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Theorem fzo1fzo0n0 9522
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 9489 . . 3  |-  ( K  e.  ( 1..^ N )  <->  ( K  e.  ( ZZ>= `  1 )  /\  N  e.  ZZ  /\  K  <  N ) )
2 elnnuz 8987 . . . . . . 7  |-  ( K  e.  NN  <->  K  e.  ( ZZ>= `  1 )
)
3 nnnn0 8613 . . . . . . . . . . 11  |-  ( K  e.  NN  ->  K  e.  NN0 )
43adantr 270 . . . . . . . . . 10  |-  ( ( K  e.  NN  /\  N  e.  ZZ )  ->  K  e.  NN0 )
54adantr 270 . . . . . . . . 9  |-  ( ( ( K  e.  NN  /\  N  e.  ZZ )  /\  K  <  N
)  ->  K  e.  NN0 )
6 nngt0 8382 . . . . . . . . . . 11  |-  ( K  e.  NN  ->  0  <  K )
7 0red 7433 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  0  e.  RR )
8 nnre 8364 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN  ->  K  e.  RR )
98adantl 271 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  K  e.  RR )
10 zre 8687 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ZZ  ->  N  e.  RR )
1110adantr 270 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  N  e.  RR )
12 lttr 7503 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  K  e.  RR  /\  N  e.  RR )  ->  (
( 0  <  K  /\  K  <  N )  ->  0  <  N
) )
137, 9, 11, 12syl3anc 1172 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  ( ( 0  < 
K  /\  K  <  N )  ->  0  <  N ) )
14 elnnz 8693 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  < 
N ) )
1514simplbi2 377 . . . . . . . . . . . . . . 15  |-  ( N  e.  ZZ  ->  (
0  <  N  ->  N  e.  NN ) )
1615adantr 270 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  ( 0  <  N  ->  N  e.  NN ) )
1713, 16syld 44 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  ( ( 0  < 
K  /\  K  <  N )  ->  N  e.  NN ) )
1817exp4b 359 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  ( K  e.  NN  ->  ( 0  <  K  -> 
( K  <  N  ->  N  e.  NN ) ) ) )
1918com13 79 . . . . . . . . . . 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 252 . . . . . . . . 9  |-  ( ( ( K  e.  NN  /\  N  e.  ZZ )  /\  K  <  N
)  ->  N  e.  NN )
22 simpr 108 . . . . . . . . 9  |-  ( ( ( K  e.  NN  /\  N  e.  ZZ )  /\  K  <  N
)  ->  K  <  N )
235, 21, 223jca 1121 . . . . . . . 8  |-  ( ( ( K  e.  NN  /\  N  e.  ZZ )  /\  K  <  N
)  ->  ( K  e.  NN0  /\  N  e.  NN  /\  K  < 
N ) )
2423exp31 356 . . . . . . 7  |-  ( K  e.  NN  ->  ( N  e.  ZZ  ->  ( K  <  N  -> 
( K  e.  NN0  /\  N  e.  NN  /\  K  <  N ) ) ) )
252, 24sylbir 133 . . . . . 6  |-  ( K  e.  ( ZZ>= `  1
)  ->  ( N  e.  ZZ  ->  ( K  <  N  ->  ( K  e.  NN0  /\  N  e.  NN  /\  K  < 
N ) ) ) )
26253imp 1135 . . . . 5  |-  ( ( K  e.  ( ZZ>= ` 
1 )  /\  N  e.  ZZ  /\  K  < 
N )  ->  ( K  e.  NN0  /\  N  e.  NN  /\  K  < 
N ) )
27 elfzo0 9521 . . . . 5  |-  ( K  e.  ( 0..^ N )  <->  ( K  e. 
NN0  /\  N  e.  NN  /\  K  <  N
) )
2826, 27sylibr 132 . . . 4  |-  ( ( K  e.  ( ZZ>= ` 
1 )  /\  N  e.  ZZ  /\  K  < 
N )  ->  K  e.  ( 0..^ N ) )
29 nnne0 8385 . . . . . 6  |-  ( K  e.  NN  ->  K  =/=  0 )
302, 29sylbir 133 . . . . 5  |-  ( K  e.  ( ZZ>= `  1
)  ->  K  =/=  0 )
31303ad2ant1 962 . . . 4  |-  ( ( K  e.  ( ZZ>= ` 
1 )  /\  N  e.  ZZ  /\  K  < 
N )  ->  K  =/=  0 )
3228, 31jca 300 . . 3  |-  ( ( K  e.  ( ZZ>= ` 
1 )  /\  N  e.  ZZ  /\  K  < 
N )  ->  ( K  e.  ( 0..^ N )  /\  K  =/=  0 ) )
331, 32sylbi 119 . 2  |-  ( K  e.  ( 1..^ N )  ->  ( K  e.  ( 0..^ N )  /\  K  =/=  0
) )
34 elnnne0 8620 . . . . . 6  |-  ( K  e.  NN  <->  ( K  e.  NN0  /\  K  =/=  0 ) )
35 nnge1 8380 . . . . . 6  |-  ( K  e.  NN  ->  1  <_  K )
3634, 35sylbir 133 . . . . 5  |-  ( ( K  e.  NN0  /\  K  =/=  0 )  -> 
1  <_  K )
37363ad2antl1 1103 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  -> 
1  <_  K )
38 simpl3 946 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  ->  K  <  N )
39 nn0z 8703 . . . . . . . . 9  |-  ( K  e.  NN0  ->  K  e.  ZZ )
4039adantr 270 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  K  e.  ZZ )
41 1zzd 8710 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  1  e.  ZZ )
42 nnz 8702 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  ZZ )
4342adantl 271 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  N  e.  ZZ )
4440, 41, 433jca 1121 . . . . . . 7  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  ( K  e.  ZZ  /\  1  e.  ZZ  /\  N  e.  ZZ )
)
45443adant3 961 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  ->  ( K  e.  ZZ  /\  1  e.  ZZ  /\  N  e.  ZZ ) )
4645adantr 270 . . . . 5  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  -> 
( K  e.  ZZ  /\  1  e.  ZZ  /\  N  e.  ZZ )
)
47 elfzo 9488 . . . . 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 888 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  ->  K  e.  ( 1..^ N ) )
5027, 49sylanb 278 . 2  |-  ( ( K  e.  ( 0..^ N )  /\  K  =/=  0 )  ->  K  e.  ( 1..^ N ) )
5133, 50impbii 124 1  |-  ( K  e.  ( 1..^ N )  <->  ( K  e.  ( 0..^ N )  /\  K  =/=  0
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 922    e. wcel 1436    =/= wne 2251   class class class wbr 3820   ` cfv 4981  (class class class)co 5613   RRcr 7293   0cc0 7294   1c1 7295    < clt 7466    <_ cle 7467   NNcn 8357   NN0cn0 8606   ZZcz 8683   ZZ>=cuz 8951  ..^cfzo 9481
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-cnex 7380  ax-resscn 7381  ax-1cn 7382  ax-1re 7383  ax-icn 7384  ax-addcl 7385  ax-addrcl 7386  ax-mulcl 7387  ax-addcom 7389  ax-addass 7391  ax-distr 7393  ax-i2m1 7394  ax-0lt1 7395  ax-0id 7397  ax-rnegex 7398  ax-cnre 7400  ax-pre-ltirr 7401  ax-pre-ltwlin 7402  ax-pre-lttrn 7403  ax-pre-ltadd 7405
This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-id 4094  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-fv 4989  df-riota 5569  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869  df-pnf 7468  df-mnf 7469  df-xr 7470  df-ltxr 7471  df-le 7472  df-sub 7599  df-neg 7600  df-inn 8358  df-n0 8607  df-z 8684  df-uz 8952  df-fz 9357  df-fzo 9482
This theorem is referenced by: (None)
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