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Theorem frecfzen2 9799
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 8993 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
2 eluzelz 8997 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3 1z 8746 . . . . 5  |-  1  e.  ZZ
4 zsubcl 8761 . . . . 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 9426 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  (
1  -  M )  e.  ZZ )  -> 
( M ... N
)  ~~  ( ( M  +  ( 1  -  M ) ) ... ( N  +  ( 1  -  M
) ) ) )
71, 2, 5, 6syl3anc 1174 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  ~~  (
( M  +  ( 1  -  M ) ) ... ( N  +  ( 1  -  M ) ) ) )
81zcnd 8839 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  CC )
9 ax-1cn 7417 . . . . 5  |-  1  e.  CC
10 pncan3 7669 . . . . 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 8725 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  CC )
13 zcn 8725 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
14 addsubass 7671 . . . . . . . 8  |-  ( ( N  e.  CC  /\  1  e.  CC  /\  M  e.  CC )  ->  (
( N  +  1 )  -  M )  =  ( N  +  ( 1  -  M
) ) )
159, 14mp3an2 1261 . . . . . . 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 2093 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  ( 1  -  M ) )  =  ( ( N  + 
1 )  -  M
) )
1911, 18oveq12d 5652 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( M  +  ( 1  -  M ) ) ... ( N  +  ( 1  -  M
) ) )  =  ( 1 ... (
( N  +  1 )  -  M ) ) )
207, 19breqtrd 3861 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  ~~  (
1 ... ( ( N  +  1 )  -  M ) ) )
21 peano2uz 9040 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
22 uznn0sub 9019 . . 3  |-  ( ( N  +  1 )  e.  ( ZZ>= `  M
)  ->  ( ( N  +  1 )  -  M )  e. 
NN0 )
23 frecfzennn.1 . . . 4  |-  G  = frec ( ( x  e.  ZZ  |->  ( x  + 
1 ) ) ,  0 )
2423frecfzennn 9798 . . 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 6481 . 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 1289    e. wcel 1438   class class class wbr 3837    |-> cmpt 3891   `'ccnv 4427   ` cfv 5002  (class class class)co 5634  freccfrec 6137    ~~ cen 6435   CCcc 7327   0cc0 7329   1c1 7330    + caddc 7332    - cmin 7632   NN0cn0 8643   ZZcz 8720   ZZ>=cuz 8988   ...cfz 9393
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-1o 6163  df-er 6272  df-en 6438  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721  df-uz 8989  df-fz 9394
This theorem is referenced by:  fzfig  9802
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