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Metamath Proof Explorer

Theorem nn0subm

Description: The nonnegative integers form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 18-Jun-2015)

Ref Expression
Assertion nn0subm ℕ0 ∈ ( SubMnd ‘ ℂfld )

Proof

Step Hyp Ref Expression
1 nn0cn ⊢ ( 𝑥 ∈ ℕ0 → 𝑥 ∈ ℂ )
2 nn0addcl ⊢ ( ( 𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0 ) → ( 𝑥 + 𝑦 ) ∈ ℕ0 )
3 0nn0 ⊢ 0 ∈ ℕ0
4 1 2 3 cnsubmlem ⊢ ℕ0 ∈ ( SubMnd ‘ ℂfld )