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Metamath Proof Explorer

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 ) )