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Theorem binom1dif 13296
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 457 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  A  e.  CC )
2 ax-1cn 9340 . . . . . 6  |-  1  e.  CC
3 addcom 9555 . . . . . 6  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
41, 2, 3sylancl 662 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A  +  1 )  =  ( 1  +  A ) )
54oveq1d 6106 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( A  + 
1 ) ^ N
)  =  ( ( 1  +  A ) ^ N ) )
6 binom1p 13294 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( 1  +  A ) ^ N
)  =  sum_ k  e.  ( 0 ... N
) ( ( N  _C  k )  x.  ( A ^ k
) ) )
7 simpr 461 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  N  e.  NN0 )
8 nn0uz 10895 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
97, 8syl6eleq 2533 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  N  e.  ( ZZ>= ` 
0 ) )
10 bccl2 12099 . . . . . . . . 9  |-  ( k  e.  ( 0 ... N )  ->  ( N  _C  k )  e.  NN )
1110adantl 466 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
0 ... N ) )  ->  ( N  _C  k )  e.  NN )
1211nncnd 10338 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
0 ... N ) )  ->  ( N  _C  k )  e.  CC )
13 elfznn0 11481 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
14 expcl 11883 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
151, 13, 14syl2an 477 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
0 ... N ) )  ->  ( A ^
k )  e.  CC )
1612, 15mulcld 9406 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
0 ... N ) )  ->  ( ( N  _C  k )  x.  ( A ^ k
) )  e.  CC )
17 oveq2 6099 . . . . . . 7  |-  ( k  =  N  ->  ( N  _C  k )  =  ( N  _C  N
) )
18 oveq2 6099 . . . . . . 7  |-  ( k  =  N  ->  ( A ^ k )  =  ( A ^ N
) )
1917, 18oveq12d 6109 . . . . . 6  |-  ( k  =  N  ->  (
( N  _C  k
)  x.  ( A ^ k ) )  =  ( ( N  _C  N )  x.  ( A ^ N
) ) )
209, 16, 19fsumm1 13220 . . . . 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 12088 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  _C  N )  =  1 )
2221adantl 466 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( N  _C  N
)  =  1 )
2322oveq1d 6106 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( N  _C  N )  x.  ( A ^ N ) )  =  ( 1  x.  ( A ^ N
) ) )
24 expcl 11883 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  CC )
2524mulid2d 9404 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( 1  x.  ( A ^ N ) )  =  ( A ^ N ) )
2623, 25eqtrd 2475 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( N  _C  N )  x.  ( A ^ N ) )  =  ( A ^ N ) )
2726oveq2d 6107 . . . . 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 2475 . . . 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 2479 . . 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 6106 . 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 11795 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( 0 ... ( N  -  1 ) )  e.  Fin )
32 fzssp1 11501 . . . . . . 7  |-  ( 0 ... ( N  - 
1 ) )  C_  ( 0 ... (
( N  -  1 )  +  1 ) )
33 nn0cn 10589 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e.  CC )
3433adantl 466 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  N  e.  CC )
35 npcan 9619 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
3634, 2, 35sylancl 662 . . . . . . . 8  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( N  - 
1 )  +  1 )  =  N )
3736oveq2d 6107 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( 0 ... (
( N  -  1 )  +  1 ) )  =  ( 0 ... N ) )
3832, 37syl5sseq 3404 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( 0 ... ( N  -  1 ) )  C_  ( 0 ... N ) )
3938sselda 3356 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
0 ... ( N  - 
1 ) ) )  ->  k  e.  ( 0 ... N ) )
4039, 16syldan 470 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
0 ... ( N  - 
1 ) ) )  ->  ( ( N  _C  k )  x.  ( A ^ k
) )  e.  CC )
4131, 40fsumcl 13210 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( A ^ k ) )  e.  CC )
4241, 24pncand 9720 . 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 2475 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091   CCcc 9280   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287    - cmin 9595   NNcn 10322   NN0cn0 10579   ZZ>=cuz 10861   ...cfz 11437   ^cexp 11865    _C cbc 12078   sum_csu 13163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-sum 13164
This theorem is referenced by: (None)
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