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Theorem fzopth 11089
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 11064 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
21adantr 452 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  e.  ( M ... N
) )
3 simpr 448 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( M ... N )  =  ( J ... K
) )
42, 3eleqtrd 2512 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  e.  ( J ... K
) )
5 elfzuz 11055 . . . . . . 7  |-  ( M  e.  ( J ... K )  ->  M  e.  ( ZZ>= `  J )
)
6 uzss 10506 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  J
)  ->  ( ZZ>= `  M )  C_  ( ZZ>=
`  J ) )
74, 5, 63syl 19 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  M )  C_  ( ZZ>= `  J )
)
8 elfzuz2 11062 . . . . . . . . 9  |-  ( M  e.  ( J ... K )  ->  K  e.  ( ZZ>= `  J )
)
9 eluzfz1 11064 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  J
)  ->  J  e.  ( J ... K ) )
104, 8, 93syl 19 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  J  e.  ( J ... K
) )
1110, 3eleqtrrd 2513 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  J  e.  ( M ... N
) )
12 elfzuz 11055 . . . . . . 7  |-  ( J  e.  ( M ... N )  ->  J  e.  ( ZZ>= `  M )
)
13 uzss 10506 . . . . . . 7  |-  ( J  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  J )  C_  ( ZZ>=
`  M ) )
1411, 12, 133syl 19 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  J )  C_  ( ZZ>= `  M )
)
157, 14eqssd 3365 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  M )  =  ( ZZ>= `  J )
)
16 eluzel2 10493 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
1716adantr 452 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  e.  ZZ )
18 uz11 10508 . . . . . 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 202 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  =  J )
21 eluzfz2 11065 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  J
)  ->  K  e.  ( J ... K ) )
224, 8, 213syl 19 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  K  e.  ( J ... K
) )
2322, 3eleqtrrd 2513 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  K  e.  ( M ... N
) )
24 elfzuz3 11056 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K )
)
25 uzss 10506 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( ZZ>= `  N )  C_  ( ZZ>=
`  K ) )
2623, 24, 253syl 19 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  N )  C_  ( ZZ>= `  K )
)
27 eluzfz2 11065 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
2827adantr 452 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  e.  ( M ... N
) )
2928, 3eleqtrd 2512 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  e.  ( J ... K
) )
30 elfzuz3 11056 . . . . . . 7  |-  ( N  e.  ( J ... K )  ->  K  e.  ( ZZ>= `  N )
)
31 uzss 10506 . . . . . . 7  |-  ( K  e.  ( ZZ>= `  N
)  ->  ( ZZ>= `  K )  C_  ( ZZ>=
`  N ) )
3229, 30, 313syl 19 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  K )  C_  ( ZZ>= `  N )
)
3326, 32eqssd 3365 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  N )  =  ( ZZ>= `  K )
)
34 eluzelz 10496 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3534adantr 452 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  e.  ZZ )
36 uz11 10508 . . . . . 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 202 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  =  K )
3920, 38jca 519 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( M  =  J  /\  N  =  K )
)
4039ex 424 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( M ... N )  =  ( J ... K
)  ->  ( M  =  J  /\  N  =  K ) ) )
41 oveq12 6090 . 2  |-  ( ( M  =  J  /\  N  =  K )  ->  ( M ... N
)  =  ( J ... K ) )
4240, 41impbid1 195 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( M ... N )  =  ( J ... K
)  <->  ( M  =  J  /\  N  =  K ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3320   ` cfv 5454  (class class class)co 6081   ZZcz 10282   ZZ>=cuz 10488   ...cfz 11043
This theorem is referenced by:  gsumval2a  14782  usgraexvlem  21414  eedimeq  25837  sdclem2  26446  2ffzeq  28121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-pre-lttri 9064  ax-pre-lttrn 9065
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-neg 9294  df-z 10283  df-uz 10489  df-fz 11044
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