Theorem mulclnq

Description: Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995) (Revised by Mario Carneiro, 8-May-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	mulclnq	⊢ ( ( 𝐴 ∈ Q ∧ 𝐵 ∈ Q ) → ( 𝐴 ·Q 𝐵 ) ∈ Q )

Proof

Step	Hyp	Ref	Expression
1		mulpqnq	⊢ ( ( 𝐴 ∈ Q ∧ 𝐵 ∈ Q ) → ( 𝐴 ·Q 𝐵 ) = ( [Q] ‘ ( 𝐴 ·pQ 𝐵 ) ) )
2		elpqn	⊢ ( 𝐴 ∈ Q → 𝐴 ∈ ( N × N ) )
3		elpqn	⊢ ( 𝐵 ∈ Q → 𝐵 ∈ ( N × N ) )
4		mulpqf	⊢ ·pQ : ( ( N × N ) × ( N × N ) ) ⟶ ( N × N )
5	4	fovcl	⊢ ( ( 𝐴 ∈ ( N × N ) ∧ 𝐵 ∈ ( N × N ) ) → ( 𝐴 ·pQ 𝐵 ) ∈ ( N × N ) )
6	2 3 5	syl2an	⊢ ( ( 𝐴 ∈ Q ∧ 𝐵 ∈ Q ) → ( 𝐴 ·pQ 𝐵 ) ∈ ( N × N ) )
7		nqercl	⊢ ( ( 𝐴 ·pQ 𝐵 ) ∈ ( N × N ) → ( [Q] ‘ ( 𝐴 ·pQ 𝐵 ) ) ∈ Q )
8	6 7	syl	⊢ ( ( 𝐴 ∈ Q ∧ 𝐵 ∈ Q ) → ( [Q] ‘ ( 𝐴 ·pQ 𝐵 ) ) ∈ Q )
9	1 8	eqeltrd	⊢ ( ( 𝐴 ∈ Q ∧ 𝐵 ∈ Q ) → ( 𝐴 ·Q 𝐵 ) ∈ Q )