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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
StepHypRef Expression
1 rpgt0 8878 . . . 4  |-  ( A  e.  RR+  ->  0  < 
A )
213ad2ant3 962 . . 3  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  0  <  A )
3 rpre 8873 . . . . 5  |-  ( A  e.  RR+  ->  A  e.  RR )
433ad2ant3 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 )
64, 5ltaddposd 7748 . . 3  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  (
0  <  A  <->  M  <  ( M  +  A ) ) )
72, 6mpbid 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 )
93adantl 271 . . . . 5  |-  ( ( M  e.  RR  /\  A  e.  RR+ )  ->  A  e.  RR )
108, 9readdcld 7262 . . . 4  |-  ( ( M  e.  RR  /\  A  e.  RR+ )  -> 
( M  +  A
)  e.  RR )
11103adant2 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 ) )
145, 11, 12, 13syl3anc 1170 . 2  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  (
( M  <  ( M  +  A )  /\  ( M  +  A
)  <_  N )  ->  M  <  N ) )
157, 14mpand 420 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 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
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