Description: The span of a Hilbert space singleton belongs to the Hilbert lattice. (Contributed by NM, 9-Jun-2004) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | spansnch | ⊢ ( 𝐴 ∈ ℋ → ( span ‘ { 𝐴 } ) ∈ Cℋ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spansn | ⊢ ( 𝐴 ∈ ℋ → ( span ‘ { 𝐴 } ) = ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ) | |
2 | snssi | ⊢ ( 𝐴 ∈ ℋ → { 𝐴 } ⊆ ℋ ) | |
3 | occl | ⊢ ( { 𝐴 } ⊆ ℋ → ( ⊥ ‘ { 𝐴 } ) ∈ Cℋ ) | |
4 | choccl | ⊢ ( ( ⊥ ‘ { 𝐴 } ) ∈ Cℋ → ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ∈ Cℋ ) | |
5 | 2 3 4 | 3syl | ⊢ ( 𝐴 ∈ ℋ → ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ∈ Cℋ ) |
6 | 1 5 | eqeltrd | ⊢ ( 𝐴 ∈ ℋ → ( span ‘ { 𝐴 } ) ∈ Cℋ ) |