Theorem shseli

Description: Membership in subspace sum. (Contributed by NM, 4-May-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	shscl.1	⊢ 𝐴 ∈ Sℋ
	shscl.2	⊢ 𝐵 ∈ Sℋ
Assertion	shseli	⊢ ( 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		shscl.1	⊢ 𝐴 ∈ Sℋ
2		shscl.2	⊢ 𝐵 ∈ Sℋ
3		shsel	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) ) )
4	1 2 3	mp2an	⊢ ( 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐶 = ( 𝑥 +ℎ 𝑦 ) )