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Metamath Proof Explorer

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 ∈ C_ℋ
	Assertion	chocini	⊢ ( 𝐴 ∩ ( ⊥ ‘ 𝐴 ) ) = 0_ℋ

Proof

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