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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elfzelfzlble Structured version   Unicode version
Theorem elfzelfzlble 37969
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 11733 . . . . . . 7 |- ( K e. ( 0 ... N ) <-> ( ( 0 e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( 0 <_ K /\ K <_ N ) ) )
2 3simpc 996 . . . . . . . 8 |- ( ( 0 e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( N e. ZZ /\ K e. ZZ ) )
32adantr 463 . . . . . . 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 567 . . . . 5 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) ) -> ( M e. ZZ /\ ( N e. ZZ /\ K e. ZZ ) ) )
6 simpl 455 . . . . . . . . 9 |- ( ( N e. ZZ /\ K e. ZZ ) -> N e. ZZ )
76anim2i 567 . . . . . . . 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 10947 . . . . . . 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 464 . . . . . 6 |- ( ( M e. ZZ /\ ( N e. ZZ /\ K e. ZZ ) ) -> N e. ZZ )
12 simprr 758 . . . . . 6 |- ( ( M e. ZZ /\ ( N e. ZZ /\ K e. ZZ ) ) -> K e. ZZ )
1310, 11, 123jca 1177 . . . . 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 1017 . . 3 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) /\ N < ( M + K ) ) -> ( ( N - M ) e. ZZ /\ N e. ZZ /\ K e. ZZ ) )
16 elfzel2 11740 . . . . . . . 8 |- ( K e. ( 0 ... N ) -> N e. ZZ )
1716zred 11008 . . . . . . 7 |- ( K e. ( 0 ... N ) -> N e. RR )
1817adantl 464 . . . . . 6 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) ) -> N e. RR )
19 zre 10909 . . . . . . 7 |- ( M e. ZZ -> M e. RR )
2019adantr 463 . . . . . 6 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) ) -> M e. RR )
21 elfzelz 11742 . . . . . . . 8 |- ( K e. ( 0 ... N ) -> K e. ZZ )
2221zred 11008 . . . . . . 7 |- ( K e. ( 0 ... N ) -> K e. RR )
2322adantl 464 . . . . . 6 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) ) -> K e. RR )
2418, 20, 233jca 1177 . . . . 5 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) ) -> ( N e. RR /\ M e. RR /\ K e. RR ) )
25 simp1 997 . . . . . . 7 |- ( ( N e. RR /\ M e. RR /\ K e. RR ) -> N e. RR )
26 readdcl 9605 . . . . . . . 8 |- ( ( M e. RR /\ K e. RR ) -> ( M + K ) e. RR )
27263adant1 1015 . . . . . . 7 |- ( ( N e. RR /\ M e. RR /\ K e. RR ) -> ( M + K ) e. RR )
28 ltle 9704 . . . . . . 7 |- ( ( N e. RR /\ ( M + K ) e. RR ) -> ( N < ( M + K ) -> N <_ ( M + K ) ) )
2925, 27, 28syl2anc 659 . . . . . 6 |- ( ( N e. RR /\ M e. RR /\ K e. RR ) -> ( N < ( M + K ) -> N <_ ( M + K ) ) )
30 lesubadd2 10066 . . . . . 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 17 . . . 4 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) ) -> ( N < ( M + K ) -> ( N - M ) <_ K ) )
33323impia 1194 . . 3 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) /\ N < ( M + K ) ) -> ( N - M ) <_ K )
34 elfzle2 11744 . . . 4 |- ( K e. ( 0 ... N ) -> K <_ N )
35343ad2ant2 1019 . . 3 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) /\ N < ( M + K ) ) -> K <_ N )
3615, 33, 35jca32 533 . 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 11733 . 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 367   /\ w3a 974   e. wcel 1842   class class class wbr 4395  (class class class)co 6278   RRcr 9521   0cc0 9522   + caddc 9525   < clt 9658   <_ cle 9659   - cmin 9841   ZZcz 10905   ...cfz 11726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727
This theorem is referenced by: (None)
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