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Theorem fzm1 10132
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 5904 . . . . . . 7  |-  ( N  =  M  ->  ( N ... N )  =  ( M ... N
) )
21eleq2d 2259 . . . . . 6  |-  ( N  =  M  ->  ( K  e.  ( N ... N )  <->  K  e.  ( M ... N ) ) )
3 elfz1eq 10067 . . . . . 6  |-  ( K  e.  ( N ... N )  ->  K  =  N )
42, 3biimtrrdi 164 . . . . 5  |-  ( N  =  M  ->  ( K  e.  ( M ... N )  ->  K  =  N ) )
5 olc 712 . . . . 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 277 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  e.  ( M ... N )  ->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
8 noel 3441 . . . . . 6  |-  -.  K  e.  (/)
9 eluzelz 9568 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
109adantr 276 . . . . . . . . . . 11  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  N  e.  ZZ )
1110zred 9406 . . . . . . . . . 10  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  N  e.  RR )
1211ltm1d 8920 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( N  -  1 )  <  N )
13 breq2 4022 . . . . . . . . . 10  |-  ( N  =  M  ->  (
( N  -  1 )  <  N  <->  ( N  -  1 )  < 
M ) )
1413adantl 277 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  (
( N  -  1 )  <  N  <->  ( N  -  1 )  < 
M ) )
1512, 14mpbid 147 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( N  -  1 )  <  M )
16 eluzel2 9564 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
1716adantr 276 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  M  e.  ZZ )
18 1zzd 9311 . . . . . . . . . 10  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  1  e.  ZZ )
1910, 18zsubcld 9411 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( N  -  1 )  e.  ZZ )
20 fzn 10074 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  ( N  -  1
)  e.  ZZ )  ->  ( ( N  -  1 )  < 
M  <->  ( M ... ( N  -  1
) )  =  (/) ) )
2117, 19, 20syl2anc 411 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  (
( N  -  1 )  <  M  <->  ( M ... ( N  -  1 ) )  =  (/) ) )
2215, 21mpbid 147 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( M ... ( N  - 
1 ) )  =  (/) )
2322eleq2d 2259 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  e.  ( M ... ( N  -  1 ) )  <->  K  e.  (/) ) )
248, 23mtbiri 676 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  -.  K  e.  ( M ... ( N  -  1 ) ) )
2524pm2.21d 620 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  e.  ( M ... ( N  -  1 ) )  ->  K  e.  ( M ... N
) ) )
26 eluzfz2 10064 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
2726ad2antrr 488 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  M )  /\  N  =  M )  /\  K  =  N
)  ->  N  e.  ( M ... N ) )
28 eleq1 2252 . . . . . . 7  |-  ( K  =  N  ->  ( K  e.  ( M ... N )  <->  N  e.  ( M ... N ) ) )
2928adantl 277 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  M )  /\  N  =  M )  /\  K  =  N
)  ->  ( K  e.  ( M ... N
)  <->  N  e.  ( M ... N ) ) )
3027, 29mpbird 167 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  M )  /\  N  =  M )  /\  K  =  N
)  ->  K  e.  ( M ... N ) )
3130ex 115 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  =  N  ->  K  e.  ( M ... N ) ) )
3225, 31jaod 718 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  (
( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N )  ->  K  e.  ( M ... N ) ) )
337, 32impbid 129 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  e.  ( M ... N )  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
34 elfzp1 10104 . . . 4  |-  ( ( N  -  1 )  e.  ( ZZ>= `  M
)  ->  ( K  e.  ( M ... (
( N  -  1 )  +  1 ) )  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  ( ( N  - 
1 )  +  1 ) ) ) )
3534adantl 277 . . 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 276 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  N  e.  ZZ )
3736zcnd 9407 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  N  e.  CC )
38 npcan1 8366 . . . . . 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 5913 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  ( M ... ( ( N  -  1 )  +  1 ) )  =  ( M ... N
) )
4140eleq2d 2259 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  ( K  e.  ( M ... ( ( N  - 
1 )  +  1 ) )  <->  K  e.  ( M ... N ) ) )
4239eqeq2d 2201 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  ( K  =  ( ( N  -  1 )  +  1 )  <->  K  =  N ) )
4342orbi2d 791 . . 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 218 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  ( K  e.  ( M ... N )  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
45 uzm1 9590 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  =  M  \/  ( N  -  1 )  e.  ( ZZ>= `  M
) ) )
4633, 44, 45mpjaodan 799 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 104    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2160   (/)c0 3437   class class class wbr 4018   ` cfv 5235  (class class class)co 5897   CCcc 7840   1c1 7843    + caddc 7845    < clt 8023    - cmin 8159   ZZcz 9284   ZZ>=cuz 9559   ...cfz 10040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-0id 7950  ax-rnegex 7951  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-inn 8951  df-n0 9208  df-z 9285  df-uz 9560  df-fz 10041
This theorem is referenced by:  bcpasc  10781  phibndlem  12251  lgsdir2lem2  14908
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