Theorem sn-addgt0d

Description: The sum of positive numbers is positive. Proof of addgt0d without ax-mulcom . (Contributed by SN, 25-Jan-2025)

	Ref	Expression
Hypotheses	sn-addgt0d.a	|- ( ph -> A e. RR )
	sn-addgt0d.b	|- ( ph -> B e. RR )
	sn-addgt0d.1	|- ( ph -> 0 < A )
	sn-addgt0d.2	|- ( ph -> 0 < B )
Assertion	sn-addgt0d	|- ( ph -> 0 < ( A + B ) )

Proof

Step	Hyp	Ref	Expression
1		sn-addgt0d.a	|- ( ph -> A e. RR )
2		sn-addgt0d.b	|- ( ph -> B e. RR )
3		sn-addgt0d.1	|- ( ph -> 0 < A )
4		sn-addgt0d.2	|- ( ph -> 0 < B )
5		0red	|- ( ph -> 0 e. RR )
6	1 2	readdcld	|- ( ph -> ( A + B ) e. RR )
7		sn-ltaddpos	|- ( ( B e. RR /\ A e. RR ) -> ( 0 < B <-> A < ( A + B ) ) )
8	2 1 7	syl2anc	|- ( ph -> ( 0 < B <-> A < ( A + B ) ) )
9	4 8	mpbid	|- ( ph -> A < ( A + B ) )
10	5 1 6 3 9	lttrd	|- ( ph -> 0 < ( A + B ) )