Theorem spansnscl

Description: The subspace sum of a closed subspace and a one-dimensional subspace is closed. (Contributed by NM, 17-Dec-2004) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		spansnj	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 +ℋ ( span ‘ { 𝐵 } ) ) = ( 𝐴 ∨ℋ ( span ‘ { 𝐵 } ) ) )
2		spansnch	⊢ ( 𝐵 ∈ ℋ → ( span ‘ { 𝐵 } ) ∈ Cℋ )
3		chjcl	⊢ ( ( 𝐴 ∈ Cℋ ∧ ( span ‘ { 𝐵 } ) ∈ Cℋ ) → ( 𝐴 ∨ℋ ( span ‘ { 𝐵 } ) ) ∈ Cℋ )
4	2 3	sylan2	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ∨ℋ ( span ‘ { 𝐵 } ) ) ∈ Cℋ )
5	1 4	eqeltrd	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 +ℋ ( span ‘ { 𝐵 } ) ) ∈ Cℋ )