Theorem binom1dif 11428

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 9203	. . . 4 |- ( ( A e. CC /\ N e. NN0 ) -> 0 e. ZZ )
2		simpr 109	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> N e. NN0 )
3	2	nn0zd 9311	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> N e. ZZ )
4		peano2zm 9229	. . . . 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 10366	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> ( 0 ... ( N - 1 ) ) e. Fin )
7		fzssp1 10002	. . . . . 6 |- ( 0 ... ( N - 1 ) ) C_ ( 0 ... ( ( N - 1 ) + 1 ) )
8		nn0cn 9124	. . . . . . . . 9 |- ( N e. NN0 -> N e. CC )
9	8	adantl 275	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN0 ) -> N e. CC )
10		ax-1cn 7846	. . . . . . . 8 |- 1 e. CC
11		npcan 8107	. . . . . . . 8 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
12	9, 10, 11	sylancl 410	. . . . . . 7 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( N - 1 ) + 1 ) = N )
13	12	oveq2d 5858	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( 0 ... ( ( N - 1 ) + 1 ) ) = ( 0 ... N ) )
14	7, 13	sseqtrid 3192	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> ( 0 ... ( N - 1 ) ) C_ ( 0 ... N ) )
15	14	sselda 3142	. . . 4 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. ( 0 ... N ) )
16		bccl2 10681	. . . . . . 7 |- ( k e. ( 0 ... N ) -> ( N _C k ) e. NN )
17	16	adantl 275	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( N _C k ) e. NN )
18	17	nncnd 8871	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( N _C k ) e. CC )
19		simpl 108	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> A e. CC )
20		elfznn0 10049	. . . . . 6 |- ( k e. ( 0 ... N ) -> k e. NN0 )
21		expcl 10473	. . . . . 6 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
22	19, 20, 21	syl2an 287	. . . . 5 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( A ^ k ) e. CC )
23	18, 22	mulcld 7919	. . . 4 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( ( N _C k ) x. ( A ^ k ) ) e. CC )
24	15, 23	syldan 280	. . 3 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( ( N _C k ) x. ( A ^ k ) ) e. CC )
25	6, 24	fsumcl 11341	. 2 |- ( ( A e. CC /\ N e. NN0 ) -> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( A ^ k ) ) e. CC )
26		expcl 10473	. 2 |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
27		addcom 8035	. . . . 5 |- ( ( A e. CC /\ 1 e. CC ) -> ( A + 1 ) = ( 1 + A ) )
28	19, 10, 27	sylancl 410	. . . 4 |- ( ( A e. CC /\ N e. NN0 ) -> ( A + 1 ) = ( 1 + A ) )
29	28	oveq1d 5857	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( A + 1 ) ^ N ) = ( ( 1 + A ) ^ N ) )
30		binom1p 11426	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( 1 + A ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( A ^ k ) ) )
31		nn0uz 9500	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
32	2, 31	eleqtrdi 2259	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> N e. ( ZZ>= ` 0 ) )
33		oveq2 5850	. . . . . 6 |- ( k = N -> ( N _C k ) = ( N _C N ) )
34		oveq2 5850	. . . . . 6 |- ( k = N -> ( A ^ k ) = ( A ^ N ) )
35	33, 34	oveq12d 5860	. . . . 5 |- ( k = N -> ( ( N _C k ) x. ( A ^ k ) ) = ( ( N _C N ) x. ( A ^ N ) ) )
36	32, 23, 35	fsumm1 11357	. . . 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 10670	. . . . . . . 8 |- ( N e. NN0 -> ( N _C N ) = 1 )
38	37	adantl 275	. . . . . . 7 |- ( ( A e. CC /\ N e. NN0 ) -> ( N _C N ) = 1 )
39	38	oveq1d 5857	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( N _C N ) x. ( A ^ N ) ) = ( 1 x. ( A ^ N ) ) )
40	26	mulid2d 7917	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( 1 x. ( A ^ N ) ) = ( A ^ N ) )
41	39, 40	eqtrd 2198	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( N _C N ) x. ( A ^ N ) ) = ( A ^ N ) )
42	41	oveq2d 5858	. . . 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 2198	. . 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 2202	. 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 8264	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 103   = wceq 1343   e. wcel 2136   ` cfv 5188  (class class class)co 5842   CCcc 7751   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   - cmin 8069   NNcn 8857   NN0cn0 9114   ZZcz 9191   ZZ>=cuz 9466   ...cfz 9944   ^cexp 10454   _C cbc 10660   sum_csu 11294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-fac 10639  df-bc 10661  df-ihash 10689  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295

This theorem is referenced by: (None)