Theorem fzopth 9624

Description: A finite set of sequential integers can represent an ordered pair. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

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

Proof of Theorem fzopth

Step	Hyp	Ref	Expression
1		eluzfz1 9594	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
2	1	adantr 271	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M e. ( M ... N ) )
3		simpr 109	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( M ... N ) = ( J ... K ) )
4	2, 3	eleqtrd 2173	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M e. ( J ... K ) )
5		elfzuz 9585	. . . . . . 7 |- ( M e. ( J ... K ) -> M e. ( ZZ>= ` J ) )
6		uzss 9138	. . . . . . 7 |- ( M e. ( ZZ>= ` J ) -> ( ZZ>= ` M ) C_ ( ZZ>= ` J ) )
7	4, 5, 6	3syl 17	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` M ) C_ ( ZZ>= ` J ) )
8		elfzuz2 9592	. . . . . . . . 9 |- ( M e. ( J ... K ) -> K e. ( ZZ>= ` J ) )
9		eluzfz1 9594	. . . . . . . . 9 |- ( K e. ( ZZ>= ` J ) -> J e. ( J ... K ) )
10	4, 8, 9	3syl 17	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> J e. ( J ... K ) )
11	10, 3	eleqtrrd 2174	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> J e. ( M ... N ) )
12		elfzuz 9585	. . . . . . 7 |- ( J e. ( M ... N ) -> J e. ( ZZ>= ` M ) )
13		uzss 9138	. . . . . . 7 |- ( J e. ( ZZ>= ` M ) -> ( ZZ>= ` J ) C_ ( ZZ>= ` M ) )
14	11, 12, 13	3syl 17	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` J ) C_ ( ZZ>= ` M ) )
15	7, 14	eqssd 3056	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` M ) = ( ZZ>= ` J ) )
16		eluzel2 9123	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
17	16	adantr 271	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M e. ZZ )
18		uz11 9140	. . . . . 6 |- ( M e. ZZ -> ( ( ZZ>= ` M ) = ( ZZ>= ` J ) <-> M = J ) )
19	17, 18	syl 14	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ( ZZ>= ` M ) = ( ZZ>= ` J ) <-> M = J ) )
20	15, 19	mpbid 146	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M = J )
21		eluzfz2 9595	. . . . . . . . 9 |- ( K e. ( ZZ>= ` J ) -> K e. ( J ... K ) )
22	4, 8, 21	3syl 17	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> K e. ( J ... K ) )
23	22, 3	eleqtrrd 2174	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> K e. ( M ... N ) )
24		elfzuz3 9586	. . . . . . 7 |- ( K e. ( M ... N ) -> N e. ( ZZ>= ` K ) )
25		uzss 9138	. . . . . . 7 |- ( N e. ( ZZ>= ` K ) -> ( ZZ>= ` N ) C_ ( ZZ>= ` K ) )
26	23, 24, 25	3syl 17	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` N ) C_ ( ZZ>= ` K ) )
27		eluzfz2 9595	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
28	27	adantr 271	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N e. ( M ... N ) )
29	28, 3	eleqtrd 2173	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N e. ( J ... K ) )
30		elfzuz3 9586	. . . . . . 7 |- ( N e. ( J ... K ) -> K e. ( ZZ>= ` N ) )
31		uzss 9138	. . . . . . 7 |- ( K e. ( ZZ>= ` N ) -> ( ZZ>= ` K ) C_ ( ZZ>= ` N ) )
32	29, 30, 31	3syl 17	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` K ) C_ ( ZZ>= ` N ) )
33	26, 32	eqssd 3056	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` N ) = ( ZZ>= ` K ) )
34		eluzelz 9127	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
35	34	adantr 271	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N e. ZZ )
36		uz11 9140	. . . . . 6 |- ( N e. ZZ -> ( ( ZZ>= ` N ) = ( ZZ>= ` K ) <-> N = K ) )
37	35, 36	syl 14	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ( ZZ>= ` N ) = ( ZZ>= ` K ) <-> N = K ) )
38	33, 37	mpbid 146	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N = K )
39	20, 38	jca 301	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( M = J /\ N = K ) )
40	39	ex 114	. 2 |- ( N e. ( ZZ>= ` M ) -> ( ( M ... N ) = ( J ... K ) -> ( M = J /\ N = K ) ) )
41		oveq12 5699	. 2 |- ( ( M = J /\ N = K ) -> ( M ... N ) = ( J ... K ) )
42	40, 41	impbid1 141	1 |- ( N e. ( ZZ>= ` M ) -> ( ( M ... N ) = ( J ... K ) <-> ( M = J /\ N = K ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1296   e. wcel 1445   C_ wss 3013   ` cfv 5049  (class class class)co 5690   ZZcz 8848   ZZ>=cuz 9118   ...cfz 9573

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-apti 7557

This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-neg 7753  df-z 8849  df-uz 9119  df-fz 9574

This theorem is referenced by:  2ffzeq  9701