Theorem chjo

Description: The join of a closed subspace and its orthocomplement is all of Hilbert space. (Contributed by NM, 31-Oct-2005) (New usage is discouraged.)

	Ref	Expression
Assertion	chjo	⊢ ( 𝐴 ∈ Cℋ → ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐴 ) ) = ℋ )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) )
2		fveq2	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ⊥ ‘ 𝐴 ) = ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) )
3	1 2	oveq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐴 ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ) )
4	3	eqeq1d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐴 ) ) = ℋ ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ) = ℋ ) )
5		ifchhv	⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∈ Cℋ
6	5	chjoi	⊢ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ) = ℋ
7	4 6	dedth	⊢ ( 𝐴 ∈ Cℋ → ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐴 ) ) = ℋ )