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Metamath Proof Explorer

Theorem mulclpi

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

Ref Expression
Assertion mulclpi ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 ·N 𝐵 ) ∈ N )

Proof

Step Hyp Ref Expression
1 mulpiord ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 ·N 𝐵 ) = ( 𝐴 ·o 𝐵 ) )
2 pinn ⊢ ( 𝐴 ∈ N → 𝐴 ∈ ω )
3 pinn ⊢ ( 𝐵 ∈ N → 𝐵 ∈ ω )
4 nnmcl ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ·o 𝐵 ) ∈ ω )
5 2 3 4 syl2an ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 ·o 𝐵 ) ∈ ω )
6 elni2 ⊢ ( 𝐵 ∈ N ↔ ( 𝐵 ∈ ω ∧ ∅ ∈ 𝐵 ) )
7 6 simprbi ⊢ ( 𝐵 ∈ N → ∅ ∈ 𝐵 )
8 7 adantl ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ∅ ∈ 𝐵 )
9 3 adantl ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → 𝐵 ∈ ω )
10 2 adantr ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → 𝐴 ∈ ω )
11 elni2 ⊢ ( 𝐴 ∈ N ↔ ( 𝐴 ∈ ω ∧ ∅ ∈ 𝐴 ) )
12 11 simprbi ⊢ ( 𝐴 ∈ N → ∅ ∈ 𝐴 )
13 12 adantr ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ∅ ∈ 𝐴 )
14 nnmordi ⊢ ( ( ( 𝐵 ∈ ω ∧ 𝐴 ∈ ω ) ∧ ∅ ∈ 𝐴 ) → ( ∅ ∈ 𝐵 → ( 𝐴 ·o ∅ ) ∈ ( 𝐴 ·o 𝐵 ) ) )
15 9 10 13 14 syl21anc ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( ∅ ∈ 𝐵 → ( 𝐴 ·o ∅ ) ∈ ( 𝐴 ·o 𝐵 ) ) )
16 8 15 mpd ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 ·o ∅ ) ∈ ( 𝐴 ·o 𝐵 ) )
17 16 ne0d ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 ·o 𝐵 ) ≠ ∅ )
18 elni ⊢ ( ( 𝐴 ·o 𝐵 ) ∈ N ↔ ( ( 𝐴 ·o 𝐵 ) ∈ ω ∧ ( 𝐴 ·o 𝐵 ) ≠ ∅ ) )
19 5 17 18 sylanbrc ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 ·o 𝐵 ) ∈ N )
20 1 19 eqeltrd ⊢ ( ( 𝐴 ∈ N ∧ 𝐵 ∈ N ) → ( 𝐴 ·N 𝐵 ) ∈ N )