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Theorem fzm1 11783
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 6303 . . . . . . 7  |-  ( N  =  M  ->  ( N ... N )  =  ( M ... N
) )
21eleq2d 2527 . . . . . 6  |-  ( N  =  M  ->  ( K  e.  ( N ... N )  <->  K  e.  ( M ... N ) ) )
3 elfz1eq 11722 . . . . . 6  |-  ( K  e.  ( N ... N )  ->  K  =  N )
42, 3syl6bir 229 . . . . 5  |-  ( N  =  M  ->  ( K  e.  ( M ... N )  ->  K  =  N ) )
5 olc 384 . . . . 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 466 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  e.  ( M ... N )  ->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
8 noel 3797 . . . . . 6  |-  -.  K  e.  (/)
9 eluzelz 11115 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
109adantr 465 . . . . . . . . . . 11  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  N  e.  ZZ )
1110zred 10990 . . . . . . . . . 10  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  N  e.  RR )
1211ltm1d 10498 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( N  -  1 )  <  N )
13 breq2 4460 . . . . . . . . . 10  |-  ( N  =  M  ->  (
( N  -  1 )  <  N  <->  ( N  -  1 )  < 
M ) )
1413adantl 466 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  (
( N  -  1 )  <  N  <->  ( N  -  1 )  < 
M ) )
1512, 14mpbid 210 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( N  -  1 )  <  M )
16 eluzel2 11111 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
1716adantr 465 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  M  e.  ZZ )
18 1zzd 10916 . . . . . . . . . 10  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  1  e.  ZZ )
1910, 18zsubcld 10995 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( N  -  1 )  e.  ZZ )
20 fzn 11727 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  ( N  -  1
)  e.  ZZ )  ->  ( ( N  -  1 )  < 
M  <->  ( M ... ( N  -  1
) )  =  (/) ) )
2117, 19, 20syl2anc 661 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  (
( N  -  1 )  <  M  <->  ( M ... ( N  -  1 ) )  =  (/) ) )
2215, 21mpbid 210 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( M ... ( N  - 
1 ) )  =  (/) )
2322eleq2d 2527 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  e.  ( M ... ( N  -  1 ) )  <->  K  e.  (/) ) )
248, 23mtbiri 303 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  -.  K  e.  ( M ... ( N  -  1 ) ) )
2524pm2.21d 106 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  e.  ( M ... ( N  -  1 ) )  ->  K  e.  ( M ... N
) ) )
26 eluzfz2 11719 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
2726ad2antrr 725 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  M )  /\  N  =  M )  /\  K  =  N
)  ->  N  e.  ( M ... N ) )
28 eleq1 2529 . . . . . . 7  |-  ( K  =  N  ->  ( K  e.  ( M ... N )  <->  N  e.  ( M ... N ) ) )
2928adantl 466 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  M )  /\  N  =  M )  /\  K  =  N
)  ->  ( K  e.  ( M ... N
)  <->  N  e.  ( M ... N ) ) )
3027, 29mpbird 232 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  M )  /\  N  =  M )  /\  K  =  N
)  ->  K  e.  ( M ... N ) )
3130ex 434 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  =  N  ->  K  e.  ( M ... N ) ) )
3225, 31jaod 380 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  (
( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N )  ->  K  e.  ( M ... N ) ) )
337, 32impbid 191 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  e.  ( M ... N )  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
34 elfzp1 11755 . . . 4  |-  ( ( N  -  1 )  e.  ( ZZ>= `  M
)  ->  ( K  e.  ( M ... (
( N  -  1 )  +  1 ) )  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  ( ( N  - 
1 )  +  1 ) ) ) )
3534adantl 466 . . 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 465 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  N  e.  ZZ )
3736zcnd 10991 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  N  e.  CC )
38 npcan1 10005 . . . . . 6  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
3937, 38syl 16 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  (
( N  -  1 )  +  1 )  =  N )
4039oveq2d 6312 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  ( M ... ( ( N  -  1 )  +  1 ) )  =  ( M ... N
) )
4140eleq2d 2527 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  ( K  e.  ( M ... ( ( N  - 
1 )  +  1 ) )  <->  K  e.  ( M ... N ) ) )
4239eqeq2d 2471 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  ( K  =  ( ( N  -  1 )  +  1 )  <->  K  =  N ) )
4342orbi2d 701 . . 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 283 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  ( K  e.  ( M ... N )  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
45 uzm1 11136 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  =  M  \/  ( N  -  1 )  e.  ( ZZ>= `  M
) ) )
4633, 44, 45mpjaodan 786 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( K  e.  ( M ... N
)  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819   (/)c0 3793   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CCcc 9507   1c1 9510    + caddc 9512    < clt 9645    - cmin 9824   ZZcz 10885   ZZ>=cuz 11106   ...cfz 11697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698
This theorem is referenced by:  bcpasc  12401  phibndlem  14311  lgsdir2lem2  23724  acongeq  31083  jm2.26lem3  31105
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