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Theorem elfzelfzlble 39206
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 than 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 11817 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  <->  ( (
0  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  (
0  <_  K  /\  K  <_  N ) ) )
2 3simpc 1029 . . . . . . . 8  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  e.  ZZ  /\  K  e.  ZZ ) )
32adantr 472 . . . . . . 7  |-  ( ( ( 0  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( 0  <_  K  /\  K  <_  N ) )  ->  ( N  e.  ZZ  /\  K  e.  ZZ ) )
41, 3sylbi 200 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( N  e.  ZZ  /\  K  e.  ZZ ) )
54anim2i 579 . . . . 5  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) ) )
6 simpl 464 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  N  e.  ZZ )
76anim2i 579 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
87ancomd 458 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( N  e.  ZZ  /\  M  e.  ZZ ) )
9 zsubcl 11003 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  -  M
)  e.  ZZ )
108, 9syl 17 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( N  -  M )  e.  ZZ )
116adantl 473 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  N  e.  ZZ )
12 simprr 774 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  K  e.  ZZ )
1310, 11, 123jca 1210 . . . . 5  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( ( N  -  M )  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ ) )
145, 13syl 17 . . . 4  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  ( ( N  -  M )  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ ) )
15143adant3 1050 . . 3  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N )  /\  N  <  ( M  +  K ) )  -> 
( ( N  -  M )  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )
)
16 elfzel2 11824 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  N  e.  ZZ )
1716zred 11063 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  N  e.  RR )
1817adantl 473 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  N  e.  RR )
19 zre 10965 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  RR )
2019adantr 472 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  M  e.  RR )
21 elfzelz 11826 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ZZ )
2221zred 11063 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  K  e.  RR )
2322adantl 473 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  K  e.  RR )
2418, 20, 233jca 1210 . . . . 5  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR ) )
25 simp1 1030 . . . . . . 7  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR )  ->  N  e.  RR )
26 readdcl 9640 . . . . . . . 8  |-  ( ( M  e.  RR  /\  K  e.  RR )  ->  ( M  +  K
)  e.  RR )
27263adant1 1048 . . . . . . 7  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR )  ->  ( M  +  K )  e.  RR )
28 ltle 9740 . . . . . . 7  |-  ( ( N  e.  RR  /\  ( M  +  K
)  e.  RR )  ->  ( N  < 
( M  +  K
)  ->  N  <_  ( M  +  K ) ) )
2925, 27, 28syl2anc 673 . . . . . 6  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR )  ->  ( N  <  ( M  +  K )  ->  N  <_  ( M  +  K
) ) )
30 lesubadd2 10108 . . . . . 6  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR )  ->  (
( N  -  M
)  <_  K  <->  N  <_  ( M  +  K ) ) )
3129, 30sylibrd 242 . . . . 5  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR )  ->  ( N  <  ( M  +  K )  ->  ( N  -  M )  <_  K ) )
3224, 31syl 17 . . . 4  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  ( N  < 
( M  +  K
)  ->  ( N  -  M )  <_  K
) )
33323impia 1228 . . 3  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N )  /\  N  <  ( M  +  K ) )  -> 
( N  -  M
)  <_  K )
34 elfzle2 11829 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  K  <_  N )
35343ad2ant2 1052 . . 3  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N )  /\  N  <  ( M  +  K ) )  ->  K  <_  N )
3615, 33, 35jca32 544 . 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 11817 . 2  |-  ( K  e.  ( ( N  -  M ) ... N )  <->  ( (
( N  -  M
)  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  (
( N  -  M
)  <_  K  /\  K  <_  N ) ) )
3836, 37sylibr 217 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 376    /\ w3a 1007    e. wcel 1904   class class class wbr 4395  (class class class)co 6308   RRcr 9556   0cc0 9557    + caddc 9560    < clt 9693    <_ cle 9694    - cmin 9880   ZZcz 10961   ...cfz 11810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811
This theorem is referenced by: (None)
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