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Theorem eigre 22411
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 5486 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( T `  A )  =  ( T `  if ( A  e.  ~H ,  A ,  0h )
) )
2 oveq2 5828 . . . . . 6  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( B  .h  A )  =  ( B  .h  if ( A  e.  ~H ,  A ,  0h )
) )
31, 2eqeq12d 2298 . . . . 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 2455 . . . . 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 5838 . . . . . 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 5838 . . . . . 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 2298 . . . . 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 5827 . . . . . 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 2295 . . . . 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 2344 . . . . 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 21579 . . . . 5  |-  0h  e.  ~H
1918elimel 3618 . . . 4  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
20 0cn 8827 . . . . 5  |-  0  e.  CC
2120elimel 3618 . . . 4  |-  if ( B  e.  CC ,  B ,  0 )  e.  CC
2219, 21eigrei 22410 . . 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 3608 . 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 1685    =/= wne 2447   ifcif 3566   ` cfv 5221  (class class class)co 5820   CCcc 8731   RRcr 8732   0cc0 8733   ~Hchil 21495    .h csm 21497    .ih csp 21498   0hc0v 21500
This theorem is referenced by:  eighmre  22539
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-hv0cl 21579  ax-hfvmul 21581  ax-hfi 21654  ax-his1 21657  ax-his3 21659  ax-his4 21660
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-2 9800  df-cj 11580  df-re 11581  df-im 11582
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