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

Step	Hyp	Ref	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 ) )
3	2	adantr 462	. . . . . . 7 |- ( ( ( 0 e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( 0 <_ K /\ K <_ N ) ) -> ( N e. ZZ /\ K e. ZZ ) )
4	1, 3	sylbi 195	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( N e. ZZ /\ K e. ZZ ) )
5	4	anim2i 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 )
7	6	anim2i 566	. . . . . . . 8 |- ( ( M e. ZZ /\ ( N e. ZZ /\ K e. ZZ ) ) -> ( M e. ZZ /\ N e. ZZ ) )
8	7	ancomd 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 )
10	8, 9	syl 16	. . . . . 6 |- ( ( M e. ZZ /\ ( N e. ZZ /\ K e. ZZ ) ) -> ( N - M ) e. ZZ )
11	6	adantl 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 )
13	10, 11, 12	3jca 1163	. . . . 5 |- ( ( M e. ZZ /\ ( N e. ZZ /\ K e. ZZ ) ) -> ( ( N - M ) e. ZZ /\ N e. ZZ /\ K e. ZZ ) )
14	5, 13	syl 16	. . . 4 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) ) -> ( ( N - M ) e. ZZ /\ N e. ZZ /\ K e. ZZ ) )
15	14	3adant3 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 )
17	16	zred 10743	. . . . . . 7 |- ( K e. ( 0 ... N ) -> N e. RR )
18	17	adantl 463	. . . . . 6 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) ) -> N e. RR )
19		zre 10646	. . . . . . 7 |- ( M e. ZZ -> M e. RR )
20	19	adantr 462	. . . . . 6 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) ) -> M e. RR )
21		elfzelz 11449	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> K e. ZZ )
22	21	zred 10743	. . . . . . 7 |- ( K e. ( 0 ... N ) -> K e. RR )
23	22	adantl 463	. . . . . 6 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) ) -> K e. RR )
24	18, 20, 23	3jca 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 )
27	26	3adant1 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 ) ) )
29	25, 27, 28	syl2anc 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 ) ) )
31	29, 30	sylibrd 234	. . . . 5 |- ( ( N e. RR /\ M e. RR /\ K e. RR ) -> ( N < ( M + K ) -> ( N - M ) <_ K ) )
32	24, 31	syl 16	. . . 4 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) ) -> ( N < ( M + K ) -> ( N - M ) <_ K ) )
33	32	3impia 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 )
35	34	3ad2ant2 1005	. . 3 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) /\ N < ( M + K ) ) -> K <_ N )
36	15, 33, 35	jca32 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 ) ) )
38	36, 37	sylibr 212	1 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) /\ N < ( M + K ) ) -> K e. ( ( N - M ) ... N ) )

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)