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Metamath Proof Explorer

Theorem nn0addcl

Description: Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002) (Proof shortened by Mario Carneiro, 17-Jul-2014)

		Ref	Expression
	Assertion	nn0addcl	$⊢ (M \in ℕ_{0} \land N \in ℕ_{0}) \to M + N \in ℕ_{0}$

Proof

Step	Hyp	Ref	Expression
1		nnsscn	$⊢ ℕ \subseteq ℂ$
2		id	$⊢ ℕ \subseteq ℂ \to ℕ \subseteq ℂ$
3		df-n0	$⊢ ℕ_{0} = ℕ \cup \{0\}$
4		nnaddcl	$⊢ (M \in ℕ \land N \in ℕ) \to M + N \in ℕ$
5	4	adantl	$⊢ (ℕ \subseteq ℂ \land (M \in ℕ \land N \in ℕ)) \to M + N \in ℕ$
6	2 3 5	un0addcl	$⊢ (ℕ \subseteq ℂ \land (M \in ℕ_{0} \land N \in ℕ_{0})) \to M + N \in ℕ_{0}$
7	1 6	mpan	$⊢ (M \in ℕ_{0} \land N \in ℕ_{0}) \to M + N \in ℕ_{0}$