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Theorem fzm1 11871
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 6295 . . . . . . 7  |-  ( N  =  M  ->  ( N ... N )  =  ( M ... N
) )
21eleq2d 2513 . . . . . 6  |-  ( N  =  M  ->  ( K  e.  ( N ... N )  <->  K  e.  ( M ... N ) ) )
3 elfz1eq 11807 . . . . . 6  |-  ( K  e.  ( N ... N )  ->  K  =  N )
42, 3syl6bir 233 . . . . 5  |-  ( N  =  M  ->  ( K  e.  ( M ... N )  ->  K  =  N ) )
5 olc 386 . . . . 5  |-  ( K  =  N  ->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) )
64, 5syl6 34 . . . 4  |-  ( N  =  M  ->  ( K  e.  ( M ... N )  ->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
76adantl 468 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  e.  ( M ... N )  ->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
8 noel 3734 . . . . . 6  |-  -.  K  e.  (/)
9 eluzelz 11165 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
109adantr 467 . . . . . . . . . . 11  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  N  e.  ZZ )
1110zred 11037 . . . . . . . . . 10  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  N  e.  RR )
1211ltm1d 10536 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( N  -  1 )  <  N )
13 breq2 4405 . . . . . . . . . 10  |-  ( N  =  M  ->  (
( N  -  1 )  <  N  <->  ( N  -  1 )  < 
M ) )
1413adantl 468 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  (
( N  -  1 )  <  N  <->  ( N  -  1 )  < 
M ) )
1512, 14mpbid 214 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( N  -  1 )  <  M )
16 eluzel2 11161 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
1716adantr 467 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  M  e.  ZZ )
18 1zzd 10965 . . . . . . . . . 10  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  1  e.  ZZ )
1910, 18zsubcld 11042 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( N  -  1 )  e.  ZZ )
20 fzn 11812 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  ( N  -  1
)  e.  ZZ )  ->  ( ( N  -  1 )  < 
M  <->  ( M ... ( N  -  1
) )  =  (/) ) )
2117, 19, 20syl2anc 666 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  (
( N  -  1 )  <  M  <->  ( M ... ( N  -  1 ) )  =  (/) ) )
2215, 21mpbid 214 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( M ... ( N  - 
1 ) )  =  (/) )
2322eleq2d 2513 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  e.  ( M ... ( N  -  1 ) )  <->  K  e.  (/) ) )
248, 23mtbiri 305 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  -.  K  e.  ( M ... ( N  -  1 ) ) )
2524pm2.21d 110 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  e.  ( M ... ( N  -  1 ) )  ->  K  e.  ( M ... N
) ) )
26 eluzfz2 11804 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
2726ad2antrr 731 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  M )  /\  N  =  M )  /\  K  =  N
)  ->  N  e.  ( M ... N ) )
28 eleq1 2516 . . . . . . 7  |-  ( K  =  N  ->  ( K  e.  ( M ... N )  <->  N  e.  ( M ... N ) ) )
2928adantl 468 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  M )  /\  N  =  M )  /\  K  =  N
)  ->  ( K  e.  ( M ... N
)  <->  N  e.  ( M ... N ) ) )
3027, 29mpbird 236 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  M )  /\  N  =  M )  /\  K  =  N
)  ->  K  e.  ( M ... N ) )
3130ex 436 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  =  N  ->  K  e.  ( M ... N ) ) )
3225, 31jaod 382 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  (
( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N )  ->  K  e.  ( M ... N ) ) )
337, 32impbid 194 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  N  =  M )  ->  ( K  e.  ( M ... N )  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
34 elfzp1 11843 . . . 4  |-  ( ( N  -  1 )  e.  ( ZZ>= `  M
)  ->  ( K  e.  ( M ... (
( N  -  1 )  +  1 ) )  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  ( ( N  - 
1 )  +  1 ) ) ) )
3534adantl 468 . . 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 467 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  N  e.  ZZ )
3736zcnd 11038 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  N  e.  CC )
38 npcan1 10041 . . . . . 6  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
3937, 38syl 17 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  (
( N  -  1 )  +  1 )  =  N )
4039oveq2d 6304 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  ( M ... ( ( N  -  1 )  +  1 ) )  =  ( M ... N
) )
4140eleq2d 2513 . . 3  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  ( K  e.  ( M ... ( ( N  - 
1 )  +  1 ) )  <->  K  e.  ( M ... N ) ) )
4239eqeq2d 2460 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  ( K  =  ( ( N  -  1 )  +  1 )  <->  K  =  N ) )
4342orbi2d 707 . . 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 287 . 2  |-  ( ( N  e.  ( ZZ>= `  M )  /\  ( N  -  1 )  e.  ( ZZ>= `  M
) )  ->  ( K  e.  ( M ... N )  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
45 uzm1 11186 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  =  M  \/  ( N  -  1 )  e.  ( ZZ>= `  M
) ) )
4633, 44, 45mpjaodan 794 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 188    \/ wo 370    /\ wa 371    = wceq 1443    e. wcel 1886   (/)c0 3730   class class class wbr 4401   ` cfv 5581  (class class class)co 6288   CCcc 9534   1c1 9537    + caddc 9539    < clt 9672    - cmin 9857   ZZcz 10934   ZZ>=cuz 11156   ...cfz 11781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782
This theorem is referenced by:  bcpasc  12503  phibndlem  14711  lgsdir2lem2  24245  submateqlem2  28627  poimirlem14  31947  poimirlem23  31956  poimirlem25  31958  poimirlem27  31960  acongeq  35827  jm2.26lem3  35850
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