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Theorem binom1dif 12532
Description: A summation for the difference between  ( ( A  +  1 ) ^ N ) and  ( A ^ N ). (Contributed by Scott Fenton, 9-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
Assertion
Ref Expression
binom1dif  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( ( A  +  1 ) ^ N )  -  ( A ^ N ) )  =  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  ( A ^ k ) ) )
Distinct variable groups:    A, k    k, N

Proof of Theorem binom1dif
StepHypRef Expression
1 simpl 444 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  A  e.  CC )
2 ax-1cn 8974 . . . . . 6  |-  1  e.  CC
3 addcom 9177 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
41, 2, 3sylancl 644 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A  +  1 )  =  ( 1  +  A ) )
54oveq1d 6028 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( A  + 
1 ) ^ N
)  =  ( ( 1  +  A ) ^ N ) )
6 binom1p 12530 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( 1  +  A ) ^ N
)  =  sum_ k  e.  ( 0 ... N
) ( ( N  _C  k )  x.  ( A ^ k
) ) )
7 simpr 448 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  N  e.  NN0 )
8 nn0uz 10445 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
97, 8syl6eleq 2470 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  N  e.  ( ZZ>= ` 
0 ) )
10 bccl2 11534 . . . . . . . . 9  |-  ( k  e.  ( 0 ... N )  ->  ( N  _C  k )  e.  NN )
1110adantl 453 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
0 ... N ) )  ->  ( N  _C  k )  e.  NN )
1211nncnd 9941 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
0 ... N ) )  ->  ( N  _C  k )  e.  CC )
13 elfznn0 11008 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
14 expcl 11319 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
151, 13, 14syl2an 464 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
0 ... N ) )  ->  ( A ^
k )  e.  CC )
1612, 15mulcld 9034 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
0 ... N ) )  ->  ( ( N  _C  k )  x.  ( A ^ k
) )  e.  CC )
17 oveq2 6021 . . . . . . 7  |-  ( k  =  N  ->  ( N  _C  k )  =  ( N  _C  N
) )
18 oveq2 6021 . . . . . . 7  |-  ( k  =  N  ->  ( A ^ k )  =  ( A ^ N
) )
1917, 18oveq12d 6031 . . . . . 6  |-  ( k  =  N  ->  (
( N  _C  k
)  x.  ( A ^ k ) )  =  ( ( N  _C  N )  x.  ( A ^ N
) ) )
209, 16, 19fsumm1 12457 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  sum_ k  e.  ( 0 ... N ) ( ( N  _C  k
)  x.  ( A ^ k ) )  =  ( sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  ( A ^ k
) )  +  ( ( N  _C  N
)  x.  ( A ^ N ) ) ) )
21 bcnn 11523 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  _C  N )  =  1 )
2221adantl 453 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( N  _C  N
)  =  1 )
2322oveq1d 6028 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( N  _C  N )  x.  ( A ^ N ) )  =  ( 1  x.  ( A ^ N
) ) )
24 expcl 11319 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  CC )
2524mulid2d 9032 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( 1  x.  ( A ^ N ) )  =  ( A ^ N ) )
2623, 25eqtrd 2412 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( N  _C  N )  x.  ( A ^ N ) )  =  ( A ^ N ) )
2726oveq2d 6029 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  ( A ^ k ) )  +  ( ( N  _C  N )  x.  ( A ^ N
) ) )  =  ( sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  ( A ^ k ) )  +  ( A ^ N ) ) )
2820, 27eqtrd 2412 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  sum_ k  e.  ( 0 ... N ) ( ( N  _C  k
)  x.  ( A ^ k ) )  =  ( sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  ( A ^ k
) )  +  ( A ^ N ) ) )
295, 6, 283eqtrd 2416 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( A  + 
1 ) ^ N
)  =  ( sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( A ^ k ) )  +  ( A ^ N ) ) )
3029oveq1d 6028 . 2  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( ( A  +  1 ) ^ N )  -  ( A ^ N ) )  =  ( ( sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( A ^ k ) )  +  ( A ^ N ) )  -  ( A ^ N ) ) )
31 fzfid 11232 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( 0 ... ( N  -  1 ) )  e.  Fin )
32 fzssp1 11020 . . . . . . 7  |-  ( 0 ... ( N  - 
1 ) )  C_  ( 0 ... (
( N  -  1 )  +  1 ) )
33 nn0cn 10156 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e.  CC )
3433adantl 453 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  N  e.  CC )
35 npcan 9239 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
3634, 2, 35sylancl 644 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( N  - 
1 )  +  1 )  =  N )
3736oveq2d 6029 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( 0 ... (
( N  -  1 )  +  1 ) )  =  ( 0 ... N ) )
3832, 37syl5sseq 3332 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( 0 ... ( N  -  1 ) )  C_  ( 0 ... N ) )
3938sselda 3284 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
0 ... ( N  - 
1 ) ) )  ->  k  e.  ( 0 ... N ) )
4039, 16syldan 457 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
0 ... ( N  - 
1 ) ) )  ->  ( ( N  _C  k )  x.  ( A ^ k
) )  e.  CC )
4131, 40fsumcl 12447 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( A ^ k ) )  e.  CC )
4241, 24pncand 9337 . 2  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  ( A ^ k
) )  +  ( A ^ N ) )  -  ( A ^ N ) )  =  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  ( A ^ k ) ) )
4330, 42eqtrd 2412 1  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( ( A  +  1 ) ^ N )  -  ( A ^ N ) )  =  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  ( A ^ k ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   ` cfv 5387  (class class class)co 6013   CCcc 8914   0cc0 8916   1c1 8917    + caddc 8919    x. cmul 8921    - cmin 9216   NNcn 9925   NN0cn0 10146   ZZ>=cuz 10413   ...cfz 10968   ^cexp 11302    _C cbc 11513   sum_csu 12399
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-fz 10969  df-fzo 11059  df-seq 11244  df-exp 11303  df-fac 11487  df-bc 11514  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-sum 12400
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