Theorem addlelt 9585

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.)

Assertion

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

Proof of Theorem addlelt

Step	Hyp	Ref	Expression
1		rpgt0 9482	. . . 4 |- ( A e. RR+ -> 0 < A )
2	1	3ad2ant3 1005	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> 0 < A )
3		rpre 9477	. . . . 5 |- ( A e. RR+ -> A e. RR )
4	3	3ad2ant3 1005	. . . 4 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> A e. RR )
5		simp1 982	. . . 4 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> M e. RR )
6	4, 5	ltaddposd 8315	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( 0 < A <-> M < ( M + A ) ) )
7	2, 6	mpbid 146	. 2 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> M < ( M + A ) )
8		simpl 108	. . . . 5 |- ( ( M e. RR /\ A e. RR+ ) -> M e. RR )
9	3	adantl 275	. . . . 5 |- ( ( M e. RR /\ A e. RR+ ) -> A e. RR )
10	8, 9	readdcld 7819	. . . 4 |- ( ( M e. RR /\ A e. RR+ ) -> ( M + A ) e. RR )
11	10	3adant2 1001	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( M + A ) e. RR )
12		simp2 983	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> N e. RR )
13		ltletr 7877	. . 3 |- ( ( M e. RR /\ ( M + A ) e. RR /\ N e. RR ) -> ( ( M < ( M + A ) /\ ( M + A ) <_ N ) -> M < N ) )
14	5, 11, 12, 13	syl3anc 1217	. 2 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( ( M < ( M + A ) /\ ( M + A ) <_ N ) -> M < N ) )
15	7, 14	mpand 426	1 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( ( M + A ) <_ N -> M < N ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   e. wcel 1481   class class class wbr 3937  (class class class)co 5782   RRcr 7643   0cc0 7644   + caddc 7647   < clt 7824   <_ cle 7825   RR+crp 9470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-i2m1 7749  ax-0id 7752  ax-rnegex 7753  ax-pre-ltwlin 7757  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-cnv 4555  df-iota 5096  df-fv 5139  df-ov 5785  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-rp 9471

This theorem is referenced by:  zltaddlt1le  9820