Theorem chocini

Description: Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of Kalmbach p. 65. (Contributed by NM, 11-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypothesis	ch0le.1	⊢ 𝐴 ∈ Cℋ
Assertion	chocini	⊢ ( 𝐴 ∩ ( ⊥ ‘ 𝐴 ) ) = 0ℋ

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	⊢ 𝐴 ∈ Cℋ
2	1	chshii	⊢ 𝐴 ∈ Sℋ
3		ocin	⊢ ( 𝐴 ∈ Sℋ → ( 𝐴 ∩ ( ⊥ ‘ 𝐴 ) ) = 0ℋ )
4	2 3	ax-mp	⊢ ( 𝐴 ∩ ( ⊥ ‘ 𝐴 ) ) = 0ℋ