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Theorem zltaddlt1le 9911
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 9171 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  RR )
21adantr 274 . . . . 5  |-  ( ( M  e.  ZZ  /\  A  e.  ( 0 (,) 1 ) )  ->  M  e.  RR )
3 elioore 9816 . . . . . 6  |-  ( A  e.  ( 0 (,) 1 )  ->  A  e.  RR )
43adantl 275 . . . . 5  |-  ( ( M  e.  ZZ  /\  A  e.  ( 0 (,) 1 ) )  ->  A  e.  RR )
52, 4readdcld 7907 . . . 4  |-  ( ( M  e.  ZZ  /\  A  e.  ( 0 (,) 1 ) )  ->  ( M  +  A )  e.  RR )
653adant2 1001 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  ( M  +  A )  e.  RR )
7 zre 9171 . . . 4  |-  ( N  e.  ZZ  ->  N  e.  RR )
873ad2ant2 1004 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  N  e.  RR )
9 ltle 7964 . . 3  |-  ( ( ( M  +  A
)  e.  RR  /\  N  e.  RR )  ->  ( ( M  +  A )  <  N  ->  ( M  +  A
)  <_  N )
)
106, 8, 9syl2anc 409 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  (
( M  +  A
)  <  N  ->  ( M  +  A )  <_  N ) )
11 elioo3g 9814 . . . . . 6  |-  ( A  e.  ( 0 (,) 1 )  <->  ( (
0  e.  RR*  /\  1  e.  RR*  /\  A  e. 
RR* )  /\  (
0  <  A  /\  A  <  1 ) ) )
12 simpl 108 . . . . . 6  |-  ( ( 0  <  A  /\  A  <  1 )  -> 
0  <  A )
1311, 12simplbiim 385 . . . . 5  |-  ( A  e.  ( 0 (,) 1 )  ->  0  <  A )
143, 13elrpd 9600 . . . 4  |-  ( A  e.  ( 0 (,) 1 )  ->  A  e.  RR+ )
15 addlelt 9675 . . . 4  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  (
( M  +  A
)  <_  N  ->  M  <  N ) )
161, 7, 14, 15syl3an 1262 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  (
( M  +  A
)  <_  N  ->  M  <  N ) )
17 zltp1le 9221 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <->  ( M  +  1 )  <_  N ) )
18173adant3 1002 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  ( M  <  N  <->  ( M  +  1 )  <_  N ) )
1933ad2ant3 1005 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  A  e.  RR )
20 1red 7893 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  1  e.  RR )
2113ad2ant1 1003 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  M  e.  RR )
22 simpr 109 . . . . . . . 8  |-  ( ( 0  <  A  /\  A  <  1 )  ->  A  <  1 )
2311, 22simplbiim 385 . . . . . . 7  |-  ( A  e.  ( 0 (,) 1 )  ->  A  <  1 )
24233ad2ant3 1005 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  A  <  1 )
2519, 20, 21, 24ltadd2dd 8297 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  ( M  +  A )  <  ( M  +  1 ) )
26 peano2z 9203 . . . . . . . 8  |-  ( M  e.  ZZ  ->  ( M  +  1 )  e.  ZZ )
2726zred 9286 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M  +  1 )  e.  RR )
28273ad2ant1 1003 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  ( M  +  1 )  e.  RR )
29 ltletr 7966 . . . . . 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 1220 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  (
( ( M  +  A )  <  ( M  +  1 )  /\  ( M  + 
1 )  <_  N
)  ->  ( M  +  A )  <  N
) )
3125, 30mpand 426 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  (
( M  +  1 )  <_  N  ->  ( M  +  A )  <  N ) )
3218, 31sylbid 149 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  ( M  <  N  ->  ( M  +  A )  <  N ) )
3316, 32syld 45 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  (
( M  +  A
)  <_  N  ->  ( M  +  A )  <  N ) )
3410, 33impbid 128 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 103    <-> wb 104    /\ w3a 963    e. wcel 2128   class class class wbr 3965  (class class class)co 5824   RRcr 7731   0cc0 7732   1c1 7733    + caddc 7735   RR*cxr 7911    < clt 7912    <_ cle 7913   ZZcz 9167   RR+crp 9560   (,)cioo 9792
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-0id 7840  ax-rnegex 7841  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-ltadd 7848
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-id 4253  df-po 4256  df-iso 4257  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-inn 8834  df-n0 9091  df-z 9168  df-rp 9561  df-ioo 9796
This theorem is referenced by:  halfleoddlt  11785
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