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Theorem subfzo0 10026
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 9966 . . 3  |-  ( I  e.  ( 0..^ N )  <->  ( I  e. 
NN0  /\  N  e.  NN  /\  I  <  N
) )
2 elfzo0 9966 . . . . 5  |-  ( J  e.  ( 0..^ N )  <->  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )
3 nn0re 8993 . . . . . . . . . . . 12  |-  ( I  e.  NN0  ->  I  e.  RR )
43adantr 274 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  I  <  N )  ->  I  e.  RR )
5 nnre 8734 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  RR )
6 nn0re 8993 . . . . . . . . . . . . . 14  |-  ( J  e.  NN0  ->  J  e.  RR )
7 resubcl 8033 . . . . . . . . . . . . . 14  |-  ( ( N  e.  RR  /\  J  e.  RR )  ->  ( N  -  J
)  e.  RR )
85, 6, 7syl2an 287 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  J  e.  NN0 )  -> 
( N  -  J
)  e.  RR )
98ancoms 266 . . . . . . . . . . . 12  |-  ( ( J  e.  NN0  /\  N  e.  NN )  ->  ( N  -  J
)  e.  RR )
1093adant3 1001 . . . . . . . . . . 11  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  ( N  -  J )  e.  RR )
114, 10anim12i 336 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
I  e.  RR  /\  ( N  -  J
)  e.  RR ) )
12 nn0ge0 9009 . . . . . . . . . . . 12  |-  ( I  e.  NN0  ->  0  <_  I )
1312adantr 274 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  I  <  N )  -> 
0  <_  I )
14 posdif 8224 . . . . . . . . . . . . 13  |-  ( ( J  e.  RR  /\  N  e.  RR )  ->  ( J  <  N  <->  0  <  ( N  -  J ) ) )
156, 5, 14syl2an 287 . . . . . . . . . . . 12  |-  ( ( J  e.  NN0  /\  N  e.  NN )  ->  ( J  <  N  <->  0  <  ( N  -  J ) ) )
1615biimp3a 1323 . . . . . . . . . . 11  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  0  <  ( N  -  J
) )
1713, 16anim12i 336 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
0  <_  I  /\  0  <  ( N  -  J ) ) )
18 addgegt0 8218 . . . . . . . . . 10  |-  ( ( ( I  e.  RR  /\  ( N  -  J
)  e.  RR )  /\  ( 0  <_  I  /\  0  <  ( N  -  J )
) )  ->  0  <  ( I  +  ( N  -  J ) ) )
1911, 17, 18syl2anc 408 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  0  <  ( I  +  ( N  -  J ) ) )
20 nn0cn 8994 . . . . . . . . . . . 12  |-  ( I  e.  NN0  ->  I  e.  CC )
2120adantr 274 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  I  <  N )  ->  I  e.  CC )
2221adantr 274 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  I  e.  CC )
23 nn0cn 8994 . . . . . . . . . . . 12  |-  ( J  e.  NN0  ->  J  e.  CC )
24233ad2ant1 1002 . . . . . . . . . . 11  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  J  e.  CC )
2524adantl 275 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  J  e.  CC )
26 nncn 8735 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  CC )
27263ad2ant2 1003 . . . . . . . . . . 11  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  N  e.  CC )
2827adantl 275 . . . . . . . . . 10  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  N  e.  CC )
2922, 25, 28subadd23d 8102 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
( I  -  J
)  +  N )  =  ( I  +  ( N  -  J
) ) )
3019, 29breqtrrd 3956 . . . . . . . 8  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  0  <  ( ( I  -  J )  +  N
) )
3163ad2ant1 1002 . . . . . . . . . 10  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  J  e.  RR )
32 resubcl 8033 . . . . . . . . . 10  |-  ( ( I  e.  RR  /\  J  e.  RR )  ->  ( I  -  J
)  e.  RR )
334, 31, 32syl2an 287 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  (
I  -  J )  e.  RR )
3453ad2ant2 1003 . . . . . . . . . 10  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  N  e.  RR )
3534adantl 275 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  N  e.  RR )
3633, 35possumd 8338 . . . . . . . 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 274 . . . . . . . . . . . 12  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  I  e.  RR )
3934adantl 275 . . . . . . . . . . . 12  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  N  e.  RR )
40 readdcl 7753 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  RR  /\  N  e.  RR )  ->  ( J  +  N
)  e.  RR )
416, 5, 40syl2an 287 . . . . . . . . . . . . . 14  |-  ( ( J  e.  NN0  /\  N  e.  NN )  ->  ( J  +  N
)  e.  RR )
42413adant3 1001 . . . . . . . . . . . . 13  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  ( J  +  N )  e.  RR )
4342adantl 275 . . . . . . . . . . . 12  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  ( J  +  N )  e.  RR )
4438, 39, 433jca 1161 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  ( I  e.  RR  /\  N  e.  RR  /\  ( J  +  N )  e.  RR ) )
45 nn0ge0 9009 . . . . . . . . . . . . . 14  |-  ( J  e.  NN0  ->  0  <_  J )
46453ad2ant1 1002 . . . . . . . . . . . . 13  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  0  <_  J )
4746adantl 275 . . . . . . . . . . . 12  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  0  <_  J
)
485, 6anim12i 336 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN  /\  J  e.  NN0 )  -> 
( N  e.  RR  /\  J  e.  RR ) )
4948ancoms 266 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  NN0  /\  N  e.  NN )  ->  ( N  e.  RR  /\  J  e.  RR ) )
50493adant3 1001 . . . . . . . . . . . . . 14  |-  ( ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N )  ->  ( N  e.  RR  /\  J  e.  RR ) )
5150adantl 275 . . . . . . . . . . . . 13  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  ( N  e.  RR  /\  J  e.  RR ) )
52 addge02 8242 . . . . . . . . . . . . 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 8195 . . . . . . . . . 10  |-  ( ( I  e.  NN0  /\  ( J  e.  NN0  /\  N  e.  NN  /\  J  <  N ) )  ->  ( I  < 
N  ->  I  <  ( J  +  N ) ) )
5655impancom 258 . . . . . . . . 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 274 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  I  e.  RR )
5931adantl 275 . . . . . . . . 9  |-  ( ( ( I  e.  NN0  /\  I  <  N )  /\  ( J  e. 
NN0  /\  N  e.  NN  /\  J  <  N
) )  ->  J  e.  RR )
6058, 59, 35ltsubadd2d 8312 . . . . . . . 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 304 . . . . . 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 1000 . . 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 962    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7625   RRcr 7626   0cc0 7627    + caddc 7630    < clt 7807    <_ cle 7808    - cmin 7940   -ucneg 7941   NNcn 8727   NN0cn0 8984  ..^cfzo 9926
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-ltadd 7743
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-inn 8728  df-n0 8985  df-z 9062  df-uz 9334  df-fz 9798  df-fzo 9927
This theorem is referenced by:  addmodlteq  10178
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