Metamath Proof Explorer

Theorem unop

Description: Basic inner product property of a unitary operator. (Contributed by NM, 22-Jan-2006) (New usage is discouraged.)

Ref Expression
Assertion unop 
|- ( ( T e. UniOp /\ A e. ~H /\ B e. ~H ) -> ( ( T ` A ) .ih ( T ` B ) ) = ( A .ih B ) )

Proof

Step Hyp Ref Expression
1 elunop 
 |-  ( T e. UniOp <-> ( T : ~H -onto-> ~H /\ A. x e. ~H A. y e. ~H ( ( T ` x ) .ih ( T ` y ) ) = ( x .ih y ) ) )
2 1 simprbi 
 |-  ( T e. UniOp -> A. x e. ~H A. y e. ~H ( ( T ` x ) .ih ( T ` y ) ) = ( x .ih y ) )
3 2 3ad2ant1 
 |-  ( ( T e. UniOp /\ A e. ~H /\ B e. ~H ) -> A. x e. ~H A. y e. ~H ( ( T ` x ) .ih ( T ` y ) ) = ( x .ih y ) )
4 fveq2 
 |-  ( x = A -> ( T ` x ) = ( T ` A ) )
5 4 oveq1d 
 |-  ( x = A -> ( ( T ` x ) .ih ( T ` y ) ) = ( ( T ` A ) .ih ( T ` y ) ) )
6 oveq1 
 |-  ( x = A -> ( x .ih y ) = ( A .ih y ) )
7 5 6 eqeq12d 
 |-  ( x = A -> ( ( ( T ` x ) .ih ( T ` y ) ) = ( x .ih y ) <-> ( ( T ` A ) .ih ( T ` y ) ) = ( A .ih y ) ) )
8 fveq2 
 |-  ( y = B -> ( T ` y ) = ( T ` B ) )
9 8 oveq2d 
 |-  ( y = B -> ( ( T ` A ) .ih ( T ` y ) ) = ( ( T ` A ) .ih ( T ` B ) ) )
10 oveq2 
 |-  ( y = B -> ( A .ih y ) = ( A .ih B ) )
11 9 10 eqeq12d 
 |-  ( y = B -> ( ( ( T ` A ) .ih ( T ` y ) ) = ( A .ih y ) <-> ( ( T ` A ) .ih ( T ` B ) ) = ( A .ih B ) ) )
12 7 11 rspc2v 
 |-  ( ( A e. ~H /\ B e. ~H ) -> ( A. x e. ~H A. y e. ~H ( ( T ` x ) .ih ( T ` y ) ) = ( x .ih y ) -> ( ( T ` A ) .ih ( T ` B ) ) = ( A .ih B ) ) )
13 12 3adant1 
 |-  ( ( T e. UniOp /\ A e. ~H /\ B e. ~H ) -> ( A. x e. ~H A. y e. ~H ( ( T ` x ) .ih ( T ` y ) ) = ( x .ih y ) -> ( ( T ` A ) .ih ( T ` B ) ) = ( A .ih B ) ) )
14 3 13 mpd 
 |-  ( ( T e. UniOp /\ A e. ~H /\ B e. ~H ) -> ( ( T ` A ) .ih ( T ` B ) ) = ( A .ih B ) )