Theorem addlelt

Description: If the sum of a real number and a positive real number is less than or equal to a third real number, the first real number is less than the third real number. (Contributed by AV, 1-Jul-2021)

	Ref	Expression
Assertion	addlelt	|- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( ( M + A ) <_ N -> M < N ) )

Proof

Step	Hyp	Ref	Expression
1		rpgt0	|- ( A e. RR+ -> 0 < A )
2	1	3ad2ant3	|- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> 0 < A )
3		rpre	|- ( A e. RR+ -> A e. RR )
4	3	3ad2ant3	|- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> A e. RR )
5		simp1	|- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> M e. RR )
6	4 5	ltaddposd	|- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( 0 < A <-> M < ( M + A ) ) )
7	2 6	mpbid	|- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> M < ( M + A ) )
8		simpl	|- ( ( M e. RR /\ A e. RR+ ) -> M e. RR )
9	3	adantl	|- ( ( M e. RR /\ A e. RR+ ) -> A e. RR )
10	8 9	readdcld	|- ( ( M e. RR /\ A e. RR+ ) -> ( M + A ) e. RR )
11	10	3adant2	|- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( M + A ) e. RR )
12		simp2	|- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> N e. RR )
13		ltletr	|- ( ( M e. RR /\ ( M + A ) e. RR /\ N e. RR ) -> ( ( M < ( M + A ) /\ ( M + A ) <_ N ) -> M < N ) )
14	5 11 12 13	syl3anc	|- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( ( M < ( M + A ) /\ ( M + A ) <_ N ) -> M < N ) )
15	7 14	mpand	|- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( ( M + A ) <_ N -> M < N ) )