Theorem qsqcl

Description: The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014)

	Ref	Expression
Assertion	qsqcl	|- ( A e. QQ -> ( A ^ 2 ) e. QQ )

Proof

Step	Hyp	Ref	Expression
1		qcn	|- ( A e. QQ -> A e. CC )
2		sqval	|- ( A e. CC -> ( A ^ 2 ) = ( A x. A ) )
3	1 2	syl	|- ( A e. QQ -> ( A ^ 2 ) = ( A x. A ) )
4		qmulcl	|- ( ( A e. QQ /\ A e. QQ ) -> ( A x. A ) e. QQ )
5	4	anidms	|- ( A e. QQ -> ( A x. A ) e. QQ )
6	3 5	eqeltrd	|- ( A e. QQ -> ( A ^ 2 ) e. QQ )