Theorem fzm1 9911

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 5789	. . . . . . 7 |- ( N = M -> ( N ... N ) = ( M ... N ) )
2	1	eleq2d 2210	. . . . . 6 |- ( N = M -> ( K e. ( N ... N ) <-> K e. ( M ... N ) ) )
3		elfz1eq 9846	. . . . . 6 |- ( K e. ( N ... N ) -> K = N )
4	2, 3	syl6bir 163	. . . . 5 |- ( N = M -> ( K e. ( M ... N ) -> K = N ) )
5		olc 701	. . . . 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 3372	. . . . . 6 |- -. K e. (/)
9		eluzelz 9359	. . . . . . . . . . . 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 9197	. . . . . . . . . 10 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> N e. RR )
12	11	ltm1d 8714	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( N - 1 ) < N )
13		breq2 3941	. . . . . . . . . 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 9355	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
17	16	adantr 274	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> M e. ZZ )
18		1zzd 9105	. . . . . . . . . 10 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> 1 e. ZZ )
19	10, 18	zsubcld 9202	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( N - 1 ) e. ZZ )
20		fzn 9853	. . . . . . . . 9 |- ( ( M e. ZZ /\ ( N - 1 ) e. ZZ ) -> ( ( N - 1 ) < M <-> ( M ... ( N - 1 ) ) = (/) ) )
21	17, 19, 20	syl2anc 409	. . . . . . . 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 2210	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( K e. ( M ... ( N - 1 ) ) <-> K e. (/) ) )
24	8, 23	mtbiri 665	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> -. K e. ( M ... ( N - 1 ) ) )
25	24	pm2.21d 609	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( K e. ( M ... ( N - 1 ) ) -> K e. ( M ... N ) ) )
26		eluzfz2 9843	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
27	26	ad2antrr 480	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` M ) /\ N = M ) /\ K = N ) -> N e. ( M ... N ) )
28		eleq1 2203	. . . . . . 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 707	. . 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 9883	. . . 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 9198	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> N e. CC )
38		npcan1 8164	. . . . . 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 5798	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
41	40	eleq2d 2210	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( K e. ( M ... ( ( N - 1 ) + 1 ) ) <-> K e. ( M ... N ) ) )
42	39	eqeq2d 2152	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( K = ( ( N - 1 ) + 1 ) <-> K = N ) )
43	42	orbi2d 780	. . 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 9380	. 2 |- ( N e. ( ZZ>= ` M ) -> ( N = M \/ ( N - 1 ) e. ( ZZ>= ` M ) ) )
46	33, 44, 45	mpjaodan 788	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 698   = wceq 1332   e. wcel 1481   (/)c0 3368   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   CCcc 7642   1c1 7645   + caddc 7647   < clt 7824   - cmin 7957   ZZcz 9078   ZZ>=cuz 9350   ...cfz 9821

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-fz 9822

This theorem is referenced by:  bcpasc  10544  phibndlem  11928