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

Theorem frecfzen2 10383
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 9492 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
2 eluzelz 9496 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3 1z 9238 . . . . 5  |-  1  e.  ZZ
4 zsubcl 9253 . . . . 5  |-  ( ( 1  e.  ZZ  /\  M  e.  ZZ )  ->  ( 1  -  M
)  e.  ZZ )
53, 1, 4sylancr 412 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( 1  -  M )  e.  ZZ )
6 fzen 9999 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  (
1  -  M )  e.  ZZ )  -> 
( M ... N
)  ~~  ( ( M  +  ( 1  -  M ) ) ... ( N  +  ( 1  -  M
) ) ) )
71, 2, 5, 6syl3anc 1233 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  ~~  (
( M  +  ( 1  -  M ) ) ... ( N  +  ( 1  -  M ) ) ) )
81zcnd 9335 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  CC )
9 ax-1cn 7867 . . . . 5  |-  1  e.  CC
10 pncan3 8127 . . . . 5  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( M  +  ( 1  -  M ) )  =  1 )
118, 9, 10sylancl 411 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M  +  ( 1  -  M ) )  =  1 )
12 zcn 9217 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  CC )
13 zcn 9217 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
14 addsubass 8129 . . . . . . . 8  |-  ( ( N  e.  CC  /\  1  e.  CC  /\  M  e.  CC )  ->  (
( N  +  1 )  -  M )  =  ( N  +  ( 1  -  M
) ) )
159, 14mp3an2 1320 . . . . . . 7  |-  ( ( N  e.  CC  /\  M  e.  CC )  ->  ( ( N  + 
1 )  -  M
)  =  ( N  +  ( 1  -  M ) ) )
1612, 13, 15syl2an 287 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N  + 
1 )  -  M
)  =  ( N  +  ( 1  -  M ) ) )
172, 1, 16syl2anc 409 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( N  +  1 )  -  M )  =  ( N  +  ( 1  -  M ) ) )
1817eqcomd 2176 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  ( 1  -  M ) )  =  ( ( N  + 
1 )  -  M
) )
1911, 18oveq12d 5871 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( M  +  ( 1  -  M ) ) ... ( N  +  ( 1  -  M
) ) )  =  ( 1 ... (
( N  +  1 )  -  M ) ) )
207, 19breqtrd 4015 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  ~~  (
1 ... ( ( N  +  1 )  -  M ) ) )
21 peano2uz 9542 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
22 uznn0sub 9518 . . 3  |-  ( ( N  +  1 )  e.  ( ZZ>= `  M
)  ->  ( ( N  +  1 )  -  M )  e. 
NN0 )
23 frecfzennn.1 . . . 4  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )
2423frecfzennn 10382 . . 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 6762 . 2  |-  ( ( ( M ... N
)  ~~  ( 1 ... ( ( N  +  1 )  -  M ) )  /\  ( 1 ... (
( N  +  1 )  -  M ) )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )  ->  ( M ... N )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )
2720, 25, 26syl2anc 409 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141   class class class wbr 3989    |-> cmpt 4050   `'ccnv 4610   ` cfv 5198  (class class class)co 5853  freccfrec 6369    ~~ cen 6716   CCcc 7772   0cc0 7774   1c1 7775    + caddc 7777    - cmin 8090   NN0cn0 9135   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-1o 6395  df-er 6513  df-en 6719  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966
This theorem is referenced by:  fzfig  10386
  Copyright terms: Public domain W3C validator