ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  frecfzen2 Unicode version

Theorem frecfzen2 9498
Description: The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Jim Kingdon, 18-May-2020.)
Hypothesis
Ref Expression
frecfzennn.1  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )
Assertion
Ref Expression
frecfzen2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )

Proof of Theorem frecfzen2
StepHypRef Expression
1 eluzel2 8694 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
2 eluzelz 8698 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3 1z 8447 . . . . 5  |-  1  e.  ZZ
4 zsubcl 8462 . . . . 5  |-  ( ( 1  e.  ZZ  /\  M  e.  ZZ )  ->  ( 1  -  M
)  e.  ZZ )
53, 1, 4sylancr 405 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( 1  -  M )  e.  ZZ )
6 fzen 9127 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  (
1  -  M )  e.  ZZ )  -> 
( M ... N
)  ~~  ( ( M  +  ( 1  -  M ) ) ... ( N  +  ( 1  -  M
) ) ) )
71, 2, 5, 6syl3anc 1170 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  ~~  (
( M  +  ( 1  -  M ) ) ... ( N  +  ( 1  -  M ) ) ) )
81zcnd 8540 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  CC )
9 ax-1cn 7120 . . . . 5  |-  1  e.  CC
10 pncan3 7372 . . . . 5  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( M  +  ( 1  -  M ) )  =  1 )
118, 9, 10sylancl 404 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M  +  ( 1  -  M ) )  =  1 )
12 zcn 8426 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  CC )
13 zcn 8426 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
14 addsubass 7374 . . . . . . . 8  |-  ( ( N  e.  CC  /\  1  e.  CC  /\  M  e.  CC )  ->  (
( N  +  1 )  -  M )  =  ( N  +  ( 1  -  M
) ) )
159, 14mp3an2 1257 . . . . . . 7  |-  ( ( N  e.  CC  /\  M  e.  CC )  ->  ( ( N  + 
1 )  -  M
)  =  ( N  +  ( 1  -  M ) ) )
1612, 13, 15syl2an 283 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N  + 
1 )  -  M
)  =  ( N  +  ( 1  -  M ) ) )
172, 1, 16syl2anc 403 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( N  +  1 )  -  M )  =  ( N  +  ( 1  -  M ) ) )
1817eqcomd 2087 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  ( 1  -  M ) )  =  ( ( N  + 
1 )  -  M
) )
1911, 18oveq12d 5555 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( M  +  ( 1  -  M ) ) ... ( N  +  ( 1  -  M
) ) )  =  ( 1 ... (
( N  +  1 )  -  M ) ) )
207, 19breqtrd 3811 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  ~~  (
1 ... ( ( N  +  1 )  -  M ) ) )
21 peano2uz 8741 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
22 uznn0sub 8720 . . 3  |-  ( ( N  +  1 )  e.  ( ZZ>= `  M
)  ->  ( ( N  +  1 )  -  M )  e. 
NN0 )
23 frecfzennn.1 . . . 4  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )
2423frecfzennn 9497 . . 3  |-  ( ( ( N  +  1 )  -  M )  e.  NN0  ->  ( 1 ... ( ( N  +  1 )  -  M ) )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )
2521, 22, 243syl 17 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( 1 ... ( ( N  +  1 )  -  M ) )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )
26 entr 6323 . 2  |-  ( ( ( M ... N
)  ~~  ( 1 ... ( ( N  +  1 )  -  M ) )  /\  ( 1 ... (
( N  +  1 )  -  M ) )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )  ->  ( M ... N )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )
2720, 25, 26syl2anc 403 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   class class class wbr 3787    |-> cmpt 3841   `'ccnv 4364   ` cfv 4926  (class class class)co 5537  freccfrec 6033    ~~ cen 6278   CCcc 7030   0cc0 7032   1c1 7033    + caddc 7035    - cmin 7335   NN0cn0 8344   ZZcz 8421   ZZ>=cuz 8689   ...cfz 9094
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331  ax-cnex 7118  ax-resscn 7119  ax-1cn 7120  ax-1re 7121  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-addcom 7127  ax-addass 7129  ax-distr 7131  ax-i2m1 7132  ax-0lt1 7133  ax-0id 7135  ax-rnegex 7136  ax-cnre 7138  ax-pre-ltirr 7139  ax-pre-ltwlin 7140  ax-pre-lttrn 7141  ax-pre-apti 7142  ax-pre-ltadd 7143
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-id 4050  df-iord 4123  df-on 4125  df-ilim 4126  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-frec 6034  df-1o 6059  df-er 6165  df-en 6281  df-pnf 7206  df-mnf 7207  df-xr 7208  df-ltxr 7209  df-le 7210  df-sub 7337  df-neg 7338  df-inn 8096  df-n0 8345  df-z 8422  df-uz 8690  df-fz 9095
This theorem is referenced by:  fzfig  9501
  Copyright terms: Public domain W3C validator