Theorem chjoi

Description: The join of a closed subspace and its orthocomplement. (Contributed by NM, 24-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypothesis	ch0le.1	⊢ 𝐴 ∈ Cℋ
Assertion	chjoi	⊢ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐴 ) ) = ℋ

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	⊢ 𝐴 ∈ Cℋ
2	1	chssii	⊢ 𝐴 ⊆ ℋ
3		ssjo	⊢ ( 𝐴 ⊆ ℋ → ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐴 ) ) = ℋ )
4	2 3	ax-mp	⊢ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐴 ) ) = ℋ