Theorem ch1dle

Description: A 1-dimensional subspace is less than or equal to any member of CH containing its generating vector. (Contributed by NM, 30-May-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	ch1dle	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ { 𝐵 } ) ) ⊆ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		chsh	⊢ ( 𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ )
2		sh1dle	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ { 𝐵 } ) ) ⊆ 𝐴 )
3	1 2	sylan	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ { 𝐵 } ) ) ⊆ 𝐴 )