Theorem fzopth 9841

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 9811	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
2	1	adantr 274	. . . . . . . 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 2218	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M e. ( J ... K ) )
5		elfzuz 9802	. . . . . . 7 |- ( M e. ( J ... K ) -> M e. ( ZZ>= ` J ) )
6		uzss 9346	. . . . . . 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 9809	. . . . . . . . 9 |- ( M e. ( J ... K ) -> K e. ( ZZ>= ` J ) )
9		eluzfz1 9811	. . . . . . . . 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 2219	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> J e. ( M ... N ) )
12		elfzuz 9802	. . . . . . 7 |- ( J e. ( M ... N ) -> J e. ( ZZ>= ` M ) )
13		uzss 9346	. . . . . . 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 3114	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` M ) = ( ZZ>= ` J ) )
16		eluzel2 9331	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
17	16	adantr 274	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> M e. ZZ )
18		uz11 9348	. . . . . 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 9812	. . . . . . . . 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 2219	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> K e. ( M ... N ) )
24		elfzuz3 9803	. . . . . . 7 |- ( K e. ( M ... N ) -> N e. ( ZZ>= ` K ) )
25		uzss 9346	. . . . . . 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 9812	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
28	27	adantr 274	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N e. ( M ... N ) )
29	28, 3	eleqtrd 2218	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N e. ( J ... K ) )
30		elfzuz3 9803	. . . . . . 7 |- ( N e. ( J ... K ) -> K e. ( ZZ>= ` N ) )
31		uzss 9346	. . . . . . 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 3114	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> ( ZZ>= ` N ) = ( ZZ>= ` K ) )
34		eluzelz 9335	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
35	34	adantr 274	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ ( M ... N ) = ( J ... K ) ) -> N e. ZZ )
36		uz11 9348	. . . . . 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 304	. . 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 5783	. 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 1331   e. wcel 1480   C_ wss 3071   ` cfv 5123  (class class class)co 5774   ZZcz 9054   ZZ>=cuz 9326   ...cfz 9790

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-apti 7735

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-neg 7936  df-z 9055  df-uz 9327  df-fz 9791

This theorem is referenced by:  2ffzeq  9918