Theorem addcli 7546

Description: Closure law for addition. (Contributed by NM, 23-Nov-1994.)

Hypotheses

Ref	Expression
axi.1	⊢ 𝐴 ∈ ℂ
axi.2	⊢ 𝐵 ∈ ℂ

Assertion

Ref	Expression
addcli	⊢ (𝐴 + 𝐵) ∈ ℂ

Proof of Theorem addcli

Step	Hyp	Ref	Expression
1		axi.1	. 2 ⊢ 𝐴 ∈ ℂ
2		axi.2	. 2 ⊢ 𝐵 ∈ ℂ
3		addcl 7521	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
4	1, 2, 3	mp2an 418	1 ⊢ (𝐴 + 𝐵) ∈ ℂ

Syntax hints:   ∈ wcel 1439  (class class class)co 5666  ℂcc 7402   + caddc 7407

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia3 107  ax-addcl 7495

This theorem is referenced by:  negdii  7820  eqneg  8253  nummac  8975  binom2i  10117  3dvds2dec  11198