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Metamath Proof Explorer

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 )