Theorem chocin

Description: Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of Kalmbach p. 65. (Contributed by NM, 13-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	chocin	⊢ ( 𝐴 ∈ Cℋ → ( 𝐴 ∩ ( ⊥ ‘ 𝐴 ) ) = 0ℋ )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) )
2		fveq2	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( ⊥ ‘ 𝐴 ) = ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) )
3	1 2	ineq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( 𝐴 ∩ ( ⊥ ‘ 𝐴 ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∩ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) ) )
4	3	eqeq1d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) → ( ( 𝐴 ∩ ( ⊥ ‘ 𝐴 ) ) = 0ℋ ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∩ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) ) = 0ℋ ) )
5		h0elch	⊢ 0ℋ ∈ Cℋ
6	5	elimel	⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∈ Cℋ
7	6	chocini	⊢ ( if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ∩ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , 0ℋ ) ) ) = 0ℋ
8	4 7	dedth	⊢ ( 𝐴 ∈ Cℋ → ( 𝐴 ∩ ( ⊥ ‘ 𝐴 ) ) = 0ℋ )