ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  subfzo0 Unicode version

Theorem subfzo0 10274
Description: The difference between two elements in a half-open range of nonnegative integers is greater than the negation of the upper bound and less than the upper bound of the range. (Contributed by AV, 20-Mar-2021.)
Assertion
Ref Expression
subfzo0  |-  ( ( I  e.  ( 0..^ N )  /\  J  e.  ( 0..^ N ) )  ->  ( -u N  <  ( I  -  J
)  /\  ( I  -  J )  <  N
) )

Proof of Theorem subfzo0
StepHypRef Expression
1 elfzo0 10214 . . 3  |-  ( I  e.  ( 0..^ N )  <->  ( I  e. 
NN0  /\  N  e.  NN  /\  I  <  N
) )
2 elfzo0 10214 . . . . 5  |-  ( J  e.  ( 0..^ N )  <->  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )
3 nn0re 9216 . . . . . . . . . . . 12  |-  ( I  e.  NN0  ->  I  e.  RR )
43adantr 276 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  I  <  N )  ->  I  e.  RR )
5 nnre 8957 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  RR )
6 nn0re 9216 . . . . . . . . . . . . . 14  |-  ( J  e.  NN0  ->  J  e.  RR )
7 resubcl 8252 . . . . . . . . . . . . . 14  |-  ( ( N  e.  RR  /\  J  e.  RR )  ->  ( N  -  J
)  e.  RR )
85, 6, 7syl2an 289 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  J  e.  NN0 )  -> 
( N  -  J
)  e.  RR )
98ancoms 268 . . . . . . . . . . . 12  |-  ( ( J  e.  NN0  /\  N  e.  NN )  ->  ( N  -  J
)  e.  RR )
1093adant3 1019 . . . . . . . . . . 11  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  ( N  -  J )  e.  RR )
114, 10anim12i 338 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
I  e.  RR  /\  ( N  -  J
)  e.  RR ) )
12 nn0ge0 9232 . . . . . . . . . . . 12  |-  ( I  e.  NN0  ->  0  <_  I )
1312adantr 276 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  I  <  N )  -> 
0  <_  I )
14 posdif 8443 . . . . . . . . . . . . 13  |-  ( ( J  e.  RR  /\  N  e.  RR )  ->  ( J  <  N  <->  0  <  ( N  -  J ) ) )
156, 5, 14syl2an 289 . . . . . . . . . . . 12  |-  ( ( J  e.  NN0  /\  N  e.  NN )  ->  ( J  <  N  <->  0  <  ( N  -  J ) ) )
1615biimp3a 1356 . . . . . . . . . . 11  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  0  <  ( N  -  J
) )
1713, 16anim12i 338 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
0  <_  I  /\  0  <  ( N  -  J ) ) )
18 addgegt0 8437 . . . . . . . . . 10  |-  ( ( ( I  e.  RR  /\  ( N  -  J
)  e.  RR )  /\  ( 0  <_  I  /\  0  <  ( N  -  J )
) )  ->  0  <  ( I  +  ( N  -  J ) ) )
1911, 17, 18syl2anc 411 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  0  <  ( I  +  ( N  -  J ) ) )
20 nn0cn 9217 . . . . . . . . . . . 12  |-  ( I  e.  NN0  ->  I  e.  CC )
2120adantr 276 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  I  <  N )  ->  I  e.  CC )
2221adantr 276 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  I  e.  CC )
23 nn0cn 9217 . . . . . . . . . . . 12  |-  ( J  e.  NN0  ->  J  e.  CC )
24233ad2ant1 1020 . . . . . . . . . . 11  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  J  e.  CC )
2524adantl 277 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  J  e.  CC )
26 nncn 8958 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  CC )
27263ad2ant2 1021 . . . . . . . . . . 11  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  N  e.  CC )
2827adantl 277 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  N  e.  CC )
2922, 25, 28subadd23d 8321 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
( I  -  J
)  +  N )  =  ( I  +  ( N  -  J
) ) )
3019, 29breqtrrd 4046 . . . . . . . 8  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  0  <  ( ( I  -  J )  +  N
) )
3163ad2ant1 1020 . . . . . . . . . 10  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  J  e.  RR )
32 resubcl 8252 . . . . . . . . . 10  |-  ( ( I  e.  RR  /\  J  e.  RR )  ->  ( I  -  J
)  e.  RR )
334, 31, 32syl2an 289 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
I  -  J )  e.  RR )
3453ad2ant2 1021 . . . . . . . . . 10  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  N  e.  RR )
3534adantl 277 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  N  e.  RR )
3633, 35possumd 8557 . . . . . . . 8  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
0  <  ( (
I  -  J )  +  N )  <->  -u N  < 
( I  -  J
) ) )
3730, 36mpbid 147 . . . . . . 7  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  -u N  <  ( I  -  J
) )
383adantr 276 . . . . . . . . . . . 12  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  I  e.  RR )
3934adantl 277 . . . . . . . . . . . 12  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  N  e.  RR )
40 readdcl 7968 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  RR  /\  N  e.  RR )  ->  ( J  +  N
)  e.  RR )
416, 5, 40syl2an 289 . . . . . . . . . . . . . 14  |-  ( ( J  e.  NN0  /\  N  e.  NN )  ->  ( J  +  N
)  e.  RR )
42413adant3 1019 . . . . . . . . . . . . 13  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  ( J  +  N )  e.  RR )
4342adantl 277 . . . . . . . . . . . 12  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  ( J  +  N )  e.  RR )
4438, 39, 433jca 1179 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  ( I  e.  RR  /\  N  e.  RR  /\  ( J  +  N )  e.  RR ) )
45 nn0ge0 9232 . . . . . . . . . . . . . 14  |-  ( J  e.  NN0  ->  0  <_  J )
46453ad2ant1 1020 . . . . . . . . . . . . 13  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  0  <_  J )
4746adantl 277 . . . . . . . . . . . 12  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  0  <_  J
)
485, 6anim12i 338 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN  /\  J  e.  NN0 )  -> 
( N  e.  RR  /\  J  e.  RR ) )
4948ancoms 268 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  NN0  /\  N  e.  NN )  ->  ( N  e.  RR  /\  J  e.  RR ) )
50493adant3 1019 . . . . . . . . . . . . . 14  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  ( N  e.  RR  /\  J  e.  RR ) )
5150adantl 277 . . . . . . . . . . . . 13  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  ( N  e.  RR  /\  J  e.  RR ) )
52 addge02 8461 . . . . . . . . . . . . 13  |-  ( ( N  e.  RR  /\  J  e.  RR )  ->  ( 0  <_  J  <->  N  <_  ( J  +  N ) ) )
5351, 52syl 14 . . . . . . . . . . . 12  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  ( 0  <_  J 
<->  N  <_  ( J  +  N ) ) )
5447, 53mpbid 147 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  N  <_  ( J  +  N )
)
5544, 54lelttrdi 8414 . . . . . . . . . 10  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  ( I  < 
N  ->  I  <  ( J  +  N ) ) )
5655impancom 260 . . . . . . . . 9  |-  ( ( I  e.  NN0  /\  I  <  N )  -> 
( ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
)  ->  I  <  ( J  +  N ) ) )
5756imp 124 . . . . . . . 8  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  I  <  ( J  +  N
) )
584adantr 276 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  I  e.  RR )
5931adantl 277 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  J  e.  RR )
6058, 59, 35ltsubadd2d 8531 . . . . . . . 8  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
( I  -  J
)  <  N  <->  I  <  ( J  +  N ) ) )
6157, 60mpbird 167 . . . . . . 7  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
I  -  J )  <  N )
6237, 61jca 306 . . . . . 6  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  ( -u N  <  ( I  -  J )  /\  ( I  -  J
)  <  N )
)
6362ex 115 . . . . 5  |-  ( ( I  e.  NN0  /\  I  <  N )  -> 
( ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
)  ->  ( -u N  <  ( I  -  J
)  /\  ( I  -  J )  <  N
) ) )
642, 63biimtrid 152 . . . 4  |-  ( ( I  e.  NN0  /\  I  <  N )  -> 
( J  e.  ( 0..^ N )  -> 
( -u N  <  (
I  -  J )  /\  ( I  -  J )  <  N
) ) )
65643adant2 1018 . . 3  |-  ( ( I  e.  NN0  /\  N  e.  NN  /\  I  <  N )  ->  ( J  e.  ( 0..^ N )  ->  ( -u N  <  ( I  -  J )  /\  ( I  -  J
)  <  N )
) )
661, 65sylbi 121 . 2  |-  ( I  e.  ( 0..^ N )  ->  ( J  e.  ( 0..^ N )  ->  ( -u N  <  ( I  -  J
)  /\  ( I  -  J )  <  N
) ) )
6766imp 124 1  |-  ( ( I  e.  ( 0..^ N )  /\  J  e.  ( 0..^ N ) )  ->  ( -u N  <  ( I  -  J
)  /\  ( I  -  J )  <  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    e. wcel 2160   class class class wbr 4018  (class class class)co 5897   CCcc 7840   RRcr 7841   0cc0 7842    + caddc 7845    < clt 8023    <_ cle 8024    - cmin 8159   -ucneg 8160   NNcn 8950   NN0cn0 9207  ..^cfzo 10174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-0id 7950  ax-rnegex 7951  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-ltadd 7958
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-inn 8951  df-n0 9208  df-z 9285  df-uz 9560  df-fz 10041  df-fzo 10175
This theorem is referenced by:  addmodlteq  10431
  Copyright terms: Public domain W3C validator