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Theorem fzo1fzo0n0 9972
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 9939 . . 3  |-  ( K  e.  ( 1..^ N )  <->  ( K  e.  ( ZZ>= `  1 )  /\  N  e.  ZZ  /\  K  <  N ) )
2 elnnuz 9374 . . . . . . 7  |-  ( K  e.  NN  <->  K  e.  ( ZZ>= `  1 )
)
3 nnnn0 8996 . . . . . . . . . . 11  |-  ( K  e.  NN  ->  K  e.  NN0 )
43adantr 274 . . . . . . . . . 10  |-  ( ( K  e.  NN  /\  N  e.  ZZ )  ->  K  e.  NN0 )
54adantr 274 . . . . . . . . 9  |-  ( ( ( K  e.  NN  /\  N  e.  ZZ )  /\  K  <  N
)  ->  K  e.  NN0 )
6 nngt0 8757 . . . . . . . . . . 11  |-  ( K  e.  NN  ->  0  <  K )
7 0red 7779 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  0  e.  RR )
8 nnre 8739 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN  ->  K  e.  RR )
98adantl 275 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  K  e.  RR )
10 zre 9070 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ZZ  ->  N  e.  RR )
1110adantr 274 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  N  e.  RR )
12 lttr 7850 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  K  e.  RR  /\  N  e.  RR )  ->  (
( 0  <  K  /\  K  <  N )  ->  0  <  N
) )
137, 9, 11, 12syl3anc 1216 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  ( ( 0  < 
K  /\  K  <  N )  ->  0  <  N ) )
14 elnnz 9076 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  < 
N ) )
1514simplbi2 382 . . . . . . . . . . . . . . 15  |-  ( N  e.  ZZ  ->  (
0  <  N  ->  N  e.  NN ) )
1615adantr 274 . . . . . . . . . . . . . 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 364 . . . . . . . . . . . 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 254 . . . . . . . . 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 1161 . . . . . . . 8  |-  ( ( ( K  e.  NN  /\  N  e.  ZZ )  /\  K  <  N
)  ->  ( K  e.  NN0  /\  N  e.  NN  /\  K  < 
N ) )
2423exp31 361 . . . . . . 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 1175 . . . . 5  |-  ( ( K  e.  ( ZZ>= ` 
1 )  /\  N  e.  ZZ  /\  K  < 
N )  ->  ( K  e.  NN0  /\  N  e.  NN  /\  K  < 
N ) )
27 elfzo0 9971 . . . . 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 8760 . . . . . 6  |-  ( K  e.  NN  ->  K  =/=  0 )
302, 29sylbir 134 . . . . 5  |-  ( K  e.  ( ZZ>= `  1
)  ->  K  =/=  0 )
31303ad2ant1 1002 . . . 4  |-  ( ( K  e.  ( ZZ>= ` 
1 )  /\  N  e.  ZZ  /\  K  < 
N )  ->  K  =/=  0 )
3228, 31jca 304 . . 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 9003 . . . . . 6  |-  ( K  e.  NN  <->  ( K  e.  NN0  /\  K  =/=  0 ) )
35 nnge1 8755 . . . . . 6  |-  ( K  e.  NN  ->  1  <_  K )
3634, 35sylbir 134 . . . . 5  |-  ( ( K  e.  NN0  /\  K  =/=  0 )  -> 
1  <_  K )
37363ad2antl1 1143 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  -> 
1  <_  K )
38 simpl3 986 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  ->  K  <  N )
39 nn0z 9086 . . . . . . . . 9  |-  ( K  e.  NN0  ->  K  e.  ZZ )
4039adantr 274 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  K  e.  ZZ )
41 1zzd 9093 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  1  e.  ZZ )
42 nnz 9085 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  ZZ )
4342adantl 275 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  N  e.  ZZ )
4440, 41, 433jca 1161 . . . . . . 7  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  ( K  e.  ZZ  /\  1  e.  ZZ  /\  N  e.  ZZ )
)
45443adant3 1001 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  ->  ( K  e.  ZZ  /\  1  e.  ZZ  /\  N  e.  ZZ ) )
4645adantr 274 . . . . 5  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  -> 
( K  e.  ZZ  /\  1  e.  ZZ  /\  N  e.  ZZ )
)
47 elfzo 9938 . . . . 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 928 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  ->  K  e.  ( 1..^ N ) )
5027, 49sylanb 282 . 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 962    e. wcel 1480    =/= wne 2308   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   RRcr 7631   0cc0 7632   1c1 7633    < clt 7812    <_ cle 7813   NNcn 8732   NN0cn0 8989   ZZcz 9066   ZZ>=cuz 9338  ..^cfzo 9931
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 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-ltadd 7748
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-nul 3364  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 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-n0 8990  df-z 9067  df-uz 9339  df-fz 9803  df-fzo 9932
This theorem is referenced by: (None)
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