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Theorem fzofzim 11711
Description: If a nonnegative integer in a finite interval of integers is not the upper bound of the interval, it is contained in the corresponding half-open integer range. (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 11598 . . . 4  |-  ( K  e.  ( 0 ... M )  <->  ( K  e.  NN0  /\  M  e. 
NN0  /\  K  <_  M ) )
2 simpl1 991 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  /\  K  =/=  M )  ->  K  e.  NN0 )
3 necom 2721 . . . . . . . . 9  |-  ( K  =/=  M  <->  M  =/=  K )
4 nn0re 10700 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  K  e.  RR )
5 nn0re 10700 . . . . . . . . . . . . 13  |-  ( M  e.  NN0  ->  M  e.  RR )
6 ltlen 9588 . . . . . . . . . . . . 13  |-  ( ( K  e.  RR  /\  M  e.  RR )  ->  ( K  <  M  <->  ( K  <_  M  /\  M  =/=  K ) ) )
74, 5, 6syl2an 477 . . . . . . . . . . . 12  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  <  M  <->  ( K  <_  M  /\  M  =/=  K ) ) )
87bicomd 201 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( ( K  <_  M  /\  M  =/=  K
)  <->  K  <  M ) )
9 elnn0z 10771 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  <->  ( K  e.  ZZ  /\  0  <_  K ) )
10 0red 9499 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
0  e.  RR )
11 zre 10762 . . . . . . . . . . . . . . . . . 18  |-  ( K  e.  ZZ  ->  K  e.  RR )
1211adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  ->  K  e.  RR )
135adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  ->  M  e.  RR )
14 lelttr 9577 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  RR  /\  K  e.  RR  /\  M  e.  RR )  ->  (
( 0  <_  K  /\  K  <  M )  ->  0  <  M
) )
1510, 12, 13, 14syl3anc 1219 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
( ( 0  <_  K  /\  K  <  M
)  ->  0  <  M ) )
16 nn0z 10781 . . . . . . . . . . . . . . . . . 18  |-  ( M  e.  NN0  ->  M  e.  ZZ )
17 elnnz 10768 . . . . . . . . . . . . . . . . . . 19  |-  ( M  e.  NN  <->  ( M  e.  ZZ  /\  0  < 
M ) )
1817simplbi2 625 . . . . . . . . . . . . . . . . . 18  |-  ( M  e.  ZZ  ->  (
0  <  M  ->  M  e.  NN ) )
1916, 18syl 16 . . . . . . . . . . . . . . . . 17  |-  ( M  e.  NN0  ->  ( 0  <  M  ->  M  e.  NN ) )
2019adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
( 0  <  M  ->  M  e.  NN ) )
2115, 20syld 44 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
( ( 0  <_  K  /\  K  <  M
)  ->  M  e.  NN ) )
2221expd 436 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
( 0  <_  K  ->  ( K  <  M  ->  M  e.  NN ) ) )
2322impancom 440 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  0  <_  K )  -> 
( M  e.  NN0  ->  ( K  <  M  ->  M  e.  NN ) ) )
249, 23sylbi 195 . . . . . . . . . . . 12  |-  ( K  e.  NN0  ->  ( M  e.  NN0  ->  ( K  <  M  ->  M  e.  NN ) ) )
2524imp 429 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  <  M  ->  M  e.  NN ) )
268, 25sylbid 215 . . . . . . . . . 10  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( ( K  <_  M  /\  M  =/=  K
)  ->  M  e.  NN ) )
2726expd 436 . . . . . . . . 9  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  <_  M  ->  ( M  =/=  K  ->  M  e.  NN ) ) )
283, 27syl7bi 230 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  <_  M  ->  ( K  =/=  M  ->  M  e.  NN ) ) )
29283impia 1185 . . . . . . 7  |-  ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  ->  ( K  =/=  M  ->  M  e.  NN ) )
3029imp 429 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  /\  K  =/=  M )  ->  M  e.  NN )
318biimpd 207 . . . . . . . . . 10  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( ( K  <_  M  /\  M  =/=  K
)  ->  K  <  M ) )
3231exp4b 607 . . . . . . . . 9  |-  ( K  e.  NN0  ->  ( M  e.  NN0  ->  ( K  <_  M  ->  ( M  =/=  K  ->  K  <  M ) ) ) )
33323imp 1182 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  ->  ( M  =/=  K  ->  K  <  M ) )
343, 33syl5bi 217 . . . . . . 7  |-  ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  ->  ( K  =/=  M  ->  K  <  M ) )
3534imp 429 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  /\  K  =/=  M )  ->  K  <  M )
362, 30, 353jca 1168 . . . . 5  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  /\  K  =/=  M )  -> 
( K  e.  NN0  /\  M  e.  NN  /\  K  <  M ) )
3736ex 434 . . . 4  |-  ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  ->  ( K  =/=  M  ->  ( K  e.  NN0  /\  M  e.  NN  /\  K  < 
M ) ) )
381, 37sylbi 195 . . 3  |-  ( K  e.  ( 0 ... M )  ->  ( K  =/=  M  ->  ( K  e.  NN0  /\  M  e.  NN  /\  K  < 
M ) ) )
3938impcom 430 . 2  |-  ( ( K  =/=  M  /\  K  e.  ( 0 ... M ) )  ->  ( K  e. 
NN0  /\  M  e.  NN  /\  K  <  M
) )
40 elfzo0 11705 . 2  |-  ( K  e.  ( 0..^ M )  <->  ( K  e. 
NN0  /\  M  e.  NN  /\  K  <  M
) )
4139, 40sylibr 212 1  |-  ( ( K  =/=  M  /\  K  e.  ( 0 ... M ) )  ->  K  e.  ( 0..^ M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1758    =/= wne 2648   class class class wbr 4401  (class class class)co 6201   RRcr 9393   0cc0 9394    < clt 9530    <_ cle 9531   NNcn 10434   NN0cn0 10691   ZZcz 10758   ...cfz 11555  ..^cfzo 11666
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 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667
This theorem is referenced by:  cshwshashlem1  14241  clwwisshclwwn  30621
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