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

Theorem fzopth 11642
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 11614 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
21adantr 463 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  e.  ( M ... N
) )
3 simpr 459 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( M ... N )  =  ( J ... K
) )
42, 3eleqtrd 2472 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  e.  ( J ... K
) )
5 elfzuz 11605 . . . . . . 7  |-  ( M  e.  ( J ... K )  ->  M  e.  ( ZZ>= `  J )
)
6 uzss 11021 . . . . . . 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 11612 . . . . . . . . 9  |-  ( M  e.  ( J ... K )  ->  K  e.  ( ZZ>= `  J )
)
9 eluzfz1 11614 . . . . . . . . 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 2473 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  J  e.  ( M ... N
) )
12 elfzuz 11605 . . . . . . 7  |-  ( J  e.  ( M ... N )  ->  J  e.  ( ZZ>= `  M )
)
13 uzss 11021 . . . . . . 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 3434 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  M )  =  ( ZZ>= `  J )
)
16 eluzel2 11006 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
1716adantr 463 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  e.  ZZ )
18 uz11 11023 . . . . . 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 11615 . . . . . . . . 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 2473 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  K  e.  ( M ... N
) )
24 elfzuz3 11606 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K )
)
25 uzss 11021 . . . . . . 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 11615 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
2827adantr 463 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  e.  ( M ... N
) )
2928, 3eleqtrd 2472 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  e.  ( J ... K
) )
30 elfzuz3 11606 . . . . . . 7  |-  ( N  e.  ( J ... K )  ->  K  e.  ( ZZ>= `  N )
)
31 uzss 11021 . . . . . . 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 3434 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  N )  =  ( ZZ>= `  K )
)
34 eluzelz 11010 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3534adantr 463 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  e.  ZZ )
36 uz11 11023 . . . . . 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 530 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( M  =  J  /\  N  =  K )
)
4039ex 432 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( M ... N )  =  ( J ... K
)  ->  ( M  =  J  /\  N  =  K ) ) )
41 oveq12 6205 . 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 367    = wceq 1399    e. wcel 1826    C_ wss 3389   ` cfv 5496  (class class class)co 6196   ZZcz 10781   ZZ>=cuz 11001   ...cfz 11593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-pre-lttri 9477  ax-pre-lttrn 9478
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-neg 9721  df-z 10782  df-uz 11002  df-fz 11594
This theorem is referenced by:  2ffzeq  11716  gsumval2a  16023  eedimeq  24322  usgraexvlem  24516  sdclem2  30401
  Copyright terms: Public domain W3C validator