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Theorem fzofzim 10131
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 10055 . . . 4  |-  ( K  e.  ( 0 ... M )  <->  ( K  e.  NN0  /\  M  e. 
NN0  /\  K  <_  M ) )
2 simpl1 995 . . . . . 6  |-  ( ( ( K  e.  NN0  /\  M  e.  NN0  /\  K  <_  M )  /\  K  =/=  M )  ->  K  e.  NN0 )
3 necom 2424 . . . . . . . . 9  |-  ( K  =/=  M  <->  M  =/=  K )
4 nn0z 9219 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  K  e.  ZZ )
5 nn0z 9219 . . . . . . . . . . . . 13  |-  ( M  e.  NN0  ->  M  e.  ZZ )
6 zltlen 9277 . . . . . . . . . . . . 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 9212 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  <->  ( K  e.  ZZ  /\  0  <_  K ) )
10 0red 7908 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
0  e.  RR )
11 zre 9203 . . . . . . . . . . . . . . . . . 18  |-  ( K  e.  ZZ  ->  K  e.  RR )
1211adantr 274 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  ->  K  e.  RR )
13 nn0re 9131 . . . . . . . . . . . . . . . . . 18  |-  ( M  e.  NN0  ->  M  e.  RR )
1413adantl 275 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  ->  M  e.  RR )
15 lelttr 7995 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  RR  /\  K  e.  RR  /\  M  e.  RR )  ->  (
( 0  <_  K  /\  K  <  M )  ->  0  <  M
) )
1610, 12, 14, 15syl3anc 1233 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
( ( 0  <_  K  /\  K  <  M
)  ->  0  <  M ) )
17 elnnz 9209 . . . . . . . . . . . . . . . . . . 19  |-  ( M  e.  NN  <->  ( M  e.  ZZ  /\  0  < 
M ) )
1817simplbi2 383 . . . . . . . . . . . . . . . . . 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 1195 . . . . . . 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 365 . . . . . . . . 9  |-  ( K  e.  NN0  ->  ( M  e.  NN0  ->  ( K  <_  M  ->  ( M  =/=  K  ->  K  <  M ) ) ) )
33323imp 1188 . . . . . . . 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 1172 . . . . 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 10125 . 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 973    e. wcel 2141    =/= wne 2340   class class class wbr 3987  (class class class)co 5850   RRcr 7760   0cc0 7761    < clt 7941    <_ cle 7942   NNcn 8865   NN0cn0 9122   ZZcz 9199   ...cfz 9952  ..^cfzo 10085
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-inn 8866  df-n0 9123  df-z 9200  df-uz 9475  df-fz 9953  df-fzo 10086
This theorem is referenced by: (None)
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