Theorem shlessi

Description: Subset implies subset of subspace sum. (Contributed by NM, 18-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	shincl.1	⊢ 𝐴 ∈ Sℋ
	shincl.2	⊢ 𝐵 ∈ Sℋ
	shless.1	⊢ 𝐶 ∈ Sℋ
Assertion	shlessi	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 +ℋ 𝐶 ) ⊆ ( 𝐵 +ℋ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		shincl.1	⊢ 𝐴 ∈ Sℋ
2		shincl.2	⊢ 𝐵 ∈ Sℋ
3		shless.1	⊢ 𝐶 ∈ Sℋ
4		shless	⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 +ℋ 𝐶 ) ⊆ ( 𝐵 +ℋ 𝐶 ) )
5	4	ex	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 +ℋ 𝐶 ) ⊆ ( 𝐵 +ℋ 𝐶 ) ) )
6	1 2 3 5	mp3an	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 +ℋ 𝐶 ) ⊆ ( 𝐵 +ℋ 𝐶 ) )