Theorem resum2sqcl

Description: The sum of two squares of real numbers is a real number. (Contributed by AV, 7-Feb-2023)

	Ref	Expression
Hypothesis	resum2sqcl.q	|- Q = ( ( A ^ 2 ) + ( B ^ 2 ) )
Assertion	resum2sqcl	|- ( ( A e. RR /\ B e. RR ) -> Q e. RR )

Proof

Step	Hyp	Ref	Expression
1		resum2sqcl.q	|- Q = ( ( A ^ 2 ) + ( B ^ 2 ) )
2		simpl	|- ( ( A e. RR /\ B e. RR ) -> A e. RR )
3	2	resqcld	|- ( ( A e. RR /\ B e. RR ) -> ( A ^ 2 ) e. RR )
4		simpr	|- ( ( A e. RR /\ B e. RR ) -> B e. RR )
5	4	resqcld	|- ( ( A e. RR /\ B e. RR ) -> ( B ^ 2 ) e. RR )
6	3 5	readdcld	|- ( ( A e. RR /\ B e. RR ) -> ( ( A ^ 2 ) + ( B ^ 2 ) ) e. RR )
7	1 6	eqeltrid	|- ( ( A e. RR /\ B e. RR ) -> Q e. RR )