Theorem elspani

Description: Membership in the span of a subset of Hilbert space. (Contributed by NM, 2-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Hypothesis	elspan.1	⊢ 𝐵 ∈ V
Assertion	elspani	⊢ ( 𝐴 ⊆ ℋ → ( 𝐵 ∈ ( span ‘ 𝐴 ) ↔ ∀ 𝑥 ∈ Sℋ ( 𝐴 ⊆ 𝑥 → 𝐵 ∈ 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elspan.1	⊢ 𝐵 ∈ V
2		spanval	⊢ ( 𝐴 ⊆ ℋ → ( span ‘ 𝐴 ) = ∩ { 𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥 } )
3	2	eleq2d	⊢ ( 𝐴 ⊆ ℋ → ( 𝐵 ∈ ( span ‘ 𝐴 ) ↔ 𝐵 ∈ ∩ { 𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥 } ) )
4	1	elintrab	⊢ ( 𝐵 ∈ ∩ { 𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥 } ↔ ∀ 𝑥 ∈ Sℋ ( 𝐴 ⊆ 𝑥 → 𝐵 ∈ 𝑥 ) )
5	3 4	bitrdi	⊢ ( 𝐴 ⊆ ℋ → ( 𝐵 ∈ ( span ‘ 𝐴 ) ↔ ∀ 𝑥 ∈ Sℋ ( 𝐴 ⊆ 𝑥 → 𝐵 ∈ 𝑥 ) ) )