Theorem resum2sqgt0

Description: The sum of the square of a nonzero real number and the square of another real number is greater than zero. (Contributed by AV, 7-Feb-2023)

	Ref	Expression
Hypothesis	resum2sqcl.q	|- Q = ( ( A ^ 2 ) + ( B ^ 2 ) )
Assertion	resum2sqgt0	|- ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR ) -> 0 < Q )

Proof

Step	Hyp	Ref	Expression
1		resum2sqcl.q	|- Q = ( ( A ^ 2 ) + ( B ^ 2 ) )
2		simpl	|- ( ( A e. RR /\ A =/= 0 ) -> A e. RR )
3	2	resqcld	|- ( ( A e. RR /\ A =/= 0 ) -> ( A ^ 2 ) e. RR )
4	3	adantr	|- ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR ) -> ( A ^ 2 ) e. RR )
5		simpr	|- ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR ) -> B e. RR )
6	5	resqcld	|- ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR ) -> ( B ^ 2 ) e. RR )
7		sqgt0	|- ( ( A e. RR /\ A =/= 0 ) -> 0 < ( A ^ 2 ) )
8	7	adantr	|- ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR ) -> 0 < ( A ^ 2 ) )
9		sqge0	|- ( B e. RR -> 0 <_ ( B ^ 2 ) )
10	9	adantl	|- ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR ) -> 0 <_ ( B ^ 2 ) )
11	4 6 8 10	addgtge0d	|- ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR ) -> 0 < ( ( A ^ 2 ) + ( B ^ 2 ) ) )
12	11 1	breqtrrdi	|- ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR ) -> 0 < Q )