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Theorem fzm1 9510
Description: Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
fzm1  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( K  e.  ( M ... N
)  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )

Proof of Theorem fzm1
StepHypRef Expression
1 oveq1 5659 . . . . . . 7  |-  ( N  =  M  ->  ( N ... N )  =  ( M ... N
) )
21eleq2d 2157 . . . . . 6  |-  ( N  =  M  ->  ( K  e.  ( N ... N )  <->  K  e.  ( M ... N ) ) )
3 elfz1eq 9447 . . . . . 6  |-  ( K  e.  ( N ... N )  ->  K  =  N )
42, 3syl6bir 162 . . . . 5  |-  ( N  =  M  ->  ( K  e.  ( M ... N )  ->  K  =  N ) )
5 olc 667 . . . . 5  |-  ( K  =  N  ->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) )
64, 5syl6 33 . . . 4  |-  ( N  =  M  ->  ( K  e.  ( M ... N )  ->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
76adantl 271 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  e.  ( M ... N )  ->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
8 noel 3290 . . . . . 6  |-  -.  K  e.  (/)
9 eluzelz 9026 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
109adantr 270 . . . . . . . . . . 11  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  N  e.  ZZ )
1110zred 8866 . . . . . . . . . 10  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  N  e.  RR )
1211ltm1d 8391 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( N  -  1 )  <  N )
13 breq2 3849 . . . . . . . . . 10  |-  ( N  =  M  ->  (
( N  -  1 )  <  N  <->  ( N  -  1 )  < 
M ) )
1413adantl 271 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  (
( N  -  1 )  <  N  <->  ( N  -  1 )  < 
M ) )
1512, 14mpbid 145 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( N  -  1 )  <  M )
16 eluzel2 9022 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
1716adantr 270 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  M  e.  ZZ )
18 1zzd 8775 . . . . . . . . . 10  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  1  e.  ZZ )
1910, 18zsubcld 8871 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( N  -  1 )  e.  ZZ )
20 fzn 9454 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  ( N  -  1
)  e.  ZZ )  ->  ( ( N  -  1 )  < 
M  <->  ( M ... ( N  -  1
) )  =  (/) ) )
2117, 19, 20syl2anc 403 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  (
( N  -  1 )  <  M  <->  ( M ... ( N  -  1 ) )  =  (/) ) )
2215, 21mpbid 145 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( M ... ( N  - 
1 ) )  =  (/) )
2322eleq2d 2157 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  e.  ( M ... ( N  -  1 ) )  <->  K  e.  (/) ) )
248, 23mtbiri 635 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  -.  K  e.  ( M ... ( N  -  1 ) ) )
2524pm2.21d 584 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  e.  ( M ... ( N  -  1 ) )  ->  K  e.  ( M ... N
) ) )
26 eluzfz2 9444 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
2726ad2antrr 472 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  M )  /\  N  =  M )  /\  K  =  N
)  ->  N  e.  ( M ... N ) )
28 eleq1 2150 . . . . . . 7  |-  ( K  =  N  ->  ( K  e.  ( M ... N )  <->  N  e.  ( M ... N ) ) )
2928adantl 271 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  M )  /\  N  =  M )  /\  K  =  N
)  ->  ( K  e.  ( M ... N
)  <->  N  e.  ( M ... N ) ) )
3027, 29mpbird 165 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  M )  /\  N  =  M )  /\  K  =  N
)  ->  K  e.  ( M ... N ) )
3130ex 113 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  =  N  ->  K  e.  ( M ... N ) ) )
3225, 31jaod 672 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  (
( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N )  ->  K  e.  ( M ... N ) ) )
337, 32impbid 127 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  e.  ( M ... N )  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
34 elfzp1 9482 . . . 4  |-  ( ( N  -  1 )  e.  ( ZZ>= `  M
)  ->  ( K  e.  ( M ... (
( N  -  1 )  +  1 ) )  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  ( ( N  - 
1 )  +  1 ) ) ) )
3534adantl 271 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  ( K  e.  ( M ... ( ( N  - 
1 )  +  1 ) )  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  ( ( N  - 
1 )  +  1 ) ) ) )
369adantr 270 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  N  e.  ZZ )
3736zcnd 8867 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  N  e.  CC )
38 npcan1 7854 . . . . . 6  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
3937, 38syl 14 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  (
( N  -  1 )  +  1 )  =  N )
4039oveq2d 5668 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  ( M ... ( ( N  -  1 )  +  1 ) )  =  ( M ... N
) )
4140eleq2d 2157 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  ( K  e.  ( M ... ( ( N  - 
1 )  +  1 ) )  <->  K  e.  ( M ... N ) ) )
4239eqeq2d 2099 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  ( K  =  ( ( N  -  1 )  +  1 )  <->  K  =  N ) )
4342orbi2d 739 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  (
( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  ( ( N  -  1 )  +  1 ) )  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
4435, 41, 433bitr3d 216 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  ( K  e.  ( M ... N )  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
45 uzm1 9047 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  =  M  \/  ( N  -  1 )  e.  ( ZZ>= `  M
) ) )
4633, 44, 45mpjaodan 747 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( K  e.  ( M ... N
)  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   (/)c0 3286   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   CCcc 7346   1c1 7349    + caddc 7351    < clt 7520    - cmin 7651   ZZcz 8748   ZZ>=cuz 9017   ...cfz 9422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-0id 7451  ax-rnegex 7452  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-inn 8421  df-n0 8672  df-z 8749  df-uz 9018  df-fz 9423
This theorem is referenced by:  bcpasc  10170  phibndlem  11466
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