Theorem fzm1 9873

Description: Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
fzm1	|- ( N e. ( ZZ>= ` M ) -> ( K e. ( M ... N ) <-> ( K e. ( M ... ( N - 1 ) ) \/ K = N ) ) )

Proof of Theorem fzm1

Step	Hyp	Ref	Expression
1		oveq1 5774	. . . . . . 7 |- ( N = M -> ( N ... N ) = ( M ... N ) )
2	1	eleq2d 2207	. . . . . 6 |- ( N = M -> ( K e. ( N ... N ) <-> K e. ( M ... N ) ) )
3		elfz1eq 9808	. . . . . 6 |- ( K e. ( N ... N ) -> K = N )
4	2, 3	syl6bir 163	. . . . 5 |- ( N = M -> ( K e. ( M ... N ) -> K = N ) )
5		olc 700	. . . . 5 |- ( K = N -> ( K e. ( M ... ( N - 1 ) ) \/ K = N ) )
6	4, 5	syl6 33	. . . 4 |- ( N = M -> ( K e. ( M ... N ) -> ( K e. ( M ... ( N - 1 ) ) \/ K = N ) ) )
7	6	adantl 275	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( K e. ( M ... N ) -> ( K e. ( M ... ( N - 1 ) ) \/ K = N ) ) )
8		noel 3362	. . . . . 6 |- -. K e. (/)
9		eluzelz 9328	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
10	9	adantr 274	. . . . . . . . . . 11 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> N e. ZZ )
11	10	zred 9166	. . . . . . . . . 10 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> N e. RR )
12	11	ltm1d 8683	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( N - 1 ) < N )
13		breq2 3928	. . . . . . . . . 10 |- ( N = M -> ( ( N - 1 ) < N <-> ( N - 1 ) < M ) )
14	13	adantl 275	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( ( N - 1 ) < N <-> ( N - 1 ) < M ) )
15	12, 14	mpbid 146	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( N - 1 ) < M )
16		eluzel2 9324	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
17	16	adantr 274	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> M e. ZZ )
18		1zzd 9074	. . . . . . . . . 10 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> 1 e. ZZ )
19	10, 18	zsubcld 9171	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( N - 1 ) e. ZZ )
20		fzn 9815	. . . . . . . . 9 |- ( ( M e. ZZ /\ ( N - 1 ) e. ZZ ) -> ( ( N - 1 ) < M <-> ( M ... ( N - 1 ) ) = (/) ) )
21	17, 19, 20	syl2anc 408	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( ( N - 1 ) < M <-> ( M ... ( N - 1 ) ) = (/) ) )
22	15, 21	mpbid 146	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( M ... ( N - 1 ) ) = (/) )
23	22	eleq2d 2207	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( K e. ( M ... ( N - 1 ) ) <-> K e. (/) ) )
24	8, 23	mtbiri 664	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> -. K e. ( M ... ( N - 1 ) ) )
25	24	pm2.21d 608	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( K e. ( M ... ( N - 1 ) ) -> K e. ( M ... N ) ) )
26		eluzfz2 9805	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
27	26	ad2antrr 479	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` M ) /\ N = M ) /\ K = N ) -> N e. ( M ... N ) )
28		eleq1 2200	. . . . . . 7 |- ( K = N -> ( K e. ( M ... N ) <-> N e. ( M ... N ) ) )
29	28	adantl 275	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` M ) /\ N = M ) /\ K = N ) -> ( K e. ( M ... N ) <-> N e. ( M ... N ) ) )
30	27, 29	mpbird 166	. . . . 5 |- ( ( ( N e. ( ZZ>= ` M ) /\ N = M ) /\ K = N ) -> K e. ( M ... N ) )
31	30	ex 114	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( K = N -> K e. ( M ... N ) ) )
32	25, 31	jaod 706	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( ( K e. ( M ... ( N - 1 ) ) \/ K = N ) -> K e. ( M ... N ) ) )
33	7, 32	impbid 128	. 2 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( K e. ( M ... N ) <-> ( K e. ( M ... ( N - 1 ) ) \/ K = N ) ) )
34		elfzp1 9845	. . . 4 |- ( ( N - 1 ) e. ( ZZ>= ` M ) -> ( K e. ( M ... ( ( N - 1 ) + 1 ) ) <-> ( K e. ( M ... ( N - 1 ) ) \/ K = ( ( N - 1 ) + 1 ) ) ) )
35	34	adantl 275	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( K e. ( M ... ( ( N - 1 ) + 1 ) ) <-> ( K e. ( M ... ( N - 1 ) ) \/ K = ( ( N - 1 ) + 1 ) ) ) )
36	9	adantr 274	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> N e. ZZ )
37	36	zcnd 9167	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> N e. CC )
38		npcan1 8133	. . . . . 6 |- ( N e. CC -> ( ( N - 1 ) + 1 ) = N )
39	37, 38	syl 14	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( ( N - 1 ) + 1 ) = N )
40	39	oveq2d 5783	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
41	40	eleq2d 2207	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( K e. ( M ... ( ( N - 1 ) + 1 ) ) <-> K e. ( M ... N ) ) )
42	39	eqeq2d 2149	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( K = ( ( N - 1 ) + 1 ) <-> K = N ) )
43	42	orbi2d 779	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( ( K e. ( M ... ( N - 1 ) ) \/ K = ( ( N - 1 ) + 1 ) ) <-> ( K e. ( M ... ( N - 1 ) ) \/ K = N ) ) )
44	35, 41, 43	3bitr3d 217	. 2 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( K e. ( M ... N ) <-> ( K e. ( M ... ( N - 1 ) ) \/ K = N ) ) )
45		uzm1 9349	. 2 |- ( N e. ( ZZ>= ` M ) -> ( N = M \/ ( N - 1 ) e. ( ZZ>= ` M ) ) )
46	33, 44, 45	mpjaodan 787	1 |- ( N e. ( ZZ>= ` M ) -> ( K e. ( M ... N ) <-> ( K e. ( M ... ( N - 1 ) ) \/ K = N ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   = wceq 1331   e. wcel 1480   (/)c0 3358   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   CCcc 7611   1c1 7614   + caddc 7616   < clt 7793   - cmin 7926   ZZcz 9047   ZZ>=cuz 9319   ...cfz 9783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-fz 9784

This theorem is referenced by:  bcpasc  10505  phibndlem  11881