Theorem binom1dif 10881

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

Step	Hyp	Ref	Expression
1		0zd 8762	. . . 4 |- ( ( A e. CC /\ N e. NN0 ) -> 0 e. ZZ )
2		simpr 108	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> N e. NN0 )
3	2	nn0zd 8866	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> N e. ZZ )
4		peano2zm 8788	. . . . 5 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
5	3, 4	syl 14	. . . 4 |- ( ( A e. CC /\ N e. NN0 ) -> ( N - 1 ) e. ZZ )
6	1, 5	fzfigd 9838	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> ( 0 ... ( N - 1 ) ) e. Fin )
7		fzssp1 9481	. . . . . 6 |- ( 0 ... ( N - 1 ) ) C_ ( 0 ... ( ( N - 1 ) + 1 ) )
8		nn0cn 8683	. . . . . . . . 9 |- ( N e. NN0 -> N e. CC )
9	8	adantl 271	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN0 ) -> N e. CC )
10		ax-1cn 7438	. . . . . . . 8 |- 1 e. CC
11		npcan 7691	. . . . . . . 8 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
12	9, 10, 11	sylancl 404	. . . . . . 7 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( N - 1 ) + 1 ) = N )
13	12	oveq2d 5668	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( 0 ... ( ( N - 1 ) + 1 ) ) = ( 0 ... N ) )
14	7, 13	syl5sseq 3074	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> ( 0 ... ( N - 1 ) ) C_ ( 0 ... N ) )
15	14	sselda 3025	. . . 4 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. ( 0 ... N ) )
16		bccl2 10176	. . . . . . 7 |- ( k e. ( 0 ... N ) -> ( N _C k ) e. NN )
17	16	adantl 271	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( N _C k ) e. NN )
18	17	nncnd 8436	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( N _C k ) e. CC )
19		simpl 107	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> A e. CC )
20		elfznn0 9528	. . . . . 6 |- ( k e. ( 0 ... N ) -> k e. NN0 )
21		expcl 9973	. . . . . 6 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
22	19, 20, 21	syl2an 283	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( A ^ k ) e. CC )
23	18, 22	mulcld 7508	. . . 4 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( ( N _C k ) x. ( A ^ k ) ) e. CC )
24	15, 23	syldan 276	. . 3 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( ( N _C k ) x. ( A ^ k ) ) e. CC )
25	6, 24	fsumcl 10794	. 2 |- ( ( A e. CC /\ N e. NN0 ) -> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( A ^ k ) ) e. CC )
26		expcl 9973	. 2 |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
27		addcom 7619	. . . . 5 |- ( ( A e. CC /\ 1 e. CC ) -> ( A + 1 ) = ( 1 + A ) )
28	19, 10, 27	sylancl 404	. . . 4 |- ( ( A e. CC /\ N e. NN0 ) -> ( A + 1 ) = ( 1 + A ) )
29	28	oveq1d 5667	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( A + 1 ) ^ N ) = ( ( 1 + A ) ^ N ) )
30		binom1p 10879	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( 1 + A ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( A ^ k ) ) )
31		nn0uz 9053	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
32	2, 31	syl6eleq 2180	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> N e. ( ZZ>= ` 0 ) )
33		oveq2 5660	. . . . . 6 |- ( k = N -> ( N _C k ) = ( N _C N ) )
34		oveq2 5660	. . . . . 6 |- ( k = N -> ( A ^ k ) = ( A ^ N ) )
35	33, 34	oveq12d 5670	. . . . 5 |- ( k = N -> ( ( N _C k ) x. ( A ^ k ) ) = ( ( N _C N ) x. ( A ^ N ) ) )
36	32, 23, 35	fsumm1 10810	. . . 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 ) ) + ( ( N _C N ) x. ( A ^ N ) ) ) )
37		bcnn 10165	. . . . . . . 8 |- ( N e. NN0 -> ( N _C N ) = 1 )
38	37	adantl 271	. . . . . . 7 |- ( ( A e. CC /\ N e. NN0 ) -> ( N _C N ) = 1 )
39	38	oveq1d 5667	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( N _C N ) x. ( A ^ N ) ) = ( 1 x. ( A ^ N ) ) )
40	26	mulid2d 7506	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( 1 x. ( A ^ N ) ) = ( A ^ N ) )
41	39, 40	eqtrd 2120	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( N _C N ) x. ( A ^ N ) ) = ( A ^ N ) )
42	41	oveq2d 5668	. . . 4 |- ( ( 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 ) ) )
43	36, 42	eqtrd 2120	. . 3 |- ( ( 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 ) ) )
44	29, 30, 43	3eqtrd 2124	. 2 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( A + 1 ) ^ N ) = ( sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( A ^ k ) ) + ( A ^ N ) ) )
45	25, 26, 44	mvrraddd 7846	1 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( ( A + 1 ) ^ N ) - ( A ^ N ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( A ^ k ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1289   e. wcel 1438   ` cfv 5015  (class class class)co 5652   CCcc 7348   0cc0 7350   1c1 7351   + caddc 7353   x. cmul 7355   - cmin 7653   NNcn 8422   NN0cn0 8673   ZZcz 8750   ZZ>=cuz 9019   ...cfz 9424   ^cexp 9954   _C cbc 10155   sum_csu 10742

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 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464  ax-caucvg 7465

This theorem depends on definitions:  df-bi 115  df-dc 781  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-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-frec 6156  df-1o 6181  df-oadd 6185  df-er 6292  df-en 6458  df-dom 6459  df-fin 6460  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-n0 8674  df-z 8751  df-uz 9020  df-q 9105  df-rp 9135  df-fz 9425  df-fzo 9554  df-iseq 9853  df-seq3 9854  df-exp 9955  df-fac 10134  df-bc 10156  df-ihash 10184  df-cj 10276  df-re 10277  df-im 10278  df-rsqrt 10431  df-abs 10432  df-clim 10667  df-isum 10743

This theorem is referenced by: (None)