Theorem binom1dif 12177

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 9591	. . . 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 9701	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> N e. ZZ )
4		peano2zm 9617	. . . . 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 10797	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> ( 0 ... ( N - 1 ) ) e. Fin )
7		fzssp1 10404	. . . . . 6 |- ( 0 ... ( N - 1 ) ) C_ ( 0 ... ( ( N - 1 ) + 1 ) )
8		nn0cn 9508	. . . . . . . . 9 |- ( N e. NN0 -> N e. CC )
9	8	adantl 277	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN0 ) -> N e. CC )
10		ax-1cn 8222	. . . . . . . 8 |- 1 e. CC
11		npcan 8484	. . . . . . . 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 6068	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( 0 ... ( ( N - 1 ) + 1 ) ) = ( 0 ... N ) )
14	7, 13	sseqtrid 3290	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> ( 0 ... ( N - 1 ) ) C_ ( 0 ... N ) )
15	14	sselda 3240	. . . 4 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. ( 0 ... N ) )
16		bccl2 11134	. . . . . . 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 9253	. . . . 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 10452	. . . . . 6 |- ( k e. ( 0 ... N ) -> k e. NN0 )
21		expcl 10923	. . . . . 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 8296	. . . 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 12090	. 2 |- ( ( A e. CC /\ N e. NN0 ) -> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( A ^ k ) ) e. CC )
26		expcl 10923	. 2 |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
27		addcom 8412	. . . . 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 6067	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( A + 1 ) ^ N ) = ( ( 1 + A ) ^ N ) )
30		binom1p 12175	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( 1 + A ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( A ^ k ) ) )
31		nn0uz 9892	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
32	2, 31	eleqtrdi 2327	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> N e. ( ZZ>= ` 0 ) )
33		oveq2 6060	. . . . . 6 |- ( k = N -> ( N _C k ) = ( N _C N ) )
34		oveq2 6060	. . . . . 6 |- ( k = N -> ( A ^ k ) = ( A ^ N ) )
35	33, 34	oveq12d 6070	. . . . 5 |- ( k = N -> ( ( N _C k ) x. ( A ^ k ) ) = ( ( N _C N ) x. ( A ^ N ) ) )
36	32, 23, 35	fsumm1 12106	. . . 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 11123	. . . . . . . 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 6067	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( N _C N ) x. ( A ^ N ) ) = ( 1 x. ( A ^ N ) ) )
40	26	mullidd 8294	. . . . . 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 6068	. . . 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 8641	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 5354  (class class class)co 6052   CCcc 8127   0cc0 8129   1c1 8130   + caddc 8132   x. cmul 8134   - cmin 8446   NNcn 9239   NN0cn0 9498   ZZcz 9579   ZZ>=cuz 9856   ...cfz 10345   ^cexp 10904   _C cbc 11113   sum_csu 12042

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 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247  ax-arch 8248  ax-caucvg 8249

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 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-oadd 6653  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-n0 9499  df-z 9580  df-uz 9857  df-q 9955  df-rp 9990  df-fz 10346  df-fzo 10481  df-seqfrec 10814  df-exp 10905  df-fac 11092  df-bc 11114  df-ihash 11143  df-cj 11531  df-re 11532  df-im 11533  df-rsqrt 11687  df-abs 11688  df-clim 11968  df-sumdc 12043

This theorem is referenced by: (None)