Theorem ococ

Description: Complement of complement of a closed subspace of Hilbert space. Theorem 3.7(ii) of Beran p. 102. (Contributed by NM, 11-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	ococ	⊢ ( 𝐴 ∈ Cℋ → ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		2fveq3	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) = ( ⊥ ‘ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ) )
2		id	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) )
3	1 2	eqeq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) = 𝐴 ↔ ( ⊥ ‘ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ) = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) )
4		ifchhv	⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∈ Cℋ
5	4	ococi	⊢ ( ⊥ ‘ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ) = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ )
6	3 5	dedth	⊢ ( 𝐴 ∈ Cℋ → ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) = 𝐴 )