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Theorem fzofzim 28158
Description: If a non-negative integer in a finite interval of integers is not the upper bound of the interval, it is contained in the corresponding half-open range of integers. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
Assertion
Ref Expression
fzofzim  |-  ( ( K  =/=  M  /\  K  e.  ( 0 ... M ) )  ->  K  e.  ( 0..^ M ) )

Proof of Theorem fzofzim
StepHypRef Expression
1 elfz2nn0 11087 . . . 4  |-  ( K  e.  ( 0 ... M )  <->  ( K  e.  NN0  /\  M  e. 
NN0  /\  K  <_  M ) )
2 simpl1 961 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  /\  K  =/=  M )  ->  K  e.  NN0 )
3 necom 2687 . . . . . . . . 9  |-  ( K  =/=  M  <->  M  =/=  K )
4 nn0re 10235 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  K  e.  RR )
5 nn0re 10235 . . . . . . . . . . . . 13  |-  ( M  e.  NN0  ->  M  e.  RR )
6 ltlen 9180 . . . . . . . . . . . . 13  |-  ( ( K  e.  RR  /\  M  e.  RR )  ->  ( K  <  M  <->  ( K  <_  M  /\  M  =/=  K ) ) )
74, 5, 6syl2an 465 . . . . . . . . . . . 12  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  <  M  <->  ( K  <_  M  /\  M  =/=  K ) ) )
87bicomd 194 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( ( K  <_  M  /\  M  =/=  K
)  <->  K  <  M ) )
9 elnn0z 10299 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  <->  ( K  e.  ZZ  /\  0  <_  K ) )
10 0re 9096 . . . . . . . . . . . . . . . . . 18  |-  0  e.  RR
1110a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
0  e.  RR )
12 zre 10291 . . . . . . . . . . . . . . . . . 18  |-  ( K  e.  ZZ  ->  K  e.  RR )
1312adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  ->  K  e.  RR )
145adantl 454 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  ->  M  e.  RR )
15 lelttr 9170 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  RR  /\  K  e.  RR  /\  M  e.  RR )  ->  (
( 0  <_  K  /\  K  <  M )  ->  0  <  M
) )
1611, 13, 14, 15syl3anc 1185 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
( ( 0  <_  K  /\  K  <  M
)  ->  0  <  M ) )
17 nn0z 10309 . . . . . . . . . . . . . . . . . 18  |-  ( M  e.  NN0  ->  M  e.  ZZ )
18 elnnz 10297 . . . . . . . . . . . . . . . . . . 19  |-  ( M  e.  NN  <->  ( M  e.  ZZ  /\  0  < 
M ) )
1918simplbi2 610 . . . . . . . . . . . . . . . . . 18  |-  ( M  e.  ZZ  ->  (
0  <  M  ->  M  e.  NN ) )
2017, 19syl 16 . . . . . . . . . . . . . . . . 17  |-  ( M  e.  NN0  ->  ( 0  <  M  ->  M  e.  NN ) )
2120adantl 454 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
( 0  <  M  ->  M  e.  NN ) )
2216, 21syld 43 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
( ( 0  <_  K  /\  K  <  M
)  ->  M  e.  NN ) )
2322exp3a 427 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
( 0  <_  K  ->  ( K  <  M  ->  M  e.  NN ) ) )
2423impancom 429 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  0  <_  K )  -> 
( M  e.  NN0  ->  ( K  <  M  ->  M  e.  NN ) ) )
259, 24sylbi 189 . . . . . . . . . . . 12  |-  ( K  e.  NN0  ->  ( M  e.  NN0  ->  ( K  <  M  ->  M  e.  NN ) ) )
2625imp 420 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  <  M  ->  M  e.  NN ) )
278, 26sylbid 208 . . . . . . . . . 10  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( ( K  <_  M  /\  M  =/=  K
)  ->  M  e.  NN ) )
2827exp3a 427 . . . . . . . . 9  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  <_  M  ->  ( M  =/=  K  ->  M  e.  NN ) ) )
293, 28syl7bi 223 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  <_  M  ->  ( K  =/=  M  ->  M  e.  NN ) ) )
30293impia 1151 . . . . . . 7  |-  ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  ->  ( K  =/=  M  ->  M  e.  NN ) )
3130imp 420 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  /\  K  =/=  M )  ->  M  e.  NN )
328biimpd 200 . . . . . . . . . 10  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( ( K  <_  M  /\  M  =/=  K
)  ->  K  <  M ) )
3332exp4b 592 . . . . . . . . 9  |-  ( K  e.  NN0  ->  ( M  e.  NN0  ->  ( K  <_  M  ->  ( M  =/=  K  ->  K  <  M ) ) ) )
34333imp 1148 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  ->  ( M  =/=  K  ->  K  <  M ) )
353, 34syl5bi 210 . . . . . . 7  |-  ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  ->  ( K  =/=  M  ->  K  <  M ) )
3635imp 420 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  /\  K  =/=  M )  ->  K  <  M )
372, 31, 363jca 1135 . . . . 5  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  /\  K  =/=  M )  -> 
( K  e.  NN0  /\  M  e.  NN  /\  K  <  M ) )
3837ex 425 . . . 4  |-  ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  ->  ( K  =/=  M  ->  ( K  e.  NN0  /\  M  e.  NN  /\  K  < 
M ) ) )
391, 38sylbi 189 . . 3  |-  ( K  e.  ( 0 ... M )  ->  ( K  =/=  M  ->  ( K  e.  NN0  /\  M  e.  NN  /\  K  < 
M ) ) )
4039impcom 421 . 2  |-  ( ( K  =/=  M  /\  K  e.  ( 0 ... M ) )  ->  ( K  e. 
NN0  /\  M  e.  NN  /\  K  <  M
) )
41 elfzo0 11176 . 2  |-  ( K  e.  ( 0..^ M )  <->  ( K  e. 
NN0  /\  M  e.  NN  /\  K  <  M
) )
4240, 41sylibr 205 1  |-  ( ( K  =/=  M  /\  K  e.  ( 0 ... M ) )  ->  K  e.  ( 0..^ M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    e. wcel 1726    =/= wne 2601   class class class wbr 4215  (class class class)co 6084   RRcr 8994   0cc0 8995    < clt 9125    <_ cle 9126   NNcn 10005   NN0cn0 10226   ZZcz 10287   ...cfz 11048  ..^cfzo 11140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-fzo 11141
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