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Theorem fzofzim 10144
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 10068 . . . 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 9232 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  K  e.  ZZ )
5 nn0z 9232 . . . . . . . . . . . . 13  |-  ( M  e.  NN0  ->  M  e.  ZZ )
6 zltlen 9290 . . . . . . . . . . . . 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 9225 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  <->  ( K  e.  ZZ  /\  0  <_  K ) )
10 0red 7921 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  -> 
0  e.  RR )
11 zre 9216 . . . . . . . . . . . . . . . . . 18  |-  ( K  e.  ZZ  ->  K  e.  RR )
1211adantr 274 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  ->  K  e.  RR )
13 nn0re 9144 . . . . . . . . . . . . . . . . . 18  |-  ( M  e.  NN0  ->  M  e.  RR )
1413adantl 275 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  ZZ  /\  M  e.  NN0 )  ->  M  e.  RR )
15 lelttr 8008 . . . . . . . . . . . . . . . . 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 9222 . . . . . . . . . . . . . . . . . . 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 10138 . 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 3989  (class class class)co 5853   RRcr 7773   0cc0 7774    < clt 7954    <_ cle 7955   NNcn 8878   NN0cn0 9135   ZZcz 9212   ...cfz 9965  ..^cfzo 10098
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 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891
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 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966  df-fzo 10099
This theorem is referenced by: (None)
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