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

Theorem addlelt 9300
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 9206 . . . 4  |-  ( A  e.  RR+  ->  0  < 
A )
213ad2ant3 967 . . 3  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  0  <  A )
3 rpre 9201 . . . . 5  |-  ( A  e.  RR+  ->  A  e.  RR )
433ad2ant3 967 . . . 4  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  A  e.  RR )
5 simp1 944 . . . 4  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  M  e.  RR )
64, 5ltaddposd 8067 . . 3  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  (
0  <  A  <->  M  <  ( M  +  A ) ) )
72, 6mpbid 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 )
93adantl 272 . . . . 5  |-  ( ( M  e.  RR  /\  A  e.  RR+ )  ->  A  e.  RR )
108, 9readdcld 7578 . . . 4  |-  ( ( M  e.  RR  /\  A  e.  RR+ )  -> 
( M  +  A
)  e.  RR )
11103adant2 963 . . 3  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  ( M  +  A )  e.  RR )
12 simp2 945 . . 3  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  N  e.  RR )
13 ltletr 7635 . . 3  |-  ( ( M  e.  RR  /\  ( M  +  A
)  e.  RR  /\  N  e.  RR )  ->  ( ( M  < 
( M  +  A
)  /\  ( M  +  A )  <_  N
)  ->  M  <  N ) )
145, 11, 12, 13syl3anc 1175 . 2  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  (
( M  <  ( M  +  A )  /\  ( M  +  A
)  <_  N )  ->  M  <  N ) )
157, 14mpand 421 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 103    /\ w3a 925    e. wcel 1439   class class class wbr 3851  (class class class)co 5666   RRcr 7410   0cc0 7411    + caddc 7414    < clt 7583    <_ cle 7584   RR+crp 9195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-addcom 7506  ax-addass 7508  ax-i2m1 7511  ax-0id 7514  ax-rnegex 7515  ax-pre-ltwlin 7519  ax-pre-ltadd 7522
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-xp 4458  df-cnv 4460  df-iota 4993  df-fv 5036  df-ov 5669  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-rp 9196
This theorem is referenced by:  zltaddlt1le  9484
  Copyright terms: Public domain W3C validator