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Theorem frecfzen2 10041
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 9181 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
2 eluzelz 9185 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3 1z 8932 . . . . 5  |-  1  e.  ZZ
4 zsubcl 8947 . . . . 5  |-  ( ( 1  e.  ZZ  /\  M  e.  ZZ )  ->  ( 1  -  M
)  e.  ZZ )
53, 1, 4sylancr 408 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( 1  -  M )  e.  ZZ )
6 fzen 9664 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  (
1  -  M )  e.  ZZ )  -> 
( M ... N
)  ~~  ( ( M  +  ( 1  -  M ) ) ... ( N  +  ( 1  -  M
) ) ) )
71, 2, 5, 6syl3anc 1184 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  ~~  (
( M  +  ( 1  -  M ) ) ... ( N  +  ( 1  -  M ) ) ) )
81zcnd 9026 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  CC )
9 ax-1cn 7588 . . . . 5  |-  1  e.  CC
10 pncan3 7841 . . . . 5  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( M  +  ( 1  -  M ) )  =  1 )
118, 9, 10sylancl 407 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M  +  ( 1  -  M ) )  =  1 )
12 zcn 8911 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  CC )
13 zcn 8911 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
14 addsubass 7843 . . . . . . . 8  |-  ( ( N  e.  CC  /\  1  e.  CC  /\  M  e.  CC )  ->  (
( N  +  1 )  -  M )  =  ( N  +  ( 1  -  M
) ) )
159, 14mp3an2 1271 . . . . . . 7  |-  ( ( N  e.  CC  /\  M  e.  CC )  ->  ( ( N  + 
1 )  -  M
)  =  ( N  +  ( 1  -  M ) ) )
1612, 13, 15syl2an 285 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N  + 
1 )  -  M
)  =  ( N  +  ( 1  -  M ) ) )
172, 1, 16syl2anc 406 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( N  +  1 )  -  M )  =  ( N  +  ( 1  -  M ) ) )
1817eqcomd 2105 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  ( 1  -  M ) )  =  ( ( N  + 
1 )  -  M
) )
1911, 18oveq12d 5724 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( M  +  ( 1  -  M ) ) ... ( N  +  ( 1  -  M
) ) )  =  ( 1 ... (
( N  +  1 )  -  M ) ) )
207, 19breqtrd 3899 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  ~~  (
1 ... ( ( N  +  1 )  -  M ) ) )
21 peano2uz 9228 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
22 uznn0sub 9207 . . 3  |-  ( ( N  +  1 )  e.  ( ZZ>= `  M
)  ->  ( ( N  +  1 )  -  M )  e. 
NN0 )
23 frecfzennn.1 . . . 4  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )
2423frecfzennn 10040 . . 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 6608 . 2  |-  ( ( ( M ... N
)  ~~  ( 1 ... ( ( N  +  1 )  -  M ) )  /\  ( 1 ... (
( N  +  1 )  -  M ) )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )  ->  ( M ... N )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )
2720, 25, 26syl2anc 406 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1299    e. wcel 1448   class class class wbr 3875    |-> cmpt 3929   `'ccnv 4476   ` cfv 5059  (class class class)co 5706  freccfrec 6217    ~~ cen 6562   CCcc 7498   0cc0 7500   1c1 7501    + caddc 7503    - cmin 7804   NN0cn0 8829   ZZcz 8906   ZZ>=cuz 9176   ...cfz 9631
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611
This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-1o 6243  df-er 6359  df-en 6565  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-n0 8830  df-z 8907  df-uz 9177  df-fz 9632
This theorem is referenced by:  fzfig  10044
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