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Metamath Proof Explorer

Theorem elspansncl

Description: A member of a span of a singleton is a vector. (Contributed by NM, 17-Dec-2004) (New usage is discouraged.)

		Ref	Expression
	Assertion	elspansncl	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ( span ‘ { 𝐴 } ) ) → 𝐵 ∈ ℋ )

Step	Hyp	Ref	Expression
1		snssi	⊢ ( 𝐴 ∈ ℋ → { 𝐴 } ⊆ ℋ )
2		elspancl	⊢ ( ( { 𝐴 } ⊆ ℋ ∧ 𝐵 ∈ ( span ‘ { 𝐴 } ) ) → 𝐵 ∈ ℋ )
3	1 2	sylan	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ( span ‘ { 𝐴 } ) ) → 𝐵 ∈ ℋ )