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Theorem binom1dif 12604
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 9040 . . . . . 6  |-  1  e.  CC
3 addcom 9244 . . . . . 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 6088 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( A  + 
1 ) ^ N
)  =  ( ( 1  +  A ) ^ N ) )
6 binom1p 12602 . . . 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 10512 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
97, 8syl6eleq 2525 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  N  e.  ( ZZ>= ` 
0 ) )
10 bccl2 11606 . . . . . . . . 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 10008 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
0 ... N ) )  ->  ( N  _C  k )  e.  CC )
13 elfznn0 11075 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
14 expcl 11391 . . . . . . . 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 9100 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
0 ... N ) )  ->  ( ( N  _C  k )  x.  ( A ^ k
) )  e.  CC )
17 oveq2 6081 . . . . . . 7  |-  ( k  =  N  ->  ( N  _C  k )  =  ( N  _C  N
) )
18 oveq2 6081 . . . . . . 7  |-  ( k  =  N  ->  ( A ^ k )  =  ( A ^ N
) )
1917, 18oveq12d 6091 . . . . . 6  |-  ( k  =  N  ->  (
( N  _C  k
)  x.  ( A ^ k ) )  =  ( ( N  _C  N )  x.  ( A ^ N
) ) )
209, 16, 19fsumm1 12529 . . . . 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 11595 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  _C  N )  =  1 )
2221adantl 453 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( N  _C  N
)  =  1 )
2322oveq1d 6088 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( N  _C  N )  x.  ( A ^ N ) )  =  ( 1  x.  ( A ^ N
) ) )
24 expcl 11391 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  CC )
2524mulid2d 9098 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( 1  x.  ( A ^ N ) )  =  ( A ^ N ) )
2623, 25eqtrd 2467 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( N  _C  N )  x.  ( A ^ N ) )  =  ( A ^ N ) )
2726oveq2d 6089 . . . . 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 2467 . . . 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 2471 . . 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 6088 . 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 11304 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( 0 ... ( N  -  1 ) )  e.  Fin )
32 fzssp1 11087 . . . . . . 7  |-  ( 0 ... ( N  - 
1 ) )  C_  ( 0 ... (
( N  -  1 )  +  1 ) )
33 nn0cn 10223 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e.  CC )
3433adantl 453 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  N  e.  CC )
35 npcan 9306 . . . . . . . . 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 6089 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( 0 ... (
( N  -  1 )  +  1 ) )  =  ( 0 ... N ) )
3832, 37syl5sseq 3388 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( 0 ... ( N  -  1 ) )  C_  ( 0 ... N ) )
3938sselda 3340 . . . . 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 12519 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( A ^ k ) )  e.  CC )
4241, 24pncand 9404 . 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 2467 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 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    - cmin 9283   NNcn 9992   NN0cn0 10213   ZZ>=cuz 10480   ...cfz 11035   ^cexp 11374    _C cbc 11585   sum_csu 12471
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472
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