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Metamath Proof Explorer

Theorem shsvai

Description: Vector sum belongs to subspace sum. (Contributed by NM, 17-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypotheses shincl.1 ⊢ 𝐴 ∈ Sℋ
shincl.2 ⊢ 𝐵 ∈ Sℋ
Assertion shsvai ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐶 +ℎ 𝐷 ) ∈ ( 𝐴 +ℋ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 shincl.1 ⊢ 𝐴 ∈ Sℋ
2 shincl.2 ⊢ 𝐵 ∈ Sℋ
3 shsva ⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐶 +ℎ 𝐷 ) ∈ ( 𝐴 +ℋ 𝐵 ) ) )
4 1 2 3 mp2an ⊢ ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐶 +ℎ 𝐷 ) ∈ ( 𝐴 +ℋ 𝐵 ) )