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Theorem elfzelfzlble 30134
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 11440 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  <->  ( (
0  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  (
0  <_  K  /\  K  <_  N ) ) )
2 3simpc 982 . . . . . . . 8  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  e.  ZZ  /\  K  e.  ZZ ) )
32adantr 462 . . . . . . 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 566 . . . . 5  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) ) )
6 simpl 454 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  N  e.  ZZ )
76anim2i 566 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
87ancomd 449 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( N  e.  ZZ  /\  M  e.  ZZ ) )
9 zsubcl 10683 . . . . . . 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 463 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  N  e.  ZZ )
12 simprr 751 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  K  e.  ZZ )
1310, 11, 123jca 1163 . . . . 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 1003 . . 3  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N )  /\  N  <  ( M  +  K ) )  -> 
( ( N  -  M )  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )
)
16 elfzel2 11447 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  N  e.  ZZ )
1716zred 10743 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  N  e.  RR )
1817adantl 463 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  N  e.  RR )
19 zre 10646 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  RR )
2019adantr 462 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  M  e.  RR )
21 elfzelz 11449 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ZZ )
2221zred 10743 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  K  e.  RR )
2322adantl 463 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  K  e.  RR )
2418, 20, 233jca 1163 . . . . 5  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR ) )
25 simp1 983 . . . . . . 7  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR )  ->  N  e.  RR )
26 readdcl 9361 . . . . . . . 8  |-  ( ( M  e.  RR  /\  K  e.  RR )  ->  ( M  +  K
)  e.  RR )
27263adant1 1001 . . . . . . 7  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR )  ->  ( M  +  K )  e.  RR )
28 ltle 9459 . . . . . . 7  |-  ( ( N  e.  RR  /\  ( M  +  K
)  e.  RR )  ->  ( N  < 
( M  +  K
)  ->  N  <_  ( M  +  K ) ) )
2925, 27, 28syl2anc 656 . . . . . 6  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR )  ->  ( N  <  ( M  +  K )  ->  N  <_  ( M  +  K
) ) )
30 lesubadd2 9808 . . . . . 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 1179 . . 3  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N )  /\  N  <  ( M  +  K ) )  -> 
( N  -  M
)  <_  K )
34 elfzle2 11451 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  K  <_  N )
35343ad2ant2 1005 . . 3  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N )  /\  N  <  ( M  +  K ) )  ->  K  <_  N )
3615, 33, 35jca32 532 . 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 11440 . 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 960    e. wcel 1761   class class class wbr 4289  (class class class)co 6090   RRcr 9277   0cc0 9278    + caddc 9281    < clt 9414    <_ cle 9415    - cmin 9591   ZZcz 10642   ...cfz 11433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434
This theorem is referenced by: (None)
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