This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem shsub1i

Description: Subspace sum is an upper bound of its arguments. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypotheses shincl.1 ⊢ 𝐴 ∈ Sℋ
shincl.2 ⊢ 𝐵 ∈ Sℋ
Assertion shsub1i 𝐴 ⊆ ( 𝐴 +ℋ 𝐵 )

Proof

Step Hyp Ref Expression
1 shincl.1 ⊢ 𝐴 ∈ Sℋ
2 shincl.2 ⊢ 𝐵 ∈ Sℋ
3 1 2 shsel1i ⊢ ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ( 𝐴 +ℋ 𝐵 ) )
4 3 ssriv ⊢ 𝐴 ⊆ ( 𝐴 +ℋ 𝐵 )