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Metamath Proof Explorer

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 )