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Metamath Proof Explorer

Theorem shsub1

Description: Subspace sum is an upper bound of its arguments. (Contributed by NM, 14-Dec-2004) (New usage is discouraged.)

Ref Expression
Assertion shsub1 ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → 𝐴 ⊆ ( 𝐴 +ℋ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 shsel1 ⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ( 𝐴 +ℋ 𝐵 ) ) )
2 1 ssrdv ⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → 𝐴 ⊆ ( 𝐴 +ℋ 𝐵 ) )