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Metamath Proof Explorer

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 ) )