Theorem fzopth 10257

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 10227	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
2	1	adantr 276	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M e. ( M ... N ) )
3		simpr 110	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( M ... N ) = ( J ... K ) )
4	2, 3	eleqtrd 2308	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M e. ( J ... K ) )
5		elfzuz 10217	. . . . . . 7 |- ( M e. ( J ... K ) -> M e. ( ZZ>= ` J ) )
6		uzss 9743	. . . . . . 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 10225	. . . . . . . . 9 |- ( M e. ( J ... K ) -> K e. ( ZZ>= ` J ) )
9		eluzfz1 10227	. . . . . . . . 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 2309	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> J e. ( M ... N ) )
12		elfzuz 10217	. . . . . . 7 |- ( J e. ( M ... N ) -> J e. ( ZZ>= ` M ) )
13		uzss 9743	. . . . . . 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 3241	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` M ) = ( ZZ>= ` J ) )
16		eluzel2 9727	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
17	16	adantr 276	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M e. ZZ )
18		uz11 9745	. . . . . 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 147	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M = J )
21		eluzfz2 10228	. . . . . . . . 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 2309	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> K e. ( M ... N ) )
24		elfzuz3 10218	. . . . . . 7 |- ( K e. ( M ... N ) -> N e. ( ZZ>= ` K ) )
25		uzss 9743	. . . . . . 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 10228	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
28	27	adantr 276	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N e. ( M ... N ) )
29	28, 3	eleqtrd 2308	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N e. ( J ... K ) )
30		elfzuz3 10218	. . . . . . 7 |- ( N e. ( J ... K ) -> K e. ( ZZ>= ` N ) )
31		uzss 9743	. . . . . . 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 3241	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` N ) = ( ZZ>= ` K ) )
34		eluzelz 9731	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
35	34	adantr 276	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N e. ZZ )
36		uz11 9745	. . . . . 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 147	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N = K )
39	20, 38	jca 306	. . 3 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( M = J /\ N = K ) )
40	39	ex 115	. 2 |- ( N e. ( ZZ>= ` M ) -> ( ( M ... N ) = ( J ... K ) -> ( M = J /\ N = K ) ) )
41		oveq12 6010	. 2 |- ( ( M = J /\ N = K ) -> ( M ... N ) = ( J ... K ) )
42	40, 41	impbid1 142	1 |- ( N e. ( ZZ>= ` M ) -> ( ( M ... N ) = ( J ... K ) <-> ( M = J /\ N = K ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   C_ wss 3197   ` cfv 5318  (class class class)co 6001   ZZcz 9446   ZZ>=cuz 9722   ...cfz 10204

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-apti 8114

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-neg 8320  df-z 9447  df-uz 9723  df-fz 10205

This theorem is referenced by:  fz0to4untppr  10320  2ffzeq  10337  gsumfzval  13424  gsumval2  13430