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Theorem eigrei 22375
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, 21-Jan-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
eigre.1  |-  A  e. 
~H
eigre.2  |-  B  e.  CC
Assertion
Ref Expression
eigrei  |-  ( ( ( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h )  -> 
( ( A  .ih  ( T `  A ) )  =  ( ( T `  A ) 
.ih  A )  <->  B  e.  RR ) )

Proof of Theorem eigrei
StepHypRef Expression
1 oveq2 5800 . . . . 5  |-  ( ( T `  A )  =  ( B  .h  A )  ->  ( A  .ih  ( T `  A ) )  =  ( A  .ih  ( B  .h  A )
) )
2 eigre.2 . . . . . 6  |-  B  e.  CC
3 eigre.1 . . . . . 6  |-  A  e. 
~H
4 his5 21626 . . . . . 6  |-  ( ( B  e.  CC  /\  A  e.  ~H  /\  A  e.  ~H )  ->  ( A  .ih  ( B  .h  A ) )  =  ( ( * `  B )  x.  ( A  .ih  A ) ) )
52, 3, 3, 4mp3an 1282 . . . . 5  |-  ( A 
.ih  ( B  .h  A ) )  =  ( ( * `  B )  x.  ( A  .ih  A ) )
61, 5syl6eq 2306 . . . 4  |-  ( ( T `  A )  =  ( B  .h  A )  ->  ( A  .ih  ( T `  A ) )  =  ( ( * `  B )  x.  ( A  .ih  A ) ) )
7 oveq1 5799 . . . . 5  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( T `  A
)  .ih  A )  =  ( ( B  .h  A )  .ih  A ) )
8 ax-his3 21624 . . . . . 6  |-  ( ( B  e.  CC  /\  A  e.  ~H  /\  A  e.  ~H )  ->  (
( B  .h  A
)  .ih  A )  =  ( B  x.  ( A  .ih  A ) ) )
92, 3, 3, 8mp3an 1282 . . . . 5  |-  ( ( B  .h  A ) 
.ih  A )  =  ( B  x.  ( A  .ih  A ) )
107, 9syl6eq 2306 . . . 4  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( T `  A
)  .ih  A )  =  ( B  x.  ( A  .ih  A ) ) )
116, 10eqeq12d 2272 . . 3  |-  ( ( T `  A )  =  ( B  .h  A )  ->  (
( A  .ih  ( T `  A )
)  =  ( ( T `  A ) 
.ih  A )  <->  ( (
* `  B )  x.  ( A  .ih  A
) )  =  ( B  x.  ( A 
.ih  A ) ) ) )
123, 3hicli 21621 . . . 4  |-  ( A 
.ih  A )  e.  CC
13 ax-his4 21625 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( A  .ih  A ) )
143, 13mpan 654 . . . . 5  |-  ( A  =/=  0h  ->  0  <  ( A  .ih  A
) )
1514gt0ne0d 9305 . . . 4  |-  ( A  =/=  0h  ->  ( A  .ih  A )  =/=  0 )
162cjcli 11620 . . . . 5  |-  ( * `
 B )  e.  CC
17 mulcan2 9374 . . . . 5  |-  ( ( ( * `  B
)  e.  CC  /\  B  e.  CC  /\  (
( A  .ih  A
)  e.  CC  /\  ( A  .ih  A )  =/=  0 ) )  ->  ( ( ( * `  B )  x.  ( A  .ih  A ) )  =  ( B  x.  ( A 
.ih  A ) )  <-> 
( * `  B
)  =  B ) )
1816, 2, 17mp3an12 1272 . . . 4  |-  ( ( ( A  .ih  A
)  e.  CC  /\  ( A  .ih  A )  =/=  0 )  -> 
( ( ( * `
 B )  x.  ( A  .ih  A
) )  =  ( B  x.  ( A 
.ih  A ) )  <-> 
( * `  B
)  =  B ) )
1912, 15, 18sylancr 647 . . 3  |-  ( A  =/=  0h  ->  (
( ( * `  B )  x.  ( A  .ih  A ) )  =  ( B  x.  ( A  .ih  A ) )  <->  ( * `  B )  =  B ) )
2011, 19sylan9bb 683 . 2  |-  ( ( ( T `  A
)  =  ( B  .h  A )  /\  A  =/=  0h )  -> 
( ( A  .ih  ( T `  A ) )  =  ( ( T `  A ) 
.ih  A )  <->  ( * `  B )  =  B ) )
212cjrebi 11625 . 2  |-  ( B  e.  RR  <->  ( * `  B )  =  B )
2220, 21syl6bbr 256 1  |-  ( ( ( 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   CCcc 8703   RRcr 8704   0cc0 8705    x. cmul 8710    < clt 8835   *ccj 11547   ~Hchil 21460    .h csm 21462    .ih csp 21463   0hc0v 21465
This theorem is referenced by:  eigre  22376  eigposi  22377
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-hfvmul 21546  ax-hfi 21619  ax-his1 21622  ax-his3 21624  ax-his4 21625
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-po 4286  df-so 4287  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-iota 6225  df-riota 6272  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-2 9772  df-cj 11550  df-re 11551  df-im 11552
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