Theorem shsel2

Description: A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	shsel2	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐶 ∈ 𝐵 → 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		shsel1	⊢ ( ( 𝐵 ∈ Sℋ ∧ 𝐴 ∈ Sℋ ) → ( 𝐶 ∈ 𝐵 → 𝐶 ∈ ( 𝐵 +ℋ 𝐴 ) ) )
2	1	ancoms	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐶 ∈ 𝐵 → 𝐶 ∈ ( 𝐵 +ℋ 𝐴 ) ) )
3		shscom	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐴 +ℋ 𝐵 ) = ( 𝐵 +ℋ 𝐴 ) )
4	3	eleq2d	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ↔ 𝐶 ∈ ( 𝐵 +ℋ 𝐴 ) ) )
5	2 4	sylibrd	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐶 ∈ 𝐵 → 𝐶 ∈ ( 𝐴 +ℋ 𝐵 ) ) )