Metamath Proof Explorer

Theorem eigre

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.)

Ref Expression
Assertion 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

Step Hyp Ref Expression
1 fveq2 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( T ` A ) = ( T ` if ( A e. ~H , A , 0h ) ) )
2 oveq2 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( B .h A ) = ( B .h if ( A e. ~H , A , 0h ) ) )
3 1 2 eqeq12d 
 |-  ( 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 
 |-  ( A = if ( A e. ~H , A , 0h ) -> ( A =/= 0h <-> if ( A e. ~H , A , 0h ) =/= 0h ) )
5 3 4 anbi12d 
 |-  ( 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 
 |-  ( A = if ( A e. ~H , A , 0h ) -> A = if ( A e. ~H , A , 0h ) )
7 6 1 oveq12d 
 |-  ( 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 ) ) ) )
8 1 6 oveq12d 
 |-  ( 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 ) ) )
9 7 8 eqeq12d 
 |-  ( 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 ) ) ) )
10 9 bibi1d 
 |-  ( 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 ) ) )
11 5 10 imbi12d 
 |-  ( 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 
 |-  ( 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 ) ) )
13 12 eqeq2d 
 |-  ( 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 ) ) ) )
14 13 anbi1d 
 |-  ( 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 
 |-  ( B = if ( B e. CC , B , 0 ) -> ( B e. RR <-> if ( B e. CC , B , 0 ) e. RR ) )
16 15 bibi2d 
 |-  ( 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 ) ) )
17 14 16 imbi12d 
 |-  ( 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 
 |-  if ( A e. ~H , A , 0h ) e. ~H
19 0cn 
 |-  0 e. CC
20 19 elimel 
 |-  if ( B e. CC , B , 0 ) e. CC
21 18 20 eigrei 
 |-  ( ( ( 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 ) )
22 11 17 21 dedth2h 
 |-  ( ( A e. ~H /\ B e. CC ) -> ( ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) -> ( ( A .ih ( T ` A ) ) = ( ( T ` A ) .ih A ) <-> B e. RR ) ) )
23 22 imp 
 |-  ( ( ( A e. ~H /\ B e. CC ) /\ ( ( T ` A ) = ( B .h A ) /\ A =/= 0h ) ) -> ( ( A .ih ( T ` A ) ) = ( ( T ` A ) .ih A ) <-> B e. RR ) )