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Theorem zltaddlt1le 9104
Description: The sum of an integer and a real number between 0 and 1 is less than or equal to a second integer iff the sum is less than the second integer. (Contributed by AV, 1-Jul-2021.)
Assertion
Ref Expression
zltaddlt1le  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  (
( M  +  A
)  <  N  <->  ( M  +  A )  <_  N
) )

Proof of Theorem zltaddlt1le
StepHypRef Expression
1 zre 8436 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  RR )
21adantr 270 . . . . 5  |-  ( ( M  e.  ZZ  /\  A  e.  ( 0 (,) 1 ) )  ->  M  e.  RR )
3 elioore 9011 . . . . . 6  |-  ( A  e.  ( 0 (,) 1 )  ->  A  e.  RR )
43adantl 271 . . . . 5  |-  ( ( M  e.  ZZ  /\  A  e.  ( 0 (,) 1 ) )  ->  A  e.  RR )
52, 4readdcld 7210 . . . 4  |-  ( ( M  e.  ZZ  /\  A  e.  ( 0 (,) 1 ) )  ->  ( M  +  A )  e.  RR )
653adant2 958 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  ( M  +  A )  e.  RR )
7 zre 8436 . . . 4  |-  ( N  e.  ZZ  ->  N  e.  RR )
873ad2ant2 961 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  N  e.  RR )
9 ltle 7265 . . 3  |-  ( ( ( M  +  A
)  e.  RR  /\  N  e.  RR )  ->  ( ( M  +  A )  <  N  ->  ( M  +  A
)  <_  N )
)
106, 8, 9syl2anc 403 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  (
( M  +  A
)  <  N  ->  ( M  +  A )  <_  N ) )
11 elioo3g 9009 . . . . . 6  |-  ( A  e.  ( 0 (,) 1 )  <->  ( (
0  e.  RR*  /\  1  e.  RR*  /\  A  e. 
RR* )  /\  (
0  <  A  /\  A  <  1 ) ) )
12 simpl 107 . . . . . 6  |-  ( ( 0  <  A  /\  A  <  1 )  -> 
0  <  A )
1311, 12simplbiim 379 . . . . 5  |-  ( A  e.  ( 0 (,) 1 )  ->  0  <  A )
143, 13elrpd 8852 . . . 4  |-  ( A  e.  ( 0 (,) 1 )  ->  A  e.  RR+ )
15 addlelt 8920 . . . 4  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  (
( M  +  A
)  <_  N  ->  M  <  N ) )
161, 7, 14, 15syl3an 1212 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  (
( M  +  A
)  <_  N  ->  M  <  N ) )
17 zltp1le 8486 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <->  ( M  +  1 )  <_  N ) )
18173adant3 959 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  ( M  <  N  <->  ( M  +  1 )  <_  N ) )
1933ad2ant3 962 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  A  e.  RR )
20 1red 7196 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  1  e.  RR )
2113ad2ant1 960 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  M  e.  RR )
22 simpr 108 . . . . . . . 8  |-  ( ( 0  <  A  /\  A  <  1 )  ->  A  <  1 )
2311, 22simplbiim 379 . . . . . . 7  |-  ( A  e.  ( 0 (,) 1 )  ->  A  <  1 )
24233ad2ant3 962 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  A  <  1 )
2519, 20, 21, 24ltadd2dd 7593 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  ( M  +  A )  <  ( M  +  1 ) )
26 peano2z 8468 . . . . . . . 8  |-  ( M  e.  ZZ  ->  ( M  +  1 )  e.  ZZ )
2726zred 8550 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M  +  1 )  e.  RR )
28273ad2ant1 960 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  ( M  +  1 )  e.  RR )
29 ltletr 7267 . . . . . 6  |-  ( ( ( M  +  A
)  e.  RR  /\  ( M  +  1
)  e.  RR  /\  N  e.  RR )  ->  ( ( ( M  +  A )  < 
( M  +  1 )  /\  ( M  +  1 )  <_  N )  ->  ( M  +  A )  <  N ) )
306, 28, 8, 29syl3anc 1170 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  (
( ( M  +  A )  <  ( M  +  1 )  /\  ( M  + 
1 )  <_  N
)  ->  ( M  +  A )  <  N
) )
3125, 30mpand 420 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  (
( M  +  1 )  <_  N  ->  ( M  +  A )  <  N ) )
3218, 31sylbid 148 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  ( M  <  N  ->  ( M  +  A )  <  N ) )
3316, 32syld 44 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  (
( M  +  A
)  <_  N  ->  ( M  +  A )  <  N ) )
3410, 33impbid 127 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  (
( M  +  A
)  <  N  <->  ( M  +  A )  <_  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    e. wcel 1434   class class class wbr 3793  (class class class)co 5543   RRcr 7042   0cc0 7043   1c1 7044    + caddc 7046   RR*cxr 7214    < clt 7215    <_ cle 7216   ZZcz 8432   RR+crp 8815   (,)cioo 8987
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 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-id 4056  df-po 4059  df-iso 4060  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-rp 8816  df-ioo 8991
This theorem is referenced by:  halfleoddlt  10438
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