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Metamath Proof Explorer

Theorem shub2

Description: A subspace is a subset of its Hilbert lattice join with another. (Contributed by NM, 22-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion shub2 ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → 𝐴 ⊆ ( 𝐵 ∨ℋ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 shub1 ⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → 𝐴 ⊆ ( 𝐴 ∨ℋ 𝐵 ) )
2 shjcom ⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( 𝐵 ∨ℋ 𝐴 ) )
3 1 2 sseqtrd ⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → 𝐴 ⊆ ( 𝐵 ∨ℋ 𝐴 ) )