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Metamath Proof Explorer

Theorem addcld

Description: Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses addcld.1 ⊢ φ → A ∈ ℂ
addcld.2 ⊢ φ → B ∈ ℂ
Assertion addcld ⊢ φ → A + B ∈ ℂ

Proof

Step Hyp Ref Expression
1 addcld.1 ⊢ φ → A ∈ ℂ
2 addcld.2 ⊢ φ → B ∈ ℂ
3 addcl ⊢ A ∈ ℂ ∧ B ∈ ℂ → A + B ∈ ℂ
4 1 2 3 syl2anc ⊢ φ → A + B ∈ ℂ