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Theorem fzofzim 9933
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 9860 . . . 4  |-  ( K  e.  ( 0 ... M )  <->  ( K  e.  NN0  /\  M  e. 
NN0  /\  K  <_  M ) )
2 simpl1 969 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  /\  K  =/=  M )  ->  K  e.  NN0 )
3 necom 2369 . . . . . . . . 9  |-  ( K  =/=  M  <->  M  =/=  K )
4 nn0z 9042 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  K  e.  ZZ )
5 nn0z 9042 . . . . . . . . . . . . 13  |-  ( M  e.  NN0  ->  M  e.  ZZ )
6 zltlen 9097 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  <  M  <->  ( K  <_  M  /\  M  =/=  K ) ) )
74, 5, 6syl2an 287 . . . . . . . . . . . 12  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  <  M  <->  ( K  <_  M  /\  M  =/=  K ) ) )
87bicomd 140 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( ( K  <_  M  /\  M  =/=  K
)  <->  K  <  M ) )
9 elnn0z 9035 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  <->  ( K  e.  ZZ  /\  0  <_  K ) )
10 0red 7735 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
0  e.  RR )
11 zre 9026 . . . . . . . . . . . . . . . . . 18  |-  ( K  e.  ZZ  ->  K  e.  RR )
1211adantr 274 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  ->  K  e.  RR )
13 nn0re 8954 . . . . . . . . . . . . . . . . . 18  |-  ( M  e.  NN0  ->  M  e.  RR )
1413adantl 275 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  ->  M  e.  RR )
15 lelttr 7820 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  RR  /\  K  e.  RR  /\  M  e.  RR )  ->  (
( 0  <_  K  /\  K  <  M )  ->  0  <  M
) )
1610, 12, 14, 15syl3anc 1201 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
( ( 0  <_  K  /\  K  <  M
)  ->  0  <  M ) )
17 elnnz 9032 . . . . . . . . . . . . . . . . . . 19  |-  ( M  e.  NN  <->  ( M  e.  ZZ  /\  0  < 
M ) )
1817simplbi2 382 . . . . . . . . . . . . . . . . . 18  |-  ( M  e.  ZZ  ->  (
0  <  M  ->  M  e.  NN ) )
195, 18syl 14 . . . . . . . . . . . . . . . . 17  |-  ( M  e.  NN0  ->  ( 0  <  M  ->  M  e.  NN ) )
2019adantl 275 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
( 0  <  M  ->  M  e.  NN ) )
2116, 20syld 45 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
( ( 0  <_  K  /\  K  <  M
)  ->  M  e.  NN ) )
2221expd 256 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
( 0  <_  K  ->  ( K  <  M  ->  M  e.  NN ) ) )
2322impancom 258 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  0  <_  K )  -> 
( M  e.  NN0  ->  ( K  <  M  ->  M  e.  NN ) ) )
249, 23sylbi 120 . . . . . . . . . . . 12  |-  ( K  e.  NN0  ->  ( M  e.  NN0  ->  ( K  <  M  ->  M  e.  NN ) ) )
2524imp 123 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  <  M  ->  M  e.  NN ) )
268, 25sylbid 149 . . . . . . . . . 10  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( ( K  <_  M  /\  M  =/=  K
)  ->  M  e.  NN ) )
2726expd 256 . . . . . . . . 9  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  <_  M  ->  ( M  =/=  K  ->  M  e.  NN ) ) )
283, 27syl7bi 164 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( K  <_  M  ->  ( K  =/=  M  ->  M  e.  NN ) ) )
29283impia 1163 . . . . . . 7  |-  ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  ->  ( K  =/=  M  ->  M  e.  NN ) )
3029imp 123 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  /\  K  =/=  M )  ->  M  e.  NN )
318biimpd 143 . . . . . . . . . 10  |-  ( ( K  e.  NN0  /\  M  e.  NN0 )  -> 
( ( K  <_  M  /\  M  =/=  K
)  ->  K  <  M ) )
3231exp4b 364 . . . . . . . . 9  |-  ( K  e.  NN0  ->  ( M  e.  NN0  ->  ( K  <_  M  ->  ( M  =/=  K  ->  K  <  M ) ) ) )
33323imp 1160 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  ->  ( M  =/=  K  ->  K  <  M ) )
343, 33syl5bi 151 . . . . . . 7  |-  ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  ->  ( K  =/=  M  ->  K  <  M ) )
3534imp 123 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  /\  K  =/=  M )  ->  K  <  M )
362, 30, 353jca 1146 . . . . 5  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  /\  K  =/=  M )  -> 
( K  e.  NN0  /\  M  e.  NN  /\  K  <  M ) )
3736ex 114 . . . 4  |-  ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  ->  ( K  =/=  M  ->  ( K  e.  NN0  /\  M  e.  NN  /\  K  < 
M ) ) )
381, 37sylbi 120 . . 3  |-  ( K  e.  ( 0 ... M )  ->  ( K  =/=  M  ->  ( K  e.  NN0  /\  M  e.  NN  /\  K  < 
M ) ) )
3938impcom 124 . 2  |-  ( ( K  =/=  M  /\  K  e.  ( 0 ... M ) )  ->  ( K  e. 
NN0  /\  M  e.  NN  /\  K  <  M
) )
40 elfzo0 9927 . 2  |-  ( K  e.  ( 0..^ M )  <->  ( K  e. 
NN0  /\  M  e.  NN  /\  K  <  M
) )
4139, 40sylibr 133 1  |-  ( ( K  =/=  M  /\  K  e.  ( 0 ... M ) )  ->  K  e.  ( 0..^ M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 947    e. wcel 1465    =/= wne 2285   class class class wbr 3899  (class class class)co 5742   RRcr 7587   0cc0 7588    < clt 7768    <_ cle 7769   NNcn 8688   NN0cn0 8945   ZZcz 9022   ...cfz 9758  ..^cfzo 9887
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-inn 8689  df-n0 8946  df-z 9023  df-uz 9295  df-fz 9759  df-fzo 9888
This theorem is referenced by: (None)
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