  A ,  0h )
) )
97, 8eqeq12d 2449 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( A  .ih  ( T `  A )
)  =  ( ( T `  A ) 
.ih  A )  <->  ( if ( A  e.  ~H ,  A ,  0h )  .ih  ( T `  if ( A  e.  ~H ,  A ,  0h )
) )  =  ( ( T `  if ( A  e.  ~H ,  A ,  0h )
)  .ih  if ( A  e.  ~H ,  A ,  0h ) ) ) )
109bibi1d 311 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( A  .ih  ( T `  A ) )  =  ( ( T `  A ) 
.ih  A )  <->  B  e.  RR )  <->  ( ( if ( A  e.  ~H ,  A ,  0h )  .ih  ( T `  if ( A  e.  ~H ,  A ,  0h )
) )  =  ( ( T `  if ( A  e.  ~H ,  A ,  0h )
)  .ih  if ( A  e.  ~H ,  A ,  0h ) )  <->  B  e.  RR ) ) )
115, 10imbi12d 312 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( ( T `
 A )  =  ( B  .h  A
)  /\  A  =/=  0h )  ->  ( ( A  .ih  ( T `  A ) )  =  ( ( T `  A )  .ih  A
)  <->  B  e.  RR ) )  <->  ( (
( T `  if ( A  e.  ~H ,  A ,  0h )
)  =  ( B  .h  if ( A  e.  ~H ,  A ,  0h ) )  /\  if ( A  e.  ~H ,  A ,  0h )  =/=  0h )  ->  (
( if ( A  e.  ~H ,  A ,  0h )  .ih  ( T `  if ( A  e.  ~H ,  A ,  0h ) ) )  =  ( ( T `
 if ( A  e.  ~H ,  A ,  0h ) )  .ih  if ( A  e.  ~H ,  A ,  0h )
)  <->  B  e.  RR ) ) ) )
12 oveq1 6080 . . . . . 6  |-  ( B  =  if ( B  e.  CC ,  B ,  0 )  -> 
( B  .h  if ( A  e.  ~H ,  A ,  0h )
)  =  ( if ( B  e.  CC ,  B ,  0 )  .h  if ( A  e.  ~H ,  A ,  0h ) ) )
1312eqeq2d 2446 . . . . 5  |-  ( B  =  if ( B  e.  CC ,  B ,  0 )  -> 
( ( T `  if ( A  e.  ~H ,  A ,  0h )
)  =  ( B  .h  if ( A  e.  ~H ,  A ,  0h ) )  <->  ( T `  if ( A  e. 
~H ,  A ,  0h ) )  =  ( if ( B  e.  CC ,  B , 
0 )  .h  if ( A  e.  ~H ,  A ,  0h )
) ) )
1413anbi1d 686 . . . 4  |-  ( B  =  if ( B  e.  CC ,  B ,  0 )  -> 
( ( ( T `
 if ( A  e.  ~H ,  A ,  0h ) )  =  ( B  .h  if ( A  e.  ~H ,  A ,  0h )
)  /\  if ( A  e.  ~H ,  A ,  0h )  =/=  0h ) 
<->  ( ( T `  if ( A  e.  ~H ,  A ,  0h )
)  =  ( if ( B  e.  CC ,  B ,  0 )  .h  if ( A  e.  ~H ,  A ,  0h ) )  /\  if ( A  e.  ~H ,  A ,  0h )  =/=  0h ) ) )
15 eleq1 2495 . . . . 5  |-  ( B  =  if ( B  e.  CC ,  B ,  0 )  -> 
( B  e.  RR  <->  if ( B  e.  CC ,  B ,  0 )  e.  RR ) )
1615bibi2d 310 . . . 4  |-  ( B  =  if ( B  e.  CC ,  B ,  0 )  -> 
( ( ( if ( A  e.  ~H ,  A ,  0h )  .ih  ( T `  if ( A  e.  ~H ,  A ,  0h )
) )  =  ( ( T `  if ( A  e.  ~H ,  A ,  0h )
)  .ih  if ( A  e.  ~H ,  A ,  0h ) )  <->  B  e.  RR )  <->  ( ( if ( A  e.  ~H ,  A ,  0h )  .ih  ( T `  if ( A  e.  ~H ,  A ,  0h )
) )  =  ( ( T `  if ( A  e.  ~H ,  A ,  0h )
)  .ih  if ( A  e.  ~H ,  A ,  0h ) )  <->  if ( B  e.  CC ,  B ,  0 )  e.  RR ) ) )
1714, 16imbi12d 312 . . 3  |-  ( B  =  if ( B  e.  CC ,  B ,  0 )  -> 
( ( ( ( T `  if ( A  e.  ~H ,  A ,  0h )
)  =  ( B  .h  if ( A  e.  ~H ,  A ,  0h ) )  /\  if ( A  e.  ~H ,  A ,  0h )  =/=  0h )  ->  (
( if ( A  e.  ~H ,  A ,  0h )  .ih  ( T `  if ( A  e.  ~H ,  A ,  0h ) ) )  =  ( ( T `
 if ( A  e.  ~H ,  A ,  0h ) )  .ih  if ( A  e.  ~H ,  A ,  0h )
)  <->  B  e.  RR ) )  <->  ( (
( T `  if ( A  e.  ~H ,  A ,  0h )
)  =  ( if ( B  e.  CC ,  B ,  0 )  .h  if ( A  e.  ~H ,  A ,  0h ) )  /\  if ( A  e.  ~H ,  A ,  0h )  =/=  0h )  ->  (
( if ( A  e.  ~H ,  A ,  0h )  .ih  ( T `  if ( A  e.  ~H ,  A ,  0h ) ) )  =  ( ( T `
 if ( A  e.  ~H ,  A ,  0h ) )  .ih  if ( A  e.  ~H ,  A ,  0h )
)  <->  if ( B  e.  CC ,  B , 
0 )  e.  RR ) ) ) )
18 ax-hv0cl 22498 . . . . 5  |-  0h  e.  ~H
1918elimel 3783 . . . 4  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
20 0cn 9076 . . . . 5  |-  0  e.  CC
2120elimel 3783 . . . 4  |-  if ( B  e.  CC ,  B ,  0 )  e.  CC
2219, 21eigrei 23329 . . 3  |-  ( ( ( T `  if ( A  e.  ~H ,  A ,  0h )
)  =  ( if ( B  e.  CC ,  B ,  0 )  .h  if ( A  e.  ~H ,  A ,  0h ) )  /\  if ( A  e.  ~H ,  A ,  0h )  =/=  0h )  ->  (
( if ( A  e.  ~H ,  A ,  0h )  .ih  ( T `  if ( A  e.  ~H ,  A ,  0h ) ) )  =  ( ( T `
 if ( A  e.  ~H ,  A ,  0h ) )  .ih  if ( A  e.  ~H ,  A ,  0h )
)  <->  if ( B  e.  CC ,  B , 
0 )  e.  RR ) )
2311, 17, 22dedth2h 3773 . 2  |-  ( ( A  e.  ~H  /\  B  e.  CC )  ->  ( ( ( T `
 A )  =  ( B  .h  A
)  /\  A  =/=  0h )  ->  ( ( A  .ih  ( T `  A ) )  =  ( ( T `  A )  .ih  A
)  <->  B  e.  RR ) ) )
2423imp 419 1  |-  ( ( ( A  e.  ~H  /\  B  e.  CC )  /\  ( ( 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   ifcif 3731   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   ~Hchil 22414    .h csm 22416    .ih csp 22417   0hc0v 22419
This theorem is referenced by:  eighmre  23458
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-hv0cl 22498  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
  Copyright terms: Public domain W3C validator