Theorem fzm1 10035

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 5849	. . . . . . 7 |- ( N = M -> ( N ... N ) = ( M ... N ) )
2	1	eleq2d 2236	. . . . . 6 |- ( N = M -> ( K e. ( N ... N ) <-> K e. ( M ... N ) ) )
3		elfz1eq 9970	. . . . . 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 3413	. . . . . 6 |- -. K e. (/)
9		eluzelz 9475	. . . . . . . . . . . 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 9313	. . . . . . . . . 10 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> N e. RR )
12	11	ltm1d 8827	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( N - 1 ) < N )
13		breq2 3986	. . . . . . . . . 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 9471	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
17	16	adantr 274	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> M e. ZZ )
18		1zzd 9218	. . . . . . . . . 10 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> 1 e. ZZ )
19	10, 18	zsubcld 9318	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( N - 1 ) e. ZZ )
20		fzn 9977	. . . . . . . . 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 2236	. . . . . 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 9967	. . . . . . 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 2229	. . . . . . 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 10007	. . . 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 9314	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> N e. CC )
38		npcan1 8276	. . . . . 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 5858	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
41	40	eleq2d 2236	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( K e. ( M ... ( ( N - 1 ) + 1 ) ) <-> K e. ( M ... N ) ) )
42	39	eqeq2d 2177	. . . 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 9496	. 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 1343   e. wcel 2136   (/)c0 3409   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   1c1 7754   + caddc 7756   < clt 7933   - cmin 8069   ZZcz 9191   ZZ>=cuz 9466   ...cfz 9944

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-fz 9945

This theorem is referenced by:  bcpasc  10679  phibndlem  12148  lgsdir2lem2  13570