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Theorem zltaddlt1le 9964
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 9216 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  RR )
21adantr 274 . . . . 5  |-  ( ( M  e.  ZZ  /\  A  e.  ( 0 (,) 1 ) )  ->  M  e.  RR )
3 elioore 9869 . . . . . 6  |-  ( A  e.  ( 0 (,) 1 )  ->  A  e.  RR )
43adantl 275 . . . . 5  |-  ( ( M  e.  ZZ  /\  A  e.  ( 0 (,) 1 ) )  ->  A  e.  RR )
52, 4readdcld 7949 . . . 4  |-  ( ( M  e.  ZZ  /\  A  e.  ( 0 (,) 1 ) )  ->  ( M  +  A )  e.  RR )
653adant2 1011 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  ( M  +  A )  e.  RR )
7 zre 9216 . . . 4  |-  ( N  e.  ZZ  ->  N  e.  RR )
873ad2ant2 1014 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  N  e.  RR )
9 ltle 8007 . . 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 9867 . . . . . 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 9650 . . . 4  |-  ( A  e.  ( 0 (,) 1 )  ->  A  e.  RR+ )
15 addlelt 9725 . . . 4  |-  ( ( M  e.  RR  /\  N  e.  RR  /\  A  e.  RR+ )  ->  (
( M  +  A
)  <_  N  ->  M  <  N ) )
161, 7, 14, 15syl3an 1275 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  (
( M  +  A
)  <_  N  ->  M  <  N ) )
17 zltp1le 9266 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <->  ( M  +  1 )  <_  N ) )
18173adant3 1012 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  ( M  <  N  <->  ( M  +  1 )  <_  N ) )
1933ad2ant3 1015 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  A  e.  RR )
20 1red 7935 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  1  e.  RR )
2113ad2ant1 1013 . . . . . 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 1015 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  A  <  1 )
2519, 20, 21, 24ltadd2dd 8341 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  ( M  +  A )  <  ( M  +  1 ) )
26 peano2z 9248 . . . . . . . 8  |-  ( M  e.  ZZ  ->  ( M  +  1 )  e.  ZZ )
2726zred 9334 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M  +  1 )  e.  RR )
28273ad2ant1 1013 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  ( M  +  1 )  e.  RR )
29 ltletr 8009 . . . . . 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 1233 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  A  e.  ( 0 (,) 1
) )  ->  (
( ( M  +  A )  <  ( M  +  1 )  /\  ( M  + 
1 )  <_  N
)  ->  ( M  +  A )  <  N
) )
3125, 30mpand 427 . . . 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 973    e. wcel 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   0cc0 7774   1c1 7775    + caddc 7777   RR*cxr 7953    < clt 7954    <_ cle 7955   ZZcz 9212   RR+crp 9610   (,)cioo 9845
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-rp 9611  df-ioo 9849
This theorem is referenced by:  halfleoddlt  11853
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