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Theorem fzopth 9026
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 8997 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
21adantr 265 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  e.  ( M ... N
) )
3 simpr 107 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( M ... N )  =  ( J ... K
) )
42, 3eleqtrd 2132 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  e.  ( J ... K
) )
5 elfzuz 8988 . . . . . . 7  |-  ( M  e.  ( J ... K )  ->  M  e.  ( ZZ>= `  J )
)
6 uzss 8589 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  J
)  ->  ( ZZ>= `  M )  C_  ( ZZ>=
`  J ) )
74, 5, 63syl 17 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  M )  C_  ( ZZ>= `  J )
)
8 elfzuz2 8995 . . . . . . . . 9  |-  ( M  e.  ( J ... K )  ->  K  e.  ( ZZ>= `  J )
)
9 eluzfz1 8997 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  J
)  ->  J  e.  ( J ... K ) )
104, 8, 93syl 17 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  J  e.  ( J ... K
) )
1110, 3eleqtrrd 2133 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  J  e.  ( M ... N
) )
12 elfzuz 8988 . . . . . . 7  |-  ( J  e.  ( M ... N )  ->  J  e.  ( ZZ>= `  M )
)
13 uzss 8589 . . . . . . 7  |-  ( J  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  J )  C_  ( ZZ>=
`  M ) )
1411, 12, 133syl 17 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  J )  C_  ( ZZ>= `  M )
)
157, 14eqssd 2990 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  M )  =  ( ZZ>= `  J )
)
16 eluzel2 8574 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
1716adantr 265 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  e.  ZZ )
18 uz11 8591 . . . . . 6  |-  ( M  e.  ZZ  ->  (
( ZZ>= `  M )  =  ( ZZ>= `  J
)  <->  M  =  J
) )
1917, 18syl 14 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  (
( ZZ>= `  M )  =  ( ZZ>= `  J
)  <->  M  =  J
) )
2015, 19mpbid 139 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  M  =  J )
21 eluzfz2 8998 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  J
)  ->  K  e.  ( J ... K ) )
224, 8, 213syl 17 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  K  e.  ( J ... K
) )
2322, 3eleqtrrd 2133 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  K  e.  ( M ... N
) )
24 elfzuz3 8989 . . . . . . 7  |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K )
)
25 uzss 8589 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( ZZ>= `  N )  C_  ( ZZ>=
`  K ) )
2623, 24, 253syl 17 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  N )  C_  ( ZZ>= `  K )
)
27 eluzfz2 8998 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
2827adantr 265 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  e.  ( M ... N
) )
2928, 3eleqtrd 2132 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  e.  ( J ... K
) )
30 elfzuz3 8989 . . . . . . 7  |-  ( N  e.  ( J ... K )  ->  K  e.  ( ZZ>= `  N )
)
31 uzss 8589 . . . . . . 7  |-  ( K  e.  ( ZZ>= `  N
)  ->  ( ZZ>= `  K )  C_  ( ZZ>=
`  N ) )
3229, 30, 313syl 17 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  K )  C_  ( ZZ>= `  N )
)
3326, 32eqssd 2990 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( ZZ>=
`  N )  =  ( ZZ>= `  K )
)
34 eluzelz 8578 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3534adantr 265 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  e.  ZZ )
36 uz11 8591 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( ZZ>= `  N )  =  ( ZZ>= `  K
)  <->  N  =  K
) )
3735, 36syl 14 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  (
( ZZ>= `  N )  =  ( ZZ>= `  K
)  <->  N  =  K
) )
3833, 37mpbid 139 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  N  =  K )
3920, 38jca 294 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( M ... N )  =  ( J ... K
) )  ->  ( M  =  J  /\  N  =  K )
)
4039ex 112 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( M ... N )  =  ( J ... K
)  ->  ( M  =  J  /\  N  =  K ) ) )
41 oveq12 5549 . 2  |-  ( ( M  =  J  /\  N  =  K )  ->  ( M ... N
)  =  ( J ... K ) )
4240, 41impbid1 134 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( M ... N )  =  ( J ... K
)  <->  ( M  =  J  /\  N  =  K ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409    C_ wss 2945   ` cfv 4930  (class class class)co 5540   ZZcz 8302   ZZ>=cuz 8569   ...cfz 8976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-apti 7057
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-neg 7248  df-z 8303  df-uz 8570  df-fz 8977
This theorem is referenced by:  2ffzeq  9100
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