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Theorem eigre 25239
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 5691 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( T `  A )  =  ( T `  if ( A  e.  ~H ,  A ,  0h )
) )
2 oveq2 6099 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( B  .h  A )  =  ( B  .h  if ( A  e.  ~H ,  A ,  0h )
) )
31, 2eqeq12d 2457 . . . . 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 2616 . . . . 5  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  =/=  0h  <->  if ( A  e.  ~H ,  A ,  0h )  =/=  0h ) )
53, 4anbi12d 710 . . . 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 22 . . . . . . 7  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  A  =  if ( A  e. 
~H ,  A ,  0h ) )
76, 1oveq12d 6109 . . . . . 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 6109 . . . . . 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 2457 . . . . 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 319 . . . 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 320 . . 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 6098 . . . . . 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 2454 . . . . 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 704 . . . 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 2503 . . . . 5  |-  ( B  =  if ( B  e.  CC ,  B ,  0 )  -> 
( B  e.  RR  <->  if ( B  e.  CC ,  B ,  0 )  e.  RR ) )
1615bibi2d 318 . . . 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 320 . . 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 ifhvhv0 24424 . . . 4  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
19 0cn 9378 . . . . 5  |-  0  e.  CC
2019elimel 3852 . . . 4  |-  if ( B  e.  CC ,  B ,  0 )  e.  CC
2118, 20eigrei 25238 . . 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 ) )
2211, 17, 21dedth2h 3842 . 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 ) ) )
2322imp 429 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   ifcif 3791   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   ~Hchil 24321    .h csm 24323    .ih csp 24324   0hc0v 24326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-hv0cl 24405  ax-hfvmul 24407  ax-hfi 24481  ax-his1 24484  ax-his3 24486  ax-his4 24487
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-2 10380  df-cj 12588  df-re 12589  df-im 12590
This theorem is referenced by:  eighmre  25367
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