Metamath Proof Explorer

Theorem eigrei

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.)

Ref Expression
Hypotheses eigre.1 
|- A e. ~H
eigre.2 
|- B e. CC
Assertion eigrei 
|- ( ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) -> ( ( A .ih ( T ` A ) ) = ( ( T ` A ) .ih A ) <-> B e. RR ) )

Proof

Step Hyp Ref Expression
1 eigre.1 
 |-  A e. ~H
2 eigre.2 
 |-  B e. CC
3 oveq2 
 |-  ( ( T ` A ) = ( B .h A ) -> ( A .ih ( T ` A ) ) = ( A .ih ( B .h A ) ) )
4 his5 
 |-  ( ( B e. CC /\ A e. ~H /\ A e. ~H ) -> ( A .ih ( B .h A ) ) = ( ( * ` B ) x. ( A .ih A ) ) )
5 2 1 1 4 mp3an 
 |-  ( A .ih ( B .h A ) ) = ( ( * ` B ) x. ( A .ih A ) )
6 3 5 eqtrdi 
 |-  ( ( T ` A ) = ( B .h A ) -> ( A .ih ( T ` A ) ) = ( ( * ` B ) x. ( A .ih A ) ) )
7 oveq1 
 |-  ( ( T ` A ) = ( B .h A ) -> ( ( T ` A ) .ih A ) = ( ( B .h A ) .ih A ) )
8 ax-his3 
 |-  ( ( B e. CC /\ A e. ~H /\ A e. ~H ) -> ( ( B .h A ) .ih A ) = ( B x. ( A .ih A ) ) )
9 2 1 1 8 mp3an 
 |-  ( ( B .h A ) .ih A ) = ( B x. ( A .ih A ) )
10 7 9 eqtrdi 
 |-  ( ( T ` A ) = ( B .h A ) -> ( ( T ` A ) .ih A ) = ( B x. ( A .ih A ) ) )
11 6 10 eqeq12d 
 |-  ( ( T ` A ) = ( B .h A ) -> ( ( A .ih ( T ` A ) ) = ( ( T ` A ) .ih A ) <-> ( ( * ` B ) x. ( A .ih A ) ) = ( B x. ( A .ih A ) ) ) )
12 1 1 hicli 
 |-  ( A .ih A ) e. CC
13 ax-his4 
 |-  ( ( A e. ~H /\ A =/= 0h ) -> 0 < ( A .ih A ) )
14 1 13 mpan 
 |-  ( A =/= 0h -> 0 < ( A .ih A ) )
15 14 gt0ne0d 
 |-  ( A =/= 0h -> ( A .ih A ) =/= 0 )
16 2 cjcli 
 |-  ( * ` B ) e. CC
17 mulcan2 
 |-  ( ( ( * ` 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 ) )
18 16 2 17 mp3an12 
 |-  ( ( ( A .ih A ) e. CC /\ ( A .ih A ) =/= 0 ) -> ( ( ( * ` B ) x. ( A .ih A ) ) = ( B x. ( A .ih A ) ) <-> ( * ` B ) = B ) )
19 12 15 18 sylancr 
 |-  ( A =/= 0h -> ( ( ( * ` B ) x. ( A .ih A ) ) = ( B x. ( A .ih A ) ) <-> ( * ` B ) = B ) )
20 11 19 sylan9bb 
 |-  ( ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) -> ( ( A .ih ( T ` A ) ) = ( ( T ` A ) .ih A ) <-> ( * ` B ) = B ) )
21 2 cjrebi 
 |-  ( B e. RR <-> ( * ` B ) = B )
22 20 21 bitr4di 
 |-  ( ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) -> ( ( A .ih ( T ` A ) ) = ( ( T ` A ) .ih A ) <-> B e. RR ) )