Theorem fzm1 10397

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 6035	. . . . . . 7 |- ( N = M -> ( N ... N ) = ( M ... N ) )
2	1	eleq2d 2301	. . . . . 6 |- ( N = M -> ( K e. ( N ... N ) <-> K e. ( M ... N ) ) )
3		elfz1eq 10332	. . . . . 6 |- ( K e. ( N ... N ) -> K = N )
4	2, 3	biimtrrdi 164	. . . . 5 |- ( N = M -> ( K e. ( M ... N ) -> K = N ) )
5		olc 719	. . . . 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 277	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( K e. ( M ... N ) -> ( K e. ( M ... ( N - 1 ) ) \/ K = N ) ) )
8		noel 3500	. . . . . 6 |- -. K e. (/)
9		eluzelz 9826	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
10	9	adantr 276	. . . . . . . . . . 11 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> N e. ZZ )
11	10	zred 9663	. . . . . . . . . 10 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> N e. RR )
12	11	ltm1d 9171	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( N - 1 ) < N )
13		breq2 4097	. . . . . . . . . 10 |- ( N = M -> ( ( N - 1 ) < N <-> ( N - 1 ) < M ) )
14	13	adantl 277	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( ( N - 1 ) < N <-> ( N - 1 ) < M ) )
15	12, 14	mpbid 147	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( N - 1 ) < M )
16		eluzel2 9821	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
17	16	adantr 276	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> M e. ZZ )
18		1zzd 9567	. . . . . . . . . 10 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> 1 e. ZZ )
19	10, 18	zsubcld 9668	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( N - 1 ) e. ZZ )
20		fzn 10339	. . . . . . . . 9 |- ( ( M e. ZZ /\ ( N - 1 ) e. ZZ ) -> ( ( N - 1 ) < M <-> ( M ... ( N - 1 ) ) = (/) ) )
21	17, 19, 20	syl2anc 411	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( ( N - 1 ) < M <-> ( M ... ( N - 1 ) ) = (/) ) )
22	15, 21	mpbid 147	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( M ... ( N - 1 ) ) = (/) )
23	22	eleq2d 2301	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( K e. ( M ... ( N - 1 ) ) <-> K e. (/) ) )
24	8, 23	mtbiri 682	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> -. K e. ( M ... ( N - 1 ) ) )
25	24	pm2.21d 624	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( K e. ( M ... ( N - 1 ) ) -> K e. ( M ... N ) ) )
26		eluzfz2 10329	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
27	26	ad2antrr 488	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` M ) /\ N = M ) /\ K = N ) -> N e. ( M ... N ) )
28		eleq1 2294	. . . . . . 7 |- ( K = N -> ( K e. ( M ... N ) <-> N e. ( M ... N ) ) )
29	28	adantl 277	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` M ) /\ N = M ) /\ K = N ) -> ( K e. ( M ... N ) <-> N e. ( M ... N ) ) )
30	27, 29	mpbird 167	. . . . 5 |- ( ( ( N e. ( ZZ>= ` M ) /\ N = M ) /\ K = N ) -> K e. ( M ... N ) )
31	30	ex 115	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( K = N -> K e. ( M ... N ) ) )
32	25, 31	jaod 725	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( ( K e. ( M ... ( N - 1 ) ) \/ K = N ) -> K e. ( M ... N ) ) )
33	7, 32	impbid 129	. 2 |- ( ( N e. ( ZZ>= ` M ) /\ N = M ) -> ( K e. ( M ... N ) <-> ( K e. ( M ... ( N - 1 ) ) \/ K = N ) ) )
34		elfzp1 10369	. . . 4 |- ( ( N - 1 ) e. ( ZZ>= ` M ) -> ( K e. ( M ... ( ( N - 1 ) + 1 ) ) <-> ( K e. ( M ... ( N - 1 ) ) \/ K = ( ( N - 1 ) + 1 ) ) ) )
35	34	adantl 277	. . 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 276	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> N e. ZZ )
37	36	zcnd 9664	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> N e. CC )
38		npcan1 8616	. . . . . 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 6044	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
41	40	eleq2d 2301	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( K e. ( M ... ( ( N - 1 ) + 1 ) ) <-> K e. ( M ... N ) ) )
42	39	eqeq2d 2243	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( K = ( ( N - 1 ) + 1 ) <-> K = N ) )
43	42	orbi2d 798	. . 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 218	. 2 |- ( ( N e. ( ZZ>= ` M ) /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( K e. ( M ... N ) <-> ( K e. ( M ... ( N - 1 ) ) \/ K = N ) ) )
45		uzm1 9848	. 2 |- ( N e. ( ZZ>= ` M ) -> ( N = M \/ ( N - 1 ) e. ( ZZ>= ` M ) ) )
46	33, 44, 45	mpjaodan 806	1 |- ( N e. ( ZZ>= ` M ) -> ( K e. ( M ... N ) <-> ( K e. ( M ... ( N - 1 ) ) \/ K = N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2202   (/)c0 3496   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8090   1c1 8093   + caddc 8095   < clt 8273   - cmin 8409   ZZcz 9540   ZZ>=cuz 9816   ...cfz 10305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817  df-fz 10306

This theorem is referenced by:  bcpasc  11091  phibndlem  12868  lgsdir2lem2  15848