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Metamath Proof Explorer

Theorem numnncl

Description: Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014)

Ref Expression
Hypotheses numnncl.1 ⊢ 𝑇 ∈ ℕ0
numnncl.2 ⊢ 𝐴 ∈ ℕ0
numnncl.3 ⊢ 𝐵 ∈ ℕ
Assertion numnncl ( ( 𝑇 · 𝐴 ) + 𝐵 ) ∈ ℕ

Proof

Step Hyp Ref Expression
1 numnncl.1 ⊢ 𝑇 ∈ ℕ0
2 numnncl.2 ⊢ 𝐴 ∈ ℕ0
3 numnncl.3 ⊢ 𝐵 ∈ ℕ
4 1 2 nn0mulcli ⊢ ( 𝑇 · 𝐴 ) ∈ ℕ0
5 nn0nnaddcl ⊢ ( ( ( 𝑇 · 𝐴 ) ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( 𝑇 · 𝐴 ) + 𝐵 ) ∈ ℕ )
6 4 3 5 mp2an ⊢ ( ( 𝑇 · 𝐴 ) + 𝐵 ) ∈ ℕ