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 )