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Metamath Proof Explorer

Theorem lnopunii

Description: If a linear operator (whose range is ~H ) is idempotent in the norm, the operator is unitary. Similar to theorem in AkhiezerGlazman p. 73. (Contributed by NM, 23-Jan-2006) (New usage is discouraged.)

Ref Expression
Hypotheses lnopuni.1 
|- T e. LinOp
lnopuni.2 
|- T : ~H -onto-> ~H
lnopuni.3 
|- A. x e. ~H ( normh ` ( T ` x ) ) = ( normh ` x )
Assertion lnopunii 
|- T e. UniOp

Proof

Step Hyp Ref Expression
1 lnopuni.1 
 |-  T e. LinOp
2 lnopuni.2 
 |-  T : ~H -onto-> ~H
3 lnopuni.3 
 |-  A. x e. ~H ( normh ` ( T ` x ) ) = ( normh ` x )
4 fveq2 
 |-  ( x = if ( x e. ~H , x , 0h ) -> ( T ` x ) = ( T ` if ( x e. ~H , x , 0h ) ) )
5 4 oveq1d 
 |-  ( x = if ( x e. ~H , x , 0h ) -> ( ( T ` x ) .ih ( T ` y ) ) = ( ( T ` if ( x e. ~H , x , 0h ) ) .ih ( T ` y ) ) )
6 oveq1 
 |-  ( x = if ( x e. ~H , x , 0h ) -> ( x .ih y ) = ( if ( x e. ~H , x , 0h ) .ih y ) )
7 5 6 eqeq12d 
 |-  ( x = if ( x e. ~H , x , 0h ) -> ( ( ( T ` x ) .ih ( T ` y ) ) = ( x .ih y ) <-> ( ( T ` if ( x e. ~H , x , 0h ) ) .ih ( T ` y ) ) = ( if ( x e. ~H , x , 0h ) .ih y ) ) )
8 fveq2 
 |-  ( y = if ( y e. ~H , y , 0h ) -> ( T ` y ) = ( T ` if ( y e. ~H , y , 0h ) ) )
9 8 oveq2d 
 |-  ( y = if ( y e. ~H , y , 0h ) -> ( ( T ` if ( x e. ~H , x , 0h ) ) .ih ( T ` y ) ) = ( ( T ` if ( x e. ~H , x , 0h ) ) .ih ( T ` if ( y e. ~H , y , 0h ) ) ) )
10 oveq2 
 |-  ( y = if ( y e. ~H , y , 0h ) -> ( if ( x e. ~H , x , 0h ) .ih y ) = ( if ( x e. ~H , x , 0h ) .ih if ( y e. ~H , y , 0h ) ) )
11 9 10 eqeq12d 
 |-  ( y = if ( y e. ~H , y , 0h ) -> ( ( ( T ` if ( x e. ~H , x , 0h ) ) .ih ( T ` y ) ) = ( if ( x e. ~H , x , 0h ) .ih y ) <-> ( ( T ` if ( x e. ~H , x , 0h ) ) .ih ( T ` if ( y e. ~H , y , 0h ) ) ) = ( if ( x e. ~H , x , 0h ) .ih if ( y e. ~H , y , 0h ) ) ) )
12 ifhvhv0 
 |-  if ( x e. ~H , x , 0h ) e. ~H
13 ifhvhv0 
 |-  if ( y e. ~H , y , 0h ) e. ~H
14 1 3 12 13 lnopunilem2 
 |-  ( ( T ` if ( x e. ~H , x , 0h ) ) .ih ( T ` if ( y e. ~H , y , 0h ) ) ) = ( if ( x e. ~H , x , 0h ) .ih if ( y e. ~H , y , 0h ) )
15 7 11 14 dedth2h 
 |-  ( ( x e. ~H /\ y e. ~H ) -> ( ( T ` x ) .ih ( T ` y ) ) = ( x .ih y ) )
16 15 rgen2 
 |-  A. x e. ~H A. y e. ~H ( ( T ` x ) .ih ( T ` y ) ) = ( x .ih y )
17 elunop 
 |-  ( T e. UniOp <-> ( T : ~H -onto-> ~H /\ A. x e. ~H A. y e. ~H ( ( T ` x ) .ih ( T ` y ) ) = ( x .ih y ) ) )
18 2 16 17 mpbir2an 
 |-  T e. UniOp