Metamath Proof Explorer

Theorem cnlnadj

Description: Every continuous linear operator has an adjoint. Theorem 3.10 of Beran p. 104. (Contributed by NM, 18-Feb-2006) (New usage is discouraged.)

Ref Expression
Assertion cnlnadj 
|- ( T e. ( LinOp i^i ContOp ) -> E. t e. ( LinOp i^i ContOp ) A. x e. ~H A. y e. ~H ( ( T ` x ) .ih y ) = ( x .ih ( t ` y ) ) )

Proof

Step Hyp Ref Expression
1 cnlnadjeu 
 |-  ( T e. ( LinOp i^i ContOp ) -> E! t e. ( LinOp i^i ContOp ) A. x e. ~H A. y e. ~H ( ( T ` x ) .ih y ) = ( x .ih ( t ` y ) ) )
2 reurex 
 |-  ( E! t e. ( LinOp i^i ContOp ) A. x e. ~H A. y e. ~H ( ( T ` x ) .ih y ) = ( x .ih ( t ` y ) ) -> E. t e. ( LinOp i^i ContOp ) A. x e. ~H A. y e. ~H ( ( T ` x ) .ih y ) = ( x .ih ( t ` y ) ) )
3 1 2 syl 
 |-  ( T e. ( LinOp i^i ContOp ) -> E. t e. ( LinOp i^i ContOp ) A. x e. ~H A. y e. ~H ( ( T ` x ) .ih y ) = ( x .ih ( t ` y ) ) )