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Theorem fzo1fzo0n0 9655
Description: An integer between 1 and an upper bound of a half-open integer range is not 0 and between 0 and the upper bound of the half-open integer range. (Contributed by Alexander van der Vekens, 21-Mar-2018.)
Assertion
Ref Expression
fzo1fzo0n0  |-  ( K  e.  ( 1..^ N )  <->  ( K  e.  ( 0..^ N )  /\  K  =/=  0
) )

Proof of Theorem fzo1fzo0n0
StepHypRef Expression
1 elfzo2 9622 . . 3  |-  ( K  e.  ( 1..^ N )  <->  ( K  e.  ( ZZ>= `  1 )  /\  N  e.  ZZ  /\  K  <  N ) )
2 elnnuz 9116 . . . . . . 7  |-  ( K  e.  NN  <->  K  e.  ( ZZ>= `  1 )
)
3 nnnn0 8741 . . . . . . . . . . 11  |-  ( K  e.  NN  ->  K  e.  NN0 )
43adantr 271 . . . . . . . . . 10  |-  ( ( K  e.  NN  /\  N  e.  ZZ )  ->  K  e.  NN0 )
54adantr 271 . . . . . . . . 9  |-  ( ( ( K  e.  NN  /\  N  e.  ZZ )  /\  K  <  N
)  ->  K  e.  NN0 )
6 nngt0 8508 . . . . . . . . . . 11  |-  ( K  e.  NN  ->  0  <  K )
7 0red 7550 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  0  e.  RR )
8 nnre 8490 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN  ->  K  e.  RR )
98adantl 272 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  K  e.  RR )
10 zre 8815 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ZZ  ->  N  e.  RR )
1110adantr 271 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  N  e.  RR )
12 lttr 7620 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  K  e.  RR  /\  N  e.  RR )  ->  (
( 0  <  K  /\  K  <  N )  ->  0  <  N
) )
137, 9, 11, 12syl3anc 1175 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  ( ( 0  < 
K  /\  K  <  N )  ->  0  <  N ) )
14 elnnz 8821 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  < 
N ) )
1514simplbi2 378 . . . . . . . . . . . . . . 15  |-  ( N  e.  ZZ  ->  (
0  <  N  ->  N  e.  NN ) )
1615adantr 271 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  ( 0  <  N  ->  N  e.  NN ) )
1713, 16syld 45 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  ( ( 0  < 
K  /\  K  <  N )  ->  N  e.  NN ) )
1817exp4b 360 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  ( K  e.  NN  ->  ( 0  <  K  -> 
( K  <  N  ->  N  e.  NN ) ) ) )
1918com13 80 . . . . . . . . . . 11  |-  ( 0  <  K  ->  ( K  e.  NN  ->  ( N  e.  ZZ  ->  ( K  <  N  ->  N  e.  NN )
) ) )
206, 19mpcom 36 . . . . . . . . . 10  |-  ( K  e.  NN  ->  ( N  e.  ZZ  ->  ( K  <  N  ->  N  e.  NN )
) )
2120imp31 253 . . . . . . . . 9  |-  ( ( ( K  e.  NN  /\  N  e.  ZZ )  /\  K  <  N
)  ->  N  e.  NN )
22 simpr 109 . . . . . . . . 9  |-  ( ( ( K  e.  NN  /\  N  e.  ZZ )  /\  K  <  N
)  ->  K  <  N )
235, 21, 223jca 1124 . . . . . . . 8  |-  ( ( ( K  e.  NN  /\  N  e.  ZZ )  /\  K  <  N
)  ->  ( K  e.  NN0  /\  N  e.  NN  /\  K  < 
N ) )
2423exp31 357 . . . . . . 7  |-  ( K  e.  NN  ->  ( N  e.  ZZ  ->  ( K  <  N  -> 
( K  e.  NN0  /\  N  e.  NN  /\  K  <  N ) ) ) )
252, 24sylbir 134 . . . . . 6  |-  ( K  e.  ( ZZ>= `  1
)  ->  ( N  e.  ZZ  ->  ( K  <  N  ->  ( K  e.  NN0  /\  N  e.  NN  /\  K  < 
N ) ) ) )
26253imp 1138 . . . . 5  |-  ( ( K  e.  ( ZZ>= ` 
1 )  /\  N  e.  ZZ  /\  K  < 
N )  ->  ( K  e.  NN0  /\  N  e.  NN  /\  K  < 
N ) )
27 elfzo0 9654 . . . . 5  |-  ( K  e.  ( 0..^ N )  <->  ( K  e. 
NN0  /\  N  e.  NN  /\  K  <  N
) )
2826, 27sylibr 133 . . . 4  |-  ( ( K  e.  ( ZZ>= ` 
1 )  /\  N  e.  ZZ  /\  K  < 
N )  ->  K  e.  ( 0..^ N ) )
29 nnne0 8511 . . . . . 6  |-  ( K  e.  NN  ->  K  =/=  0 )
302, 29sylbir 134 . . . . 5  |-  ( K  e.  ( ZZ>= `  1
)  ->  K  =/=  0 )
31303ad2ant1 965 . . . 4  |-  ( ( K  e.  ( ZZ>= ` 
1 )  /\  N  e.  ZZ  /\  K  < 
N )  ->  K  =/=  0 )
3228, 31jca 301 . . 3  |-  ( ( K  e.  ( ZZ>= ` 
1 )  /\  N  e.  ZZ  /\  K  < 
N )  ->  ( K  e.  ( 0..^ N )  /\  K  =/=  0 ) )
331, 32sylbi 120 . 2  |-  ( K  e.  ( 1..^ N )  ->  ( K  e.  ( 0..^ N )  /\  K  =/=  0
) )
34 elnnne0 8748 . . . . . 6  |-  ( K  e.  NN  <->  ( K  e.  NN0  /\  K  =/=  0 ) )
35 nnge1 8506 . . . . . 6  |-  ( K  e.  NN  ->  1  <_  K )
3634, 35sylbir 134 . . . . 5  |-  ( ( K  e.  NN0  /\  K  =/=  0 )  -> 
1  <_  K )
37363ad2antl1 1106 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  -> 
1  <_  K )
38 simpl3 949 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  ->  K  <  N )
39 nn0z 8831 . . . . . . . . 9  |-  ( K  e.  NN0  ->  K  e.  ZZ )
4039adantr 271 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  K  e.  ZZ )
41 1zzd 8838 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  1  e.  ZZ )
42 nnz 8830 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  ZZ )
4342adantl 272 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  N  e.  ZZ )
4440, 41, 433jca 1124 . . . . . . 7  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  ( K  e.  ZZ  /\  1  e.  ZZ  /\  N  e.  ZZ )
)
45443adant3 964 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  ->  ( K  e.  ZZ  /\  1  e.  ZZ  /\  N  e.  ZZ ) )
4645adantr 271 . . . . 5  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  -> 
( K  e.  ZZ  /\  1  e.  ZZ  /\  N  e.  ZZ )
)
47 elfzo 9621 . . . . 5  |-  ( ( K  e.  ZZ  /\  1  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( 1..^ N )  <->  ( 1  <_  K  /\  K  <  N ) ) )
4846, 47syl 14 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  -> 
( K  e.  ( 1..^ N )  <->  ( 1  <_  K  /\  K  <  N ) ) )
4937, 38, 48mpbir2and 891 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  ->  K  e.  ( 1..^ N ) )
5027, 49sylanb 279 . 2  |-  ( ( K  e.  ( 0..^ N )  /\  K  =/=  0 )  ->  K  e.  ( 1..^ N ) )
5133, 50impbii 125 1  |-  ( K  e.  ( 1..^ N )  <->  ( K  e.  ( 0..^ N )  /\  K  =/=  0
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 925    e. wcel 1439    =/= wne 2256   class class class wbr 3851   ` cfv 5028  (class class class)co 5666   RRcr 7410   0cc0 7411   1c1 7412    < clt 7583    <_ cle 7584   NNcn 8483   NN0cn0 8734   ZZcz 8811   ZZ>=cuz 9080  ..^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-nul 3288  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: (None)
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