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Theorem subfzo0 9199
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 9140 . . 3  |-  ( I  e.  ( 0..^ N )  <->  ( I  e. 
NN0  /\  N  e.  NN  /\  I  <  N
) )
2 elfzo0 9140 . . . . 5  |-  ( J  e.  ( 0..^ N )  <->  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )
3 nn0re 8248 . . . . . . . . . . . 12  |-  ( I  e.  NN0  ->  I  e.  RR )
43adantr 265 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  I  <  N )  ->  I  e.  RR )
5 nnre 7997 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  RR )
6 nn0re 8248 . . . . . . . . . . . . . 14  |-  ( J  e.  NN0  ->  J  e.  RR )
7 resubcl 7338 . . . . . . . . . . . . . 14  |-  ( ( N  e.  RR  /\  J  e.  RR )  ->  ( N  -  J
)  e.  RR )
85, 6, 7syl2an 277 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  J  e.  NN0 )  -> 
( N  -  J
)  e.  RR )
98ancoms 259 . . . . . . . . . . . 12  |-  ( ( J  e.  NN0  /\  N  e.  NN )  ->  ( N  -  J
)  e.  RR )
1093adant3 935 . . . . . . . . . . 11  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  ( N  -  J )  e.  RR )
114, 10anim12i 325 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
I  e.  RR  /\  ( N  -  J
)  e.  RR ) )
12 nn0ge0 8264 . . . . . . . . . . . 12  |-  ( I  e.  NN0  ->  0  <_  I )
1312adantr 265 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  I  <  N )  -> 
0  <_  I )
14 posdif 7524 . . . . . . . . . . . . 13  |-  ( ( J  e.  RR  /\  N  e.  RR )  ->  ( J  <  N  <->  0  <  ( N  -  J ) ) )
156, 5, 14syl2an 277 . . . . . . . . . . . 12  |-  ( ( J  e.  NN0  /\  N  e.  NN )  ->  ( J  <  N  <->  0  <  ( N  -  J ) ) )
1615biimp3a 1251 . . . . . . . . . . 11  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  0  <  ( N  -  J
) )
1713, 16anim12i 325 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
0  <_  I  /\  0  <  ( N  -  J ) ) )
18 addgegt0 7518 . . . . . . . . . 10  |-  ( ( ( I  e.  RR  /\  ( N  -  J
)  e.  RR )  /\  ( 0  <_  I  /\  0  <  ( N  -  J )
) )  ->  0  <  ( I  +  ( N  -  J ) ) )
1911, 17, 18syl2anc 397 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  0  <  ( I  +  ( N  -  J ) ) )
20 nn0cn 8249 . . . . . . . . . . . 12  |-  ( I  e.  NN0  ->  I  e.  CC )
2120adantr 265 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  I  <  N )  ->  I  e.  CC )
2221adantr 265 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  I  e.  CC )
23 nn0cn 8249 . . . . . . . . . . . 12  |-  ( J  e.  NN0  ->  J  e.  CC )
24233ad2ant1 936 . . . . . . . . . . 11  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  J  e.  CC )
2524adantl 266 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  J  e.  CC )
26 nncn 7998 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  CC )
27263ad2ant2 937 . . . . . . . . . . 11  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  N  e.  CC )
2827adantl 266 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  N  e.  CC )
2922, 25, 28subadd23d 7407 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
( I  -  J
)  +  N )  =  ( I  +  ( N  -  J
) ) )
3019, 29breqtrrd 3818 . . . . . . . 8  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  0  <  ( ( I  -  J )  +  N
) )
3163ad2ant1 936 . . . . . . . . . 10  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  J  e.  RR )
32 resubcl 7338 . . . . . . . . . 10  |-  ( ( I  e.  RR  /\  J  e.  RR )  ->  ( I  -  J
)  e.  RR )
334, 31, 32syl2an 277 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
I  -  J )  e.  RR )
3453ad2ant2 937 . . . . . . . . . 10  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  N  e.  RR )
3534adantl 266 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  N  e.  RR )
3633, 35possumd 7634 . . . . . . . 8  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
0  <  ( (
I  -  J )  +  N )  <->  -u N  < 
( I  -  J
) ) )
3730, 36mpbid 139 . . . . . . 7  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  -u N  <  ( I  -  J
) )
383adantr 265 . . . . . . . . . . . 12  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  I  e.  RR )
3934adantl 266 . . . . . . . . . . . 12  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  N  e.  RR )
40 readdcl 7065 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  RR  /\  N  e.  RR )  ->  ( J  +  N
)  e.  RR )
416, 5, 40syl2an 277 . . . . . . . . . . . . . 14  |-  ( ( J  e.  NN0  /\  N  e.  NN )  ->  ( J  +  N
)  e.  RR )
42413adant3 935 . . . . . . . . . . . . 13  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  ( J  +  N )  e.  RR )
4342adantl 266 . . . . . . . . . . . 12  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  ( J  +  N )  e.  RR )
4438, 39, 433jca 1095 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  ( I  e.  RR  /\  N  e.  RR  /\  ( J  +  N )  e.  RR ) )
45 nn0ge0 8264 . . . . . . . . . . . . . 14  |-  ( J  e.  NN0  ->  0  <_  J )
46453ad2ant1 936 . . . . . . . . . . . . 13  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  0  <_  J )
4746adantl 266 . . . . . . . . . . . 12  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  0  <_  J
)
485, 6anim12i 325 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN  /\  J  e.  NN0 )  -> 
( N  e.  RR  /\  J  e.  RR ) )
4948ancoms 259 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  NN0  /\  N  e.  NN )  ->  ( N  e.  RR  /\  J  e.  RR ) )
50493adant3 935 . . . . . . . . . . . . . 14  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  ( N  e.  RR  /\  J  e.  RR ) )
5150adantl 266 . . . . . . . . . . . . 13  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  ( N  e.  RR  /\  J  e.  RR ) )
52 addge02 7542 . . . . . . . . . . . . 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 139 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  N  <_  ( J  +  N )
)
5544, 54lelttrdi 7495 . . . . . . . . . 10  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  ( I  < 
N  ->  I  <  ( J  +  N ) ) )
5655impancom 251 . . . . . . . . 9  |-  ( ( I  e.  NN0  /\  I  <  N )  -> 
( ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
)  ->  I  <  ( J  +  N ) ) )
5756imp 119 . . . . . . . 8  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  I  <  ( J  +  N
) )
584adantr 265 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  I  e.  RR )
5931adantl 266 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  J  e.  RR )
6058, 59, 35ltsubadd2d 7608 . . . . . . . 8  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
( I  -  J
)  <  N  <->  I  <  ( J  +  N ) ) )
6157, 60mpbird 160 . . . . . . 7  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
I  -  J )  <  N )
6237, 61jca 294 . . . . . 6  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  ( -u N  <  ( I  -  J )  /\  ( I  -  J
)  <  N )
)
6362ex 112 . . . . 5  |-  ( ( I  e.  NN0  /\  I  <  N )  -> 
( ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
)  ->  ( -u N  <  ( I  -  J
)  /\  ( I  -  J )  <  N
) ) )
642, 63syl5bi 145 . . . 4  |-  ( ( I  e.  NN0  /\  I  <  N )  -> 
( J  e.  ( 0..^ N )  -> 
( -u N  <  (
I  -  J )  /\  ( I  -  J )  <  N
) ) )
65643adant2 934 . . 3  |-  ( ( I  e.  NN0  /\  N  e.  NN  /\  I  <  N )  ->  ( J  e.  ( 0..^ N )  ->  ( -u N  <  ( I  -  J )  /\  ( I  -  J
)  <  N )
) )
661, 65sylbi 118 . 2  |-  ( I  e.  ( 0..^ N )  ->  ( J  e.  ( 0..^ N )  ->  ( -u N  <  ( I  -  J
)  /\  ( I  -  J )  <  N
) ) )
6766imp 119 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 101    <-> wb 102    /\ w3a 896    e. wcel 1409   class class class wbr 3792  (class class class)co 5540   CCcc 6945   RRcr 6946   0cc0 6947    + caddc 6950    < clt 7119    <_ cle 7120    - cmin 7245   -ucneg 7246   NNcn 7990   NN0cn0 8239  ..^cfzo 9101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-ltadd 7058
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-inn 7991  df-n0 8240  df-z 8303  df-uz 8570  df-fz 8977  df-fzo 9102
This theorem is referenced by:  addmodlteq  9348
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