MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fzopth Structured version   Unicode version

Theorem fzopth 11731
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
StepHypRef Expression
1 eluzfz1 11704 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
21adantr 465 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  e.  ( M ... N
) )
3 simpr 461 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( M ... N )  =  ( J ... K
) )
42, 3eleqtrd 2533 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  e.  ( J ... K
) )
5 elfzuz 11695 . . . . . . 7  |-  ( M  e.  ( J ... K )  ->  M  e.  ( ZZ>= `  J )
)
6 uzss 11112 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  J
)  ->  ( ZZ>= `  M )  C_  ( ZZ>=
`  J ) )
74, 5, 63syl 20 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  M )  C_  ( ZZ>= `  J )
)
8 elfzuz2 11702 . . . . . . . . 9  |-  ( M  e.  ( J ... K )  ->  K  e.  ( ZZ>= `  J )
)
9 eluzfz1 11704 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  J
)  ->  J  e.  ( J ... K ) )
104, 8, 93syl 20 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  J  e.  ( J ... K
) )
1110, 3eleqtrrd 2534 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  J  e.  ( M ... N
) )
12 elfzuz 11695 . . . . . . 7  |-  ( J  e.  ( M ... N )  ->  J  e.  ( ZZ>= `  M )
)
13 uzss 11112 . . . . . . 7  |-  ( J  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  J )  C_  ( ZZ>=
`  M ) )
1411, 12, 133syl 20 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  J )  C_  ( ZZ>= `  M )
)
157, 14eqssd 3506 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  M )  =  ( ZZ>= `  J )
)
16 eluzel2 11097 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
1716adantr 465 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  e.  ZZ )
18 uz11 11114 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( ZZ>= `  M )  =  ( ZZ>= `  J
)  <->  M  =  J
) )
1917, 18syl 16 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  (
( ZZ>= `  M )  =  ( ZZ>= `  J
)  <->  M  =  J
) )
2015, 19mpbid 210 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  =  J )
21 eluzfz2 11705 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  J
)  ->  K  e.  ( J ... K ) )
224, 8, 213syl 20 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  K  e.  ( J ... K
) )
2322, 3eleqtrrd 2534 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  K  e.  ( M ... N
) )
24 elfzuz3 11696 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K )
)
25 uzss 11112 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( ZZ>= `  N )  C_  ( ZZ>=
`  K ) )
2623, 24, 253syl 20 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  N )  C_  ( ZZ>= `  K )
)
27 eluzfz2 11705 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
2827adantr 465 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  e.  ( M ... N
) )
2928, 3eleqtrd 2533 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  e.  ( J ... K
) )
30 elfzuz3 11696 . . . . . . 7  |-  ( N  e.  ( J ... K )  ->  K  e.  ( ZZ>= `  N )
)
31 uzss 11112 . . . . . . 7  |-  ( K  e.  ( ZZ>= `  N
)  ->  ( ZZ>= `  K )  C_  ( ZZ>=
`  N ) )
3229, 30, 313syl 20 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  K )  C_  ( ZZ>= `  N )
)
3326, 32eqssd 3506 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  N )  =  ( ZZ>= `  K )
)
34 eluzelz 11101 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3534adantr 465 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  e.  ZZ )
36 uz11 11114 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( ZZ>= `  N )  =  ( ZZ>= `  K
)  <->  N  =  K
) )
3735, 36syl 16 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  (
( ZZ>= `  N )  =  ( ZZ>= `  K
)  <->  N  =  K
) )
3833, 37mpbid 210 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  =  K )
3920, 38jca 532 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( M  =  J  /\  N  =  K )
)
4039ex 434 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( M ... N )  =  ( J ... K
)  ->  ( M  =  J  /\  N  =  K ) ) )
41 oveq12 6290 . 2  |-  ( ( M  =  J  /\  N  =  K )  ->  ( M ... N
)  =  ( J ... K ) )
4240, 41impbid1 203 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( M ... N )  =  ( J ... K
)  <->  ( M  =  J  /\  N  =  K ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804    C_ wss 3461   ` cfv 5578  (class class class)co 6281   ZZcz 10871   ZZ>=cuz 11092   ...cfz 11683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-pre-lttri 9569  ax-pre-lttrn 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-neg 9813  df-z 10872  df-uz 11093  df-fz 11684
This theorem is referenced by:  2ffzeq  11805  gsumval2a  15885  eedimeq  24179  usgraexvlem  24373  sdclem2  30211
  Copyright terms: Public domain W3C validator