Theorem binom1dif 11288

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 9090	. . . 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 9195	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> N e. ZZ )
4		peano2zm 9116	. . . . 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 10235	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> ( 0 ... ( N - 1 ) ) e. Fin )
7		fzssp1 9878	. . . . . 6 |- ( 0 ... ( N - 1 ) ) C_ ( 0 ... ( ( N - 1 ) + 1 ) )
8		nn0cn 9011	. . . . . . . . 9 |- ( N e. NN0 -> N e. CC )
9	8	adantl 275	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN0 ) -> N e. CC )
10		ax-1cn 7737	. . . . . . . 8 |- 1 e. CC
11		npcan 7995	. . . . . . . 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 5798	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( 0 ... ( ( N - 1 ) + 1 ) ) = ( 0 ... N ) )
14	7, 13	sseqtrid 3152	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> ( 0 ... ( N - 1 ) ) C_ ( 0 ... N ) )
15	14	sselda 3102	. . . 4 |- ( ( ( A e. CC /\ N e. NN0 ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. ( 0 ... N ) )
16		bccl2 10546	. . . . . . 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 8758	. . . . 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 9925	. . . . . 6 |- ( k e. ( 0 ... N ) -> k e. NN0 )
21		expcl 10342	. . . . . 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 7810	. . . 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 11201	. 2 |- ( ( A e. CC /\ N e. NN0 ) -> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( A ^ k ) ) e. CC )
26		expcl 10342	. 2 |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )
27		addcom 7923	. . . . 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 5797	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( A + 1 ) ^ N ) = ( ( 1 + A ) ^ N ) )
30		binom1p 11286	. . 3 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( 1 + A ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( A ^ k ) ) )
31		nn0uz 9384	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
32	2, 31	eleqtrdi 2233	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> N e. ( ZZ>= ` 0 ) )
33		oveq2 5790	. . . . . 6 |- ( k = N -> ( N _C k ) = ( N _C N ) )
34		oveq2 5790	. . . . . 6 |- ( k = N -> ( A ^ k ) = ( A ^ N ) )
35	33, 34	oveq12d 5800	. . . . 5 |- ( k = N -> ( ( N _C k ) x. ( A ^ k ) ) = ( ( N _C N ) x. ( A ^ N ) ) )
36	32, 23, 35	fsumm1 11217	. . . 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 10535	. . . . . . . 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 5797	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( N _C N ) x. ( A ^ N ) ) = ( 1 x. ( A ^ N ) ) )
40	26	mulid2d 7808	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( 1 x. ( A ^ N ) ) = ( A ^ N ) )
41	39, 40	eqtrd 2173	. . . . 5 |- ( ( A e. CC /\ N e. NN0 ) -> ( ( N _C N ) x. ( A ^ N ) ) = ( A ^ N ) )
42	41	oveq2d 5798	. . . 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 2173	. . 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 2177	. 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 8152	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 1332   e. wcel 1481   ` cfv 5131  (class class class)co 5782   CCcc 7642   0cc0 7644   1c1 7645   + caddc 7647   x. cmul 7649   - cmin 7957   NNcn 8744   NN0cn0 9001   ZZcz 9078   ZZ>=cuz 9350   ...cfz 9821   ^cexp 10323   _C cbc 10525   sum_csu 11154

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-frec 6296  df-1o 6321  df-oadd 6325  df-er 6437  df-en 6643  df-dom 6644  df-fin 6645  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-fz 9822  df-fzo 9951  df-seqfrec 10250  df-exp 10324  df-fac 10504  df-bc 10526  df-ihash 10554  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-clim 11080  df-sumdc 11155

This theorem is referenced by: (None)