Theorem chslej

Description: Subspace sum is smaller than subspace join. Remark in Kalmbach p. 65. (Contributed by NM, 12-Jul-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chslej	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 +ℋ 𝐵 ) ⊆ ( 𝐴 ∨ℋ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		chsh	⊢ ( 𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ )
2		chsh	⊢ ( 𝐵 ∈ Cℋ → 𝐵 ∈ Sℋ )
3		shslej	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐴 +ℋ 𝐵 ) ⊆ ( 𝐴 ∨ℋ 𝐵 ) )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 +ℋ 𝐵 ) ⊆ ( 𝐴 ∨ℋ 𝐵 ) )