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Theorem eigre 23179
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 5661 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( T `  A )  =  ( T `  if ( A  e.  ~H ,  A ,  0h )
) )
2 oveq2 6021 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( B  .h  A )  =  ( B  .h  if ( A  e.  ~H ,  A ,  0h )
) )
31, 2eqeq12d 2394 . . . . 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 2551 . . . . 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 6031 . . . . . 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 6031 . . . . . 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 2394 . . . . 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 6020 . . . . . 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 2391 . . . . 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 2440 . . . . 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 22347 . . . . 5  |-  0h  e.  ~H
1918elimel 3727 . . . 4  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
20 0cn 9010 . . . . 5  |-  0  e.  CC
2120elimel 3727 . . . 4  |-  if ( B  e.  CC ,  B ,  0 )  e.  CC
2219, 21eigrei 23178 . . 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 3717 . 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 1649    e. wcel 1717    =/= wne 2543   ifcif 3675   ` cfv 5387  (class class class)co 6013   CCcc 8914   RRcr 8915   0cc0 8916   ~Hchil 22263    .h csm 22265    .ih csp 22266   0hc0v 22268
This theorem is referenced by:  eighmre  23307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-hv0cl 22347  ax-hfvmul 22349  ax-hfi 22422  ax-his1 22425  ax-his3 22427  ax-his4 22428
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-2 9983  df-cj 11824  df-re 11825  df-im 11826
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