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Theorem elfzelfzlble 39048
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 11788 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  <->  ( (
0  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  (
0  <_  K  /\  K  <_  N ) ) )
2 3simpc 1006 . . . . . . . 8  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  e.  ZZ  /\  K  e.  ZZ ) )
32adantr 467 . . . . . . 7  |-  ( ( ( 0  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( 0  <_  K  /\  K  <_  N ) )  ->  ( N  e.  ZZ  /\  K  e.  ZZ ) )
41, 3sylbi 199 . . . . . 6  |-  ( K  e.  ( 0 ... N )  ->  ( N  e.  ZZ  /\  K  e.  ZZ ) )
54anim2i 572 . . . . 5  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) ) )
6 simpl 459 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  N  e.  ZZ )
76anim2i 572 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
87ancomd 453 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( N  e.  ZZ  /\  M  e.  ZZ ) )
9 zsubcl 10976 . . . . . . 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 468 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  N  e.  ZZ )
12 simprr 765 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( N  e.  ZZ  /\  K  e.  ZZ ) )  ->  K  e.  ZZ )
1310, 11, 123jca 1187 . . . . 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 1027 . . 3  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N )  /\  N  <  ( M  +  K ) )  -> 
( ( N  -  M )  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )
)
16 elfzel2 11795 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  N  e.  ZZ )
1716zred 11037 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  N  e.  RR )
1817adantl 468 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  N  e.  RR )
19 zre 10938 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  RR )
2019adantr 467 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  M  e.  RR )
21 elfzelz 11797 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ZZ )
2221zred 11037 . . . . . . 7  |-  ( K  e.  ( 0 ... N )  ->  K  e.  RR )
2322adantl 468 . . . . . 6  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  K  e.  RR )
2418, 20, 233jca 1187 . . . . 5  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N ) )  ->  ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR ) )
25 simp1 1007 . . . . . . 7  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR )  ->  N  e.  RR )
26 readdcl 9619 . . . . . . . 8  |-  ( ( M  e.  RR  /\  K  e.  RR )  ->  ( M  +  K
)  e.  RR )
27263adant1 1025 . . . . . . 7  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR )  ->  ( M  +  K )  e.  RR )
28 ltle 9719 . . . . . . 7  |-  ( ( N  e.  RR  /\  ( M  +  K
)  e.  RR )  ->  ( N  < 
( M  +  K
)  ->  N  <_  ( M  +  K ) ) )
2925, 27, 28syl2anc 666 . . . . . 6  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR )  ->  ( N  <  ( M  +  K )  ->  N  <_  ( M  +  K
) ) )
30 lesubadd2 10084 . . . . . 6  |-  ( ( N  e.  RR  /\  M  e.  RR  /\  K  e.  RR )  ->  (
( N  -  M
)  <_  K  <->  N  <_  ( M  +  K ) ) )
3129, 30sylibrd 238 . . . . 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 1204 . . 3  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N )  /\  N  <  ( M  +  K ) )  -> 
( N  -  M
)  <_  K )
34 elfzle2 11800 . . . 4  |-  ( K  e.  ( 0 ... N )  ->  K  <_  N )
35343ad2ant2 1029 . . 3  |-  ( ( M  e.  ZZ  /\  K  e.  ( 0 ... N )  /\  N  <  ( M  +  K ) )  ->  K  <_  N )
3615, 33, 35jca32 538 . 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 11788 . 2  |-  ( K  e.  ( ( N  -  M ) ... N )  <->  ( (
( N  -  M
)  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  (
( N  -  M
)  <_  K  /\  K  <_  N ) ) )
3836, 37sylibr 216 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 371    /\ w3a 984    e. wcel 1886   class class class wbr 4401  (class class class)co 6288   RRcr 9535   0cc0 9536    + caddc 9539    < clt 9672    <_ cle 9673    - cmin 9857   ZZcz 10934   ...cfz 11781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782
This theorem is referenced by: (None)
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