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Theorem subfzo0 9714
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 9654 . . 3  |-  ( I  e.  ( 0..^ N )  <->  ( I  e. 
NN0  /\  N  e.  NN  /\  I  <  N
) )
2 elfzo0 9654 . . . . 5  |-  ( J  e.  ( 0..^ N )  <->  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )
3 nn0re 8743 . . . . . . . . . . . 12  |-  ( I  e.  NN0  ->  I  e.  RR )
43adantr 271 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  I  <  N )  ->  I  e.  RR )
5 nnre 8490 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  RR )
6 nn0re 8743 . . . . . . . . . . . . . 14  |-  ( J  e.  NN0  ->  J  e.  RR )
7 resubcl 7807 . . . . . . . . . . . . . 14  |-  ( ( N  e.  RR  /\  J  e.  RR )  ->  ( N  -  J
)  e.  RR )
85, 6, 7syl2an 284 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  J  e.  NN0 )  -> 
( N  -  J
)  e.  RR )
98ancoms 265 . . . . . . . . . . . 12  |-  ( ( J  e.  NN0  /\  N  e.  NN )  ->  ( N  -  J
)  e.  RR )
1093adant3 964 . . . . . . . . . . 11  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  ( N  -  J )  e.  RR )
114, 10anim12i 332 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
I  e.  RR  /\  ( N  -  J
)  e.  RR ) )
12 nn0ge0 8759 . . . . . . . . . . . 12  |-  ( I  e.  NN0  ->  0  <_  I )
1312adantr 271 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  I  <  N )  -> 
0  <_  I )
14 posdif 7994 . . . . . . . . . . . . 13  |-  ( ( J  e.  RR  /\  N  e.  RR )  ->  ( J  <  N  <->  0  <  ( N  -  J ) ) )
156, 5, 14syl2an 284 . . . . . . . . . . . 12  |-  ( ( J  e.  NN0  /\  N  e.  NN )  ->  ( J  <  N  <->  0  <  ( N  -  J ) ) )
1615biimp3a 1282 . . . . . . . . . . 11  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  0  <  ( N  -  J
) )
1713, 16anim12i 332 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
0  <_  I  /\  0  <  ( N  -  J ) ) )
18 addgegt0 7988 . . . . . . . . . 10  |-  ( ( ( I  e.  RR  /\  ( N  -  J
)  e.  RR )  /\  ( 0  <_  I  /\  0  <  ( N  -  J )
) )  ->  0  <  ( I  +  ( N  -  J ) ) )
1911, 17, 18syl2anc 404 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  0  <  ( I  +  ( N  -  J ) ) )
20 nn0cn 8744 . . . . . . . . . . . 12  |-  ( I  e.  NN0  ->  I  e.  CC )
2120adantr 271 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  I  <  N )  ->  I  e.  CC )
2221adantr 271 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  I  e.  CC )
23 nn0cn 8744 . . . . . . . . . . . 12  |-  ( J  e.  NN0  ->  J  e.  CC )
24233ad2ant1 965 . . . . . . . . . . 11  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  J  e.  CC )
2524adantl 272 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  J  e.  CC )
26 nncn 8491 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  CC )
27263ad2ant2 966 . . . . . . . . . . 11  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  N  e.  CC )
2827adantl 272 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  N  e.  CC )
2922, 25, 28subadd23d 7876 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
( I  -  J
)  +  N )  =  ( I  +  ( N  -  J
) ) )
3019, 29breqtrrd 3877 . . . . . . . 8  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  0  <  ( ( I  -  J )  +  N
) )
3163ad2ant1 965 . . . . . . . . . 10  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  J  e.  RR )
32 resubcl 7807 . . . . . . . . . 10  |-  ( ( I  e.  RR  /\  J  e.  RR )  ->  ( I  -  J
)  e.  RR )
334, 31, 32syl2an 284 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
I  -  J )  e.  RR )
3453ad2ant2 966 . . . . . . . . . 10  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  N  e.  RR )
3534adantl 272 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  N  e.  RR )
3633, 35possumd 8107 . . . . . . . 8  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
0  <  ( (
I  -  J )  +  N )  <->  -u N  < 
( I  -  J
) ) )
3730, 36mpbid 146 . . . . . . 7  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  -u N  <  ( I  -  J
) )
383adantr 271 . . . . . . . . . . . 12  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  I  e.  RR )
3934adantl 272 . . . . . . . . . . . 12  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  N  e.  RR )
40 readdcl 7529 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  RR  /\  N  e.  RR )  ->  ( J  +  N
)  e.  RR )
416, 5, 40syl2an 284 . . . . . . . . . . . . . 14  |-  ( ( J  e.  NN0  /\  N  e.  NN )  ->  ( J  +  N
)  e.  RR )
42413adant3 964 . . . . . . . . . . . . 13  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  ( J  +  N )  e.  RR )
4342adantl 272 . . . . . . . . . . . 12  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  ( J  +  N )  e.  RR )
4438, 39, 433jca 1124 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  ( I  e.  RR  /\  N  e.  RR  /\  ( J  +  N )  e.  RR ) )
45 nn0ge0 8759 . . . . . . . . . . . . . 14  |-  ( J  e.  NN0  ->  0  <_  J )
46453ad2ant1 965 . . . . . . . . . . . . 13  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  0  <_  J )
4746adantl 272 . . . . . . . . . . . 12  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  0  <_  J
)
485, 6anim12i 332 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN  /\  J  e.  NN0 )  -> 
( N  e.  RR  /\  J  e.  RR ) )
4948ancoms 265 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  NN0  /\  N  e.  NN )  ->  ( N  e.  RR  /\  J  e.  RR ) )
50493adant3 964 . . . . . . . . . . . . . 14  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  ( N  e.  RR  /\  J  e.  RR ) )
5150adantl 272 . . . . . . . . . . . . 13  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  ( N  e.  RR  /\  J  e.  RR ) )
52 addge02 8012 . . . . . . . . . . . . 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 146 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  N  <_  ( J  +  N )
)
5544, 54lelttrdi 7965 . . . . . . . . . 10  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  ( I  < 
N  ->  I  <  ( J  +  N ) ) )
5655impancom 257 . . . . . . . . 9  |-  ( ( I  e.  NN0  /\  I  <  N )  -> 
( ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
)  ->  I  <  ( J  +  N ) ) )
5756imp 123 . . . . . . . 8  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  I  <  ( J  +  N
) )
584adantr 271 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  I  e.  RR )
5931adantl 272 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  J  e.  RR )
6058, 59, 35ltsubadd2d 8081 . . . . . . . 8  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
( I  -  J
)  <  N  <->  I  <  ( J  +  N ) ) )
6157, 60mpbird 166 . . . . . . 7  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
I  -  J )  <  N )
6237, 61jca 301 . . . . . 6  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  ( -u N  <  ( I  -  J )  /\  ( I  -  J
)  <  N )
)
6362ex 114 . . . . 5  |-  ( ( I  e.  NN0  /\  I  <  N )  -> 
( ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
)  ->  ( -u N  <  ( I  -  J
)  /\  ( I  -  J )  <  N
) ) )
642, 63syl5bi 151 . . . 4  |-  ( ( I  e.  NN0  /\  I  <  N )  -> 
( J  e.  ( 0..^ N )  -> 
( -u N  <  (
I  -  J )  /\  ( I  -  J )  <  N
) ) )
65643adant2 963 . . 3  |-  ( ( I  e.  NN0  /\  N  e.  NN  /\  I  <  N )  ->  ( J  e.  ( 0..^ N )  ->  ( -u N  <  ( I  -  J )  /\  ( I  -  J
)  <  N )
) )
661, 65sylbi 120 . 2  |-  ( I  e.  ( 0..^ N )  ->  ( J  e.  ( 0..^ N )  ->  ( -u N  <  ( I  -  J
)  /\  ( I  -  J )  <  N
) ) )
6766imp 123 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 103    <-> wb 104    /\ w3a 925    e. wcel 1439   class class class wbr 3851  (class class class)co 5666   CCcc 7409   RRcr 7410   0cc0 7411    + caddc 7414    < clt 7583    <_ cle 7584    - cmin 7714   -ucneg 7715   NNcn 8483   NN0cn0 8734  ..^cfzo 9614
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-addcom 7506  ax-addass 7508  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-0id 7514  ax-rnegex 7515  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-ltadd 7522
This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-inn 8484  df-n0 8735  df-z 8812  df-uz 9081  df-fz 9486  df-fzo 9615
This theorem is referenced by:  addmodlteq  9866
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