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Theorem elfzelfzlble 30214
Description: Membership of an element of a finite set of sequential integers in a finite set of sequential integers with the same upper bound and a lower bound less then the upper bound. (Contributed by AV, 21-Oct-2018.)
Assertion
Ref Expression
elfzelfzlble  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N )  /\  N  <  ( M  +  K ) )  ->  K  e.  ( ( N  -  M ) ... N ) )

Proof of Theorem elfzelfzlble
StepHypRef Expression
1 elfz2 11449 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  <->  ( (
0  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  (
0  <_  K  /\  K  <_  N ) ) )
2 3simpc 987 . . . . . . . 8  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  e.  ZZ  /\  K  e.  ZZ ) )
32adantr 465 . . . . . . 7  |-  ( ( ( 0  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( 0  <_  K  /\  K  <_  N ) )  ->  ( N  e.  ZZ  /\  K  e.  ZZ ) )
41, 3sylbi 195 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( N  e.  ZZ  /\  K  e.  ZZ ) )
54anim2i 569 . . . . 5  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) ) )
6 simpl 457 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  N  e.  ZZ )
76anim2i 569 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
87ancomd 451 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( N  e.  ZZ  /\  M  e.  ZZ ) )
9 zsubcl 10692 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  -  M
)  e.  ZZ )
108, 9syl 16 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( N  -  M )  e.  ZZ )
116adantl 466 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  N  e.  ZZ )
12 simprr 756 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  K  e.  ZZ )
1310, 11, 123jca 1168 . . . . 5  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( ( N  -  M )  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ ) )
145, 13syl 16 . . . 4  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  ( ( N  -  M )  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ ) )
15143adant3 1008 . . 3  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N )  /\  N  <  ( M  +  K ) )  -> 
( ( N  -  M )  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )
)
16 elfzel2 11456 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  N  e.  ZZ )
1716zred 10752 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  N  e.  RR )
1817adantl 466 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  N  e.  RR )
19 zre 10655 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  RR )
2019adantr 465 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  M  e.  RR )
21 elfzelz 11458 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ZZ )
2221zred 10752 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  K  e.  RR )
2322adantl 466 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  K  e.  RR )
2418, 20, 233jca 1168 . . . . 5  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR ) )
25 simp1 988 . . . . . . 7  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR )  ->  N  e.  RR )
26 readdcl 9370 . . . . . . . 8  |-  ( ( M  e.  RR  /\  K  e.  RR )  ->  ( M  +  K
)  e.  RR )
27263adant1 1006 . . . . . . 7  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR )  ->  ( M  +  K )  e.  RR )
28 ltle 9468 . . . . . . 7  |-  ( ( N  e.  RR  /\  ( M  +  K
)  e.  RR )  ->  ( N  < 
( M  +  K
)  ->  N  <_  ( M  +  K ) ) )
2925, 27, 28syl2anc 661 . . . . . 6  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR )  ->  ( N  <  ( M  +  K )  ->  N  <_  ( M  +  K
) ) )
30 lesubadd2 9817 . . . . . 6  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR )  ->  (
( N  -  M
)  <_  K  <->  N  <_  ( M  +  K ) ) )
3129, 30sylibrd 234 . . . . 5  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR )  ->  ( N  <  ( M  +  K )  ->  ( N  -  M )  <_  K ) )
3224, 31syl 16 . . . 4  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  ( N  < 
( M  +  K
)  ->  ( N  -  M )  <_  K
) )
33323impia 1184 . . 3  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N )  /\  N  <  ( M  +  K ) )  -> 
( N  -  M
)  <_  K )
34 elfzle2 11460 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  K  <_  N )
35343ad2ant2 1010 . . 3  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N )  /\  N  <  ( M  +  K ) )  ->  K  <_  N )
3615, 33, 35jca32 535 . 2  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N )  /\  N  <  ( M  +  K ) )  -> 
( ( ( N  -  M )  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  (
( N  -  M
)  <_  K  /\  K  <_  N ) ) )
37 elfz2 11449 . 2  |-  ( K  e.  ( ( N  -  M ) ... N )  <->  ( (
( N  -  M
)  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  (
( N  -  M
)  <_  K  /\  K  <_  N ) ) )
3836, 37sylibr 212 1  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N )  /\  N  <  ( M  +  K ) )  ->  K  e.  ( ( N  -  M ) ... N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1756   class class class wbr 4297  (class class class)co 6096   RRcr 9286   0cc0 9287    + caddc 9290    < clt 9423    <_ cle 9424    - cmin 9600   ZZcz 10651   ...cfz 11442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443
This theorem is referenced by: (None)
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