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Metamath Proof Explorer

Theorem chjo

Description: The join of a closed subspace and its orthocomplement is all of Hilbert space. (Contributed by NM, 31-Oct-2005) (New usage is discouraged.)

Ref Expression
Assertion chjo ( 𝐴 ∈ Cℋ → ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐴 ) ) = ℋ )

Proof

Step Hyp Ref Expression
1 id ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) )
2 fveq2 ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ⊥ ‘ 𝐴 ) = ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) )
3 1 2 oveq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐴 ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ) )
4 3 eqeq1d ⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐴 ) ) = ℋ ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ) = ℋ ) )
5 ifchhv ⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∈ Cℋ
6 5 chjoi ⊢ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( ⊥ ‘ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ) ) = ℋ
7 4 6 dedth ⊢ ( 𝐴 ∈ Cℋ → ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐴 ) ) = ℋ )