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Metamath Proof Explorer

Theorem shsub2i

Description: Subspace sum is an upper bound of its arguments. (Contributed by NM, 17-Dec-2004) (New usage is discouraged.)

Ref Expression
Hypotheses shincl.1 ⊢ 𝐴 ∈ Sℋ
shincl.2 ⊢ 𝐵 ∈ Sℋ
Assertion shsub2i 𝐴 ⊆ ( 𝐵 +ℋ 𝐴 )

Proof

Step Hyp Ref Expression
1 shincl.1 ⊢ 𝐴 ∈ Sℋ
2 shincl.2 ⊢ 𝐵 ∈ Sℋ
3 2 1 shsel2i ⊢ ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ( 𝐵 +ℋ 𝐴 ) )
4 3 ssriv ⊢ 𝐴 ⊆ ( 𝐵 +ℋ 𝐴 )