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Metamath Proof Explorer

Theorem addclpi

Description: Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995) (New usage is discouraged.)

Ref Expression
Assertion addclpi ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 +N 𝐵 ) ∈ N )

Proof

Step Hyp Ref Expression
1 addpiord ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 +N 𝐵 ) = ( 𝐴 +o 𝐵 ) )
2 pinn ⊢ ( 𝐴 ∈ N → 𝐴 ∈ ω )
3 pinn ⊢ ( 𝐵 ∈ N → 𝐵 ∈ ω )
4 nnacl ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 +o 𝐵 ) ∈ ω )
5 3 4 sylan2 ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ N ) → ( 𝐴 +o 𝐵 ) ∈ ω )
6 elni2 ⊢ ( 𝐵 ∈ N ↔ ( 𝐵 ∈ ω ∧ ∅ ∈ 𝐵 ) )
7 nnaordi ⊢ ( ( 𝐵 ∈ ω ∧ 𝐴 ∈ ω ) → ( ∅ ∈ 𝐵 → ( 𝐴 +o ∅ ) ∈ ( 𝐴 +o 𝐵 ) ) )
8 ne0i ⊢ ( ( 𝐴 +o ∅ ) ∈ ( 𝐴 +o 𝐵 ) → ( 𝐴 +o 𝐵 ) ≠ ∅ )
9 7 8 syl6 ⊢ ( ( 𝐵 ∈ ω ∧ 𝐴 ∈ ω ) → ( ∅ ∈ 𝐵 → ( 𝐴 +o 𝐵 ) ≠ ∅ ) )
10 9 expcom ⊢ ( 𝐴 ∈ ω → ( 𝐵 ∈ ω → ( ∅ ∈ 𝐵 → ( 𝐴 +o 𝐵 ) ≠ ∅ ) ) )
11 10 imp32 ⊢ ( ( 𝐴 ∈ ω ∧ ( 𝐵 ∈ ω ∧ ∅ ∈ 𝐵 ) ) → ( 𝐴 +o 𝐵 ) ≠ ∅ )
12 6 11 sylan2b ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ N ) → ( 𝐴 +o 𝐵 ) ≠ ∅ )
13 elni ⊢ ( ( 𝐴 +o 𝐵 ) ∈ N ↔ ( ( 𝐴 +o 𝐵 ) ∈ ω ∧ ( 𝐴 +o 𝐵 ) ≠ ∅ ) )
14 5 12 13 sylanbrc ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ N ) → ( 𝐴 +o 𝐵 ) ∈ N )
15 2 14 sylan ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 +o 𝐵 ) ∈ N )
16 1 15 eqeltrd ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 +N 𝐵 ) ∈ N )