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Theorem eigre 22415
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 5525 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( T `  A )  =  ( T `  if ( A  e.  ~H ,  A ,  0h )
) )
2 oveq2 5866 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( B  .h  A )  =  ( B  .h  if ( A  e.  ~H ,  A ,  0h )
) )
31, 2eqeq12d 2297 . . . . 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 2454 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  =/=  0h  <->  if ( A  e.  ~H ,  A ,  0h )  =/=  0h ) )
53, 4anbi12d 691 . . . 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 19 . . . . . . 7  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  A  =  if ( A  e. 
~H ,  A ,  0h ) )
76, 1oveq12d 5876 . . . . . 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 5876 . . . . . 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 2297 . . . . 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 310 . . . 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 311 . . 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 5865 . . . . . 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 2294 . . . . 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 685 . . . 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 2343 . . . . 5  |-  ( B  =  if ( B  e.  CC ,  B ,  0 )  -> 
( B  e.  RR  <->  if ( B  e.  CC ,  B ,  0 )  e.  RR ) )
1615bibi2d 309 . . . 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 311 . . 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 21583 . . . . 5  |-  0h  e.  ~H
1918elimel 3617 . . . 4  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
20 0cn 8831 . . . . 5  |-  0  e.  CC
2120elimel 3617 . . . 4  |-  if ( B  e.  CC ,  B ,  0 )  e.  CC
2219, 21eigrei 22414 . . 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 3607 . 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 418 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   ifcif 3565   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   ~Hchil 21499    .h csm 21501    .ih csp 21502   0hc0v 21504
This theorem is referenced by:  eighmre  22543
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-hv0cl 21583  ax-hfvmul 21585  ax-hfi 21658  ax-his1 21661  ax-his3 21663  ax-his4 21664
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-cj 11584  df-re 11585  df-im 11586
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