Theorem qsubcl

Description: Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004)

	Ref	Expression
Assertion	qsubcl	⊢ ( ( 𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ) → ( 𝐴 − 𝐵 ) ∈ ℚ )

Proof

Step	Hyp	Ref	Expression
1		qcn	⊢ ( 𝐴 ∈ ℚ → 𝐴 ∈ ℂ )
2		qcn	⊢ ( 𝐵 ∈ ℚ → 𝐵 ∈ ℂ )
3		negsub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ) → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) )
5		qnegcl	⊢ ( 𝐵 ∈ ℚ → - 𝐵 ∈ ℚ )
6		qaddcl	⊢ ( ( 𝐴 ∈ ℚ ∧ - 𝐵 ∈ ℚ ) → ( 𝐴 + - 𝐵 ) ∈ ℚ )
7	5 6	sylan2	⊢ ( ( 𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ) → ( 𝐴 + - 𝐵 ) ∈ ℚ )
8	4 7	eqeltrrd	⊢ ( ( 𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ) → ( 𝐴 − 𝐵 ) ∈ ℚ )