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

Theorem fzopth 10844
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 10819 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
21adantr 451 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  e.  ( M ... N
) )
3 simpr 447 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( M ... N )  =  ( J ... K
) )
42, 3eleqtrd 2372 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  e.  ( J ... K
) )
5 elfzuz 10810 . . . . . . 7  |-  ( M  e.  ( J ... K )  ->  M  e.  ( ZZ>= `  J )
)
6 uzss 10264 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  J
)  ->  ( ZZ>= `  M )  C_  ( ZZ>=
`  J ) )
74, 5, 63syl 18 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  M )  C_  ( ZZ>= `  J )
)
8 elfzuz2 10817 . . . . . . . . 9  |-  ( M  e.  ( J ... K )  ->  K  e.  ( ZZ>= `  J )
)
9 eluzfz1 10819 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  J
)  ->  J  e.  ( J ... K ) )
104, 8, 93syl 18 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  J  e.  ( J ... K
) )
1110, 3eleqtrrd 2373 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  J  e.  ( M ... N
) )
12 elfzuz 10810 . . . . . . 7  |-  ( J  e.  ( M ... N )  ->  J  e.  ( ZZ>= `  M )
)
13 uzss 10264 . . . . . . 7  |-  ( J  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  J )  C_  ( ZZ>=
`  M ) )
1411, 12, 133syl 18 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  J )  C_  ( ZZ>= `  M )
)
157, 14eqssd 3209 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  M )  =  ( ZZ>= `  J )
)
16 eluzel2 10251 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
1716adantr 451 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  e.  ZZ )
18 uz11 10266 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( ZZ>= `  M )  =  ( ZZ>= `  J
)  <->  M  =  J
) )
1917, 18syl 15 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  (
( ZZ>= `  M )  =  ( ZZ>= `  J
)  <->  M  =  J
) )
2015, 19mpbid 201 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  =  J )
21 eluzfz2 10820 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  J
)  ->  K  e.  ( J ... K ) )
224, 8, 213syl 18 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  K  e.  ( J ... K
) )
2322, 3eleqtrrd 2373 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  K  e.  ( M ... N
) )
24 elfzuz3 10811 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K )
)
25 uzss 10264 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( ZZ>= `  N )  C_  ( ZZ>=
`  K ) )
2623, 24, 253syl 18 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  N )  C_  ( ZZ>= `  K )
)
27 eluzfz2 10820 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
2827adantr 451 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  e.  ( M ... N
) )
2928, 3eleqtrd 2372 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  e.  ( J ... K
) )
30 elfzuz3 10811 . . . . . . 7  |-  ( N  e.  ( J ... K )  ->  K  e.  ( ZZ>= `  N )
)
31 uzss 10264 . . . . . . 7  |-  ( K  e.  ( ZZ>= `  N
)  ->  ( ZZ>= `  K )  C_  ( ZZ>=
`  N ) )
3229, 30, 313syl 18 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  K )  C_  ( ZZ>= `  N )
)
3326, 32eqssd 3209 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  N )  =  ( ZZ>= `  K )
)
34 eluzelz 10254 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3534adantr 451 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  e.  ZZ )
36 uz11 10266 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( ZZ>= `  N )  =  ( ZZ>= `  K
)  <->  N  =  K
) )
3735, 36syl 15 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  (
( ZZ>= `  N )  =  ( ZZ>= `  K
)  <->  N  =  K
) )
3833, 37mpbid 201 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  =  K )
3920, 38jca 518 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( M  =  J  /\  N  =  K )
)
4039ex 423 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( M ... N )  =  ( J ... K
)  ->  ( M  =  J  /\  N  =  K ) ) )
41 oveq12 5883 . 2  |-  ( ( M  =  J  /\  N  =  K )  ->  ( M ... N
)  =  ( J ... K ) )
4240, 41impbid1 194 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( M ... N )  =  ( J ... K
)  <->  ( M  =  J  /\  N  =  K ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271  (class class class)co 5874   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798
This theorem is referenced by:  gsumval2a  14475  eedimeq  24598  dffprod  25422  sdclem2  26555  usgraexvlem  28261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-neg 9056  df-z 10041  df-uz 10247  df-fz 10799
  Copyright terms: Public domain W3C validator