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Metamath Proof Explorer

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 ∈ S ℋ
shincl.2 ⊢ B ∈ S ℋ
shless.1 ⊢ C ∈ S ℋ
Assertion shlessi ⊢ A ⊆ B → A + ℋ C ⊆ B + ℋ C

Proof

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