Theorem elfzelfzlble 30214

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 11449	. . . . . . 7 |- ( K e. ( 0 ... N ) <-> ( ( 0 e. ZZ /\ N e. ZZ /\ K e. ZZ ) /\ ( 0 <_ K /\ K <_ N ) ) )
2		3simpc 987	. . . . . . . 8 |- ( ( 0 e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( N e. ZZ /\ K e. ZZ ) )
3	2	adantr 465	. . . . . . 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 569	. . . . 5 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) ) -> ( M e. ZZ /\ ( N e. ZZ /\ K e. ZZ ) ) )
6		simpl 457	. . . . . . . . 9 |- ( ( N e. ZZ /\ K e. ZZ ) -> N e. ZZ )
7	6	anim2i 569	. . . . . . . 8 |- ( ( M e. ZZ /\ ( N e. ZZ /\ K e. ZZ ) ) -> ( M e. ZZ /\ N e. ZZ ) )
8	7	ancomd 451	. . . . . . 7 |- ( ( M e. ZZ /\ ( N e. ZZ /\ K e. ZZ ) ) -> ( N e. ZZ /\ M e. ZZ ) )
9		zsubcl 10692	. . . . . . 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 466	. . . . . 6 |- ( ( M e. ZZ /\ ( N e. ZZ /\ K e. ZZ ) ) -> N e. ZZ )
12		simprr 756	. . . . . 6 |- ( ( M e. ZZ /\ ( N e. ZZ /\ K e. ZZ ) ) -> K e. ZZ )
13	10, 11, 12	3jca 1168	. . . . 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 1008	. . 3 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) /\ N < ( M + K ) ) -> ( ( N - M ) e. ZZ /\ N e. ZZ /\ K e. ZZ ) )
16		elfzel2 11456	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> N e. ZZ )
17	16	zred 10752	. . . . . . 7 |- ( K e. ( 0 ... N ) -> N e. RR )
18	17	adantl 466	. . . . . 6 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) ) -> N e. RR )
19		zre 10655	. . . . . . 7 |- ( M e. ZZ -> M e. RR )
20	19	adantr 465	. . . . . 6 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) ) -> M e. RR )
21		elfzelz 11458	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> K e. ZZ )
22	21	zred 10752	. . . . . . 7 |- ( K e. ( 0 ... N ) -> K e. RR )
23	22	adantl 466	. . . . . 6 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) ) -> K e. RR )
24	18, 20, 23	3jca 1168	. . . . 5 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) ) -> ( N e. RR /\ M e. RR /\ K e. RR ) )
25		simp1 988	. . . . . . 7 |- ( ( N e. RR /\ M e. RR /\ K e. RR ) -> N e. RR )
26		readdcl 9370	. . . . . . . 8 |- ( ( M e. RR /\ K e. RR ) -> ( M + K ) e. RR )
27	26	3adant1 1006	. . . . . . 7 |- ( ( N e. RR /\ M e. RR /\ K e. RR ) -> ( M + K ) e. RR )
28		ltle 9468	. . . . . . 7 |- ( ( N e. RR /\ ( M + K ) e. RR ) -> ( N < ( M + K ) -> N <_ ( M + K ) ) )
29	25, 27, 28	syl2anc 661	. . . . . 6 |- ( ( N e. RR /\ M e. RR /\ K e. RR ) -> ( N < ( M + K ) -> N <_ ( M + K ) ) )
30		lesubadd2 9817	. . . . . 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 1184	. . 3 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) /\ N < ( M + K ) ) -> ( N - M ) <_ K )
34		elfzle2 11460	. . . 4 |- ( K e. ( 0 ... N ) -> K <_ N )
35	34	3ad2ant2 1010	. . 3 |- ( ( M e. ZZ /\ K e. ( 0 ... N ) /\ N < ( M + K ) ) -> K <_ N )
36	15, 33, 35	jca32 535	. 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 11449	. 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 965   e. wcel 1756   class class class wbr 4297  (class class class)co 6096   RRcr 9286   0cc0 9287   + caddc 9290   < clt 9423   <_ cle 9424   - cmin 9600   ZZcz 10651   ...cfz 11442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443

This theorem is referenced by: (None)