Theorem add20i

Description: Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999)

	Ref	Expression
Hypotheses	lt2.1	|- A e. RR
	lt2.2	|- B e. RR
Assertion	add20i	|- ( ( 0 <_ A /\ 0 <_ B ) -> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lt2.1	|- A e. RR
2		lt2.2	|- B e. RR
3		add20	|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
4	3	an4s	|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 <_ B ) ) -> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
5	1 2 4	mpanl12	|- ( ( 0 <_ A /\ 0 <_ B ) -> ( ( A + B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )