ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  addlelt Unicode version

Theorem addlelt 9797
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
StepHypRef Expression
1 rpgt0 9694 . . . 4  |-  ( A  e.  RR+  ->  0  < 
A )
213ad2ant3 1022 . . 3  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  0  <  A )
3 rpre 9689 . . . . 5  |-  ( A  e.  RR+  ->  A  e.  RR )
433ad2ant3 1022 . . . 4  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  A  e.  RR )
5 simp1 999 . . . 4  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  M  e.  RR )
64, 5ltaddposd 8515 . . 3  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  (
0  <  A  <->  M  <  ( M  +  A ) ) )
72, 6mpbid 147 . 2  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  M  <  ( M  +  A
) )
8 simpl 109 . . . . 5  |-  ( ( M  e.  RR  /\  A  e.  RR+ )  ->  M  e.  RR )
93adantl 277 . . . . 5  |-  ( ( M  e.  RR  /\  A  e.  RR+ )  ->  A  e.  RR )
108, 9readdcld 8016 . . . 4  |-  ( ( M  e.  RR  /\  A  e.  RR+ )  -> 
( M  +  A
)  e.  RR )
11103adant2 1018 . . 3  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  ( M  +  A )  e.  RR )
12 simp2 1000 . . 3  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  N  e.  RR )
13 ltletr 8076 . . 3  |-  ( ( M  e.  RR  /\  ( M  +  A
)  e.  RR  /\  N  e.  RR )  ->  ( ( M  < 
( M  +  A
)  /\  ( M  +  A )  <_  N
)  ->  M  <  N ) )
145, 11, 12, 13syl3anc 1249 . 2  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  (
( M  <  ( M  +  A )  /\  ( M  +  A
)  <_  N )  ->  M  <  N ) )
157, 14mpand 429 1  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  (
( M  +  A
)  <_  N  ->  M  <  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980    e. wcel 2160   class class class wbr 4018  (class class class)co 5895   RRcr 7839   0cc0 7840    + caddc 7843    < clt 8021    <_ cle 8022   RR+crp 9682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-addcom 7940  ax-addass 7942  ax-i2m1 7945  ax-0id 7948  ax-rnegex 7949  ax-pre-ltwlin 7953  ax-pre-ltadd 7956
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-cnv 4652  df-iota 5196  df-fv 5243  df-ov 5898  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-rp 9683
This theorem is referenced by:  zltaddlt1le  10036
  Copyright terms: Public domain W3C validator