Theorem addlelt 8972

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 8878	. . . 4 |- ( A e. RR+ -> 0 < A )
2	1	3ad2ant3 962	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> 0 < A )
3		rpre 8873	. . . . 5 |- ( A e. RR+ -> A e. RR )
4	3	3ad2ant3 962	. . . 4 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> A e. RR )
5		simp1 939	. . . 4 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> M e. RR )
6	4, 5	ltaddposd 7748	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( 0 < A <-> M < ( M + A ) ) )
7	2, 6	mpbid 145	. 2 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> M < ( M + A ) )
8		simpl 107	. . . . 5 |- ( ( M e. RR /\ A e. RR+ ) -> M e. RR )
9	3	adantl 271	. . . . 5 |- ( ( M e. RR /\ A e. RR+ ) -> A e. RR )
10	8, 9	readdcld 7262	. . . 4 |- ( ( M e. RR /\ A e. RR+ ) -> ( M + A ) e. RR )
11	10	3adant2 958	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( M + A ) e. RR )
12		simp2 940	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> N e. RR )
13		ltletr 7319	. . 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 1170	. 2 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( ( M < ( M + A ) /\ ( M + A ) <_ N ) -> M < N ) )
15	7, 14	mpand 420	1 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( ( M + A ) <_ N -> M < N ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 920   e. wcel 1434   class class class wbr 3805  (class class class)co 5563   RRcr 7094   0cc0 7095   + caddc 7098   < clt 7267   <_ cle 7268   RR+crp 8867

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-i2m1 7195  ax-0id 7198  ax-rnegex 7199  ax-pre-ltwlin 7203  ax-pre-ltadd 7206

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-xp 4397  df-cnv 4399  df-iota 4917  df-fv 4960  df-ov 5566  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-rp 8868

This theorem is referenced by:  zltaddlt1le  9156