Theorem sh1dle

Description: A 1-dimensional subspace is less than or equal to any subspace containing its generating vector. (Contributed by NM, 24-Nov-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	sh1dle	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ { 𝐵 } ) ) ⊆ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		shel	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ∈ ℋ )
2		spansn	⊢ ( 𝐵 ∈ ℋ → ( span ‘ { 𝐵 } ) = ( ⊥ ‘ ( ⊥ ‘ { 𝐵 } ) ) )
3	1 2	syl	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴 ) → ( span ‘ { 𝐵 } ) = ( ⊥ ‘ ( ⊥ ‘ { 𝐵 } ) ) )
4		spansnss	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴 ) → ( span ‘ { 𝐵 } ) ⊆ 𝐴 )
5	3 4	eqsstrrd	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ { 𝐵 } ) ) ⊆ 𝐴 )