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Theorem fzen2 11271
Description: The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Mario Carneiro, 13-Feb-2014.)
Hypothesis
Ref Expression
fzennn.1  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )
Assertion
Ref Expression
fzen2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )

Proof of Theorem fzen2
StepHypRef Expression
1 eluzel2 10457 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
2 eluzelz 10460 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
3 1z 10275 . . . . 5  |-  1  e.  ZZ
4 zsubcl 10283 . . . . 5  |-  ( ( 1  e.  ZZ  /\  M  e.  ZZ )  ->  ( 1  -  M
)  e.  ZZ )
53, 1, 4sylancr 645 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( 1  -  M )  e.  ZZ )
6 fzen 11036 . . . 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 10340 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  CC )
9 ax-1cn 9012 . . . . 5  |-  1  e.  CC
10 pncan3 9277 . . . . 5  |-  ( ( M  e.  CC  /\  1  e.  CC )  ->  ( M  +  ( 1  -  M ) )  =  1 )
118, 9, 10sylancl 644 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M  +  ( 1  -  M ) )  =  1 )
12 zcn 10251 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  CC )
13 zcn 10251 . . . . . . 7  |-  ( M  e.  ZZ  ->  M  e.  CC )
14 addsubass 9279 . . . . . . . 8  |-  ( ( N  e.  CC  /\  1  e.  CC  /\  M  e.  CC )  ->  (
( N  +  1 )  -  M )  =  ( N  +  ( 1  -  M
) ) )
159, 14mp3an2 1267 . . . . . . 7  |-  ( ( N  e.  CC  /\  M  e.  CC )  ->  ( ( N  + 
1 )  -  M
)  =  ( N  +  ( 1  -  M ) ) )
1612, 13, 15syl2an 464 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N  + 
1 )  -  M
)  =  ( N  +  ( 1  -  M ) ) )
172, 1, 16syl2anc 643 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( N  +  1 )  -  M )  =  ( N  +  ( 1  -  M ) ) )
1817eqcomd 2417 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  ( 1  -  M ) )  =  ( ( N  + 
1 )  -  M
) )
1911, 18oveq12d 6066 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( M  +  ( 1  -  M ) ) ... ( N  +  ( 1  -  M
) ) )  =  ( 1 ... (
( N  +  1 )  -  M ) ) )
207, 19breqtrd 4204 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  ~~  (
1 ... ( ( N  +  1 )  -  M ) ) )
21 peano2uz 10494 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
22 uznn0sub 10481 . . 3  |-  ( ( N  +  1 )  e.  ( ZZ>= `  M
)  ->  ( ( N  +  1 )  -  M )  e. 
NN0 )
23 fzennn.1 . . . 4  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )
2423fzennn 11270 . . 3  |-  ( ( ( N  +  1 )  -  M )  e.  NN0  ->  ( 1 ... ( ( N  +  1 )  -  M ) )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )
2521, 22, 243syl 19 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( 1 ... ( ( N  +  1 )  -  M ) )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )
26 entr 7126 . 2  |-  ( ( ( M ... N
)  ~~  ( 1 ... ( ( N  +  1 )  -  M ) )  /\  ( 1 ... (
( N  +  1 )  -  M ) )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )  ->  ( M ... N )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )
2720, 25, 26syl2anc 643 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   _Vcvv 2924   class class class wbr 4180    e. cmpt 4234   omcom 4812   `'ccnv 4844    |` cres 4847   ` cfv 5421  (class class class)co 6048   reccrdg 6634    ~~ cen 7073   CCcc 8952   0cc0 8954   1c1 8955    + caddc 8957    - cmin 9255   NN0cn0 10185   ZZcz 10246   ZZ>=cuz 10452   ...cfz 11007
This theorem is referenced by:  fzfi  11274
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008
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