MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fzo1fzo0n0 Structured version   Unicode version

Theorem fzo1fzo0n0 11691
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 11659 . . 3  |-  ( K  e.  ( 1..^ N )  <->  ( K  e.  ( ZZ>= `  1 )  /\  N  e.  ZZ  /\  K  <  N ) )
2 elnnuz 11000 . . . . . . 7  |-  ( K  e.  NN  <->  K  e.  ( ZZ>= `  1 )
)
3 nnnn0 10689 . . . . . . . . . . . 12  |-  ( K  e.  NN  ->  K  e.  NN0 )
43adantr 465 . . . . . . . . . . 11  |-  ( ( K  e.  NN  /\  N  e.  ZZ )  ->  K  e.  NN0 )
54adantr 465 . . . . . . . . . 10  |-  ( ( ( K  e.  NN  /\  N  e.  ZZ )  /\  K  <  N
)  ->  K  e.  NN0 )
6 nngt0 10454 . . . . . . . . . . . . 13  |-  ( K  e.  NN  ->  0  <  K )
7 0red 9490 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  0  e.  RR )
8 nnre 10432 . . . . . . . . . . . . . . . . . 18  |-  ( K  e.  NN  ->  K  e.  RR )
98adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  K  e.  RR )
10 zre 10753 . . . . . . . . . . . . . . . . . 18  |-  ( N  e.  ZZ  ->  N  e.  RR )
1110adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  N  e.  RR )
12 lttr 9554 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  RR  /\  K  e.  RR  /\  N  e.  RR )  ->  (
( 0  <  K  /\  K  <  N )  ->  0  <  N
) )
137, 9, 11, 12syl3anc 1219 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  ( ( 0  < 
K  /\  K  <  N )  ->  0  <  N ) )
14 elnnz 10759 . . . . . . . . . . . . . . . . . 18  |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  < 
N ) )
1514simplbi2 625 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ZZ  ->  (
0  <  N  ->  N  e.  NN ) )
1615adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  ( 0  <  N  ->  N  e.  NN ) )
1713, 16syld 44 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  ( ( 0  < 
K  /\  K  <  N )  ->  N  e.  NN ) )
1817exp4b 607 . . . . . . . . . . . . . 14  |-  ( N  e.  ZZ  ->  ( K  e.  NN  ->  ( 0  <  K  -> 
( K  <  N  ->  N  e.  NN ) ) ) )
1918com13 80 . . . . . . . . . . . . 13  |-  ( 0  <  K  ->  ( K  e.  NN  ->  ( N  e.  ZZ  ->  ( K  <  N  ->  N  e.  NN )
) ) )
206, 19mpcom 36 . . . . . . . . . . . 12  |-  ( K  e.  NN  ->  ( N  e.  ZZ  ->  ( K  <  N  ->  N  e.  NN )
) )
2120imp 429 . . . . . . . . . . 11  |-  ( ( K  e.  NN  /\  N  e.  ZZ )  ->  ( K  <  N  ->  N  e.  NN ) )
2221imp 429 . . . . . . . . . 10  |-  ( ( ( K  e.  NN  /\  N  e.  ZZ )  /\  K  <  N
)  ->  N  e.  NN )
23 simpr 461 . . . . . . . . . 10  |-  ( ( ( K  e.  NN  /\  N  e.  ZZ )  /\  K  <  N
)  ->  K  <  N )
245, 22, 233jca 1168 . . . . . . . . 9  |-  ( ( ( K  e.  NN  /\  N  e.  ZZ )  /\  K  <  N
)  ->  ( K  e.  NN0  /\  N  e.  NN  /\  K  < 
N ) )
2524ex 434 . . . . . . . 8  |-  ( ( K  e.  NN  /\  N  e.  ZZ )  ->  ( K  <  N  ->  ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N ) ) )
2625ex 434 . . . . . . 7  |-  ( K  e.  NN  ->  ( N  e.  ZZ  ->  ( K  <  N  -> 
( K  e.  NN0  /\  N  e.  NN  /\  K  <  N ) ) ) )
272, 26sylbir 213 . . . . . 6  |-  ( K  e.  ( ZZ>= `  1
)  ->  ( N  e.  ZZ  ->  ( K  <  N  ->  ( K  e.  NN0  /\  N  e.  NN  /\  K  < 
N ) ) ) )
28273imp 1182 . . . . 5  |-  ( ( K  e.  ( ZZ>= ` 
1 )  /\  N  e.  ZZ  /\  K  < 
N )  ->  ( K  e.  NN0  /\  N  e.  NN  /\  K  < 
N ) )
29 elfzo0 11690 . . . . 5  |-  ( K  e.  ( 0..^ N )  <->  ( K  e. 
NN0  /\  N  e.  NN  /\  K  <  N
) )
3028, 29sylibr 212 . . . 4  |-  ( ( K  e.  ( ZZ>= ` 
1 )  /\  N  e.  ZZ  /\  K  < 
N )  ->  K  e.  ( 0..^ N ) )
31 nnne0 10457 . . . . . 6  |-  ( K  e.  NN  ->  K  =/=  0 )
322, 31sylbir 213 . . . . 5  |-  ( K  e.  ( ZZ>= `  1
)  ->  K  =/=  0 )
33323ad2ant1 1009 . . . 4  |-  ( ( K  e.  ( ZZ>= ` 
1 )  /\  N  e.  ZZ  /\  K  < 
N )  ->  K  =/=  0 )
3430, 33jca 532 . . 3  |-  ( ( K  e.  ( ZZ>= ` 
1 )  /\  N  e.  ZZ  /\  K  < 
N )  ->  ( K  e.  ( 0..^ N )  /\  K  =/=  0 ) )
351, 34sylbi 195 . 2  |-  ( K  e.  ( 1..^ N )  ->  ( K  e.  ( 0..^ N )  /\  K  =/=  0
) )
36 elnnne0 10696 . . . . . 6  |-  ( K  e.  NN  <->  ( K  e.  NN0  /\  K  =/=  0 ) )
37 nnge1 10451 . . . . . 6  |-  ( K  e.  NN  ->  1  <_  K )
3836, 37sylbir 213 . . . . 5  |-  ( ( K  e.  NN0  /\  K  =/=  0 )  -> 
1  <_  K )
39383ad2antl1 1150 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  -> 
1  <_  K )
40 simpl3 993 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  ->  K  <  N )
41 nn0z 10772 . . . . . . . . 9  |-  ( K  e.  NN0  ->  K  e.  ZZ )
4241adantr 465 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  K  e.  ZZ )
43 1zzd 10780 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  1  e.  ZZ )
44 nnz 10771 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  ZZ )
4544adantl 466 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  N  e.  ZZ )
4642, 43, 453jca 1168 . . . . . . 7  |-  ( ( K  e.  NN0  /\  N  e.  NN )  ->  ( K  e.  ZZ  /\  1  e.  ZZ  /\  N  e.  ZZ )
)
47463adant3 1008 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  ->  ( K  e.  ZZ  /\  1  e.  ZZ  /\  N  e.  ZZ ) )
4847adantr 465 . . . . 5  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  -> 
( K  e.  ZZ  /\  1  e.  ZZ  /\  N  e.  ZZ )
)
49 elfzo 11658 . . . . 5  |-  ( ( K  e.  ZZ  /\  1  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( 1..^ N )  <->  ( 1  <_  K  /\  K  <  N ) ) )
5048, 49syl 16 . . . 4  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  -> 
( K  e.  ( 1..^ N )  <->  ( 1  <_  K  /\  K  <  N ) ) )
5139, 40, 50mpbir2and 913 . . 3  |-  ( ( ( K  e.  NN0  /\  N  e.  NN  /\  K  <  N )  /\  K  =/=  0 )  ->  K  e.  ( 1..^ N ) )
5229, 51sylanb 472 . 2  |-  ( ( K  e.  ( 0..^ N )  /\  K  =/=  0 )  ->  K  e.  ( 1..^ N ) )
5335, 52impbii 188 1  |-  ( K  e.  ( 1..^ N )  <->  ( K  e.  ( 0..^ N )  /\  K  =/=  0
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1758    =/= wne 2644   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   RRcr 9384   0cc0 9385   1c1 9386    < clt 9521    <_ cle 9522   NNcn 10425   NN0cn0 10682   ZZcz 10749   ZZ>=cuz 10964  ..^cfzo 11651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-fzo 11652
This theorem is referenced by:  modprmn0modprm0  13979  clwwisshclww  30611
  Copyright terms: Public domain W3C validator