Theorem binom1dif 12198

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 9606	. . . 4 |- ( ( A e. CC /\ N e. NN0 ) -> 0 e. ZZ )
2		simpr 110	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> N e. NN0 )
3	2	nn0zd 9716	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> N e. ZZ )
4		peano2zm 9632	. . . . 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 10817	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> ( 0 ... ( N - 1 ) ) e. Fin )
7		fzssp1 10422	. . . . . 6 |- ( 0 ... ( N - 1 ) ) C_ ( 0 ... ( ( N - 1 ) + 1 ) )
8		nn0cn 9523	. . . . . . . . 9 |- ( N e. NN0 -> N e. CC )
9	8	adantl 277	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN0 ) -> N e. CC )
10		ax-1cn 8236	. . . . . . . 8 |- 1 e. CC
11		npcan 8498	. . . . . . . 8 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
12	9, 10, 11	sylancl 413	. . . . . . 7 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( N - 1 ) + 1 ) = N )
13	12	oveq2d 6074	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( 0 ... ( ( N - 1 ) + 1 ) ) = ( 0 ... N ) )
14	7, 13	sseqtrid 3292	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> ( 0 ... ( N - 1 ) ) C_ ( 0 ... N ) )
15	14	sselda 3242	. . . 4 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. ( 0 ... N ) )
16		bccl2 11155	. . . . . . 7 |- ( k e. ( 0 ... N ) -> ( N _C k ) e. NN )
17	16	adantl 277	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( N _C k ) e. NN )
18	17	nncnd 9268	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( N _C k ) e. CC )
19		simpl 109	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> A e. CC )
20		elfznn0 10470	. . . . . 6 |- ( k e. ( 0 ... N ) -> k e. NN0 )
21		expcl 10943	. . . . . 6 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
22	19, 20, 21	syl2an 289	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( A ^ k ) e. CC )
23	18, 22	mulcld 8310	. . . 4 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( ( N _C k ) x. ( A ^ k ) ) e. CC )
24	15, 23	syldan 282	. . 3 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( ( N _C k ) x. ( A ^ k ) ) e. CC )
25	6, 24	fsumcl 12111	. 2 |- ( ( A e. CC /\ N e. NN0 ) -> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( A ^ k ) ) e. CC )
26		expcl 10943	. 2 |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
27		addcom 8426	. . . . 5 |- ( ( A e. CC /\ 1 e. CC ) -> ( A + 1 ) = ( 1 + A ) )
28	19, 10, 27	sylancl 413	. . . 4 |- ( ( A e. CC /\ N e. NN0 ) -> ( A + 1 ) = ( 1 + A ) )
29	28	oveq1d 6073	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( A + 1 ) ^ N ) = ( ( 1 + A ) ^ N ) )
30		binom1p 12196	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( 1 + A ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( A ^ k ) ) )
31		nn0uz 9907	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
32	2, 31	eleqtrdi 2327	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> N e. ( ZZ>= ` 0 ) )
33		oveq2 6066	. . . . . 6 |- ( k = N -> ( N _C k ) = ( N _C N ) )
34		oveq2 6066	. . . . . 6 |- ( k = N -> ( A ^ k ) = ( A ^ N ) )
35	33, 34	oveq12d 6076	. . . . 5 |- ( k = N -> ( ( N _C k ) x. ( A ^ k ) ) = ( ( N _C N ) x. ( A ^ N ) ) )
36	32, 23, 35	fsumm1 12127	. . . 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 11144	. . . . . . . 8 |- ( N e. NN0 -> ( N _C N ) = 1 )
38	37	adantl 277	. . . . . . 7 |- ( ( A e. CC /\ N e. NN0 ) -> ( N _C N ) = 1 )
39	38	oveq1d 6073	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( N _C N ) x. ( A ^ N ) ) = ( 1 x. ( A ^ N ) ) )
40	26	mullidd 8308	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( 1 x. ( A ^ N ) ) = ( A ^ N ) )
41	39, 40	eqtrd 2267	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( N _C N ) x. ( A ^ N ) ) = ( A ^ N ) )
42	41	oveq2d 6074	. . . 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 2267	. . 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 2271	. 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 8655	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 104   = wceq 1398   e. wcel 2205   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   - cmin 8460   NNcn 9254   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361   ^cexp 10924   _C cbc 11134   sum_csu 12063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925  df-fac 11113  df-bc 11135  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-sumdc 12064

This theorem is referenced by: (None)