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Theorem addlelt 9562
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 9460 . . . 4  |-  ( A  e.  RR+  ->  0  < 
A )
213ad2ant3 1004 . . 3  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  0  <  A )
3 rpre 9455 . . . . 5  |-  ( A  e.  RR+  ->  A  e.  RR )
433ad2ant3 1004 . . . 4  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  A  e.  RR )
5 simp1 981 . . . 4  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  M  e.  RR )
64, 5ltaddposd 8298 . . 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 275 . . . . 5  |-  ( ( M  e.  RR  /\  A  e.  RR+ )  ->  A  e.  RR )
108, 9readdcld 7802 . . . 4  |-  ( ( M  e.  RR  /\  A  e.  RR+ )  -> 
( M  +  A
)  e.  RR )
11103adant2 1000 . . 3  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  ( M  +  A )  e.  RR )
12 simp2 982 . . 3  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  N  e.  RR )
13 ltletr 7860 . . 3  |-  ( ( M  e.  RR  /\  ( M  +  A
)  e.  RR  /\  N  e.  RR )  ->  ( ( M  < 
( M  +  A
)  /\  ( M  +  A )  <_  N
)  ->  M  <  N ) )
145, 11, 12, 13syl3anc 1216 . 2  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  (
( M  <  ( M  +  A )  /\  ( M  +  A
)  <_  N )  ->  M  <  N ) )
157, 14mpand 425 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 962    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   RRcr 7626   0cc0 7627    + caddc 7630    < clt 7807    <_ cle 7808   RR+crp 9448
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-i2m1 7732  ax-0id 7735  ax-rnegex 7736  ax-pre-ltwlin 7740  ax-pre-ltadd 7743
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-iota 5088  df-fv 5131  df-ov 5777  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-rp 9449
This theorem is referenced by:  zltaddlt1le  9796
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