Theorem bcp1nk 10501

Description: The proportion of one binomial coefficient to another with N and K increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.)

Assertion

Ref	Expression
bcp1nk	|- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C ( K + 1 ) ) = ( ( N _C K ) x. ( ( N + 1 ) / ( K + 1 ) ) ) )

Proof of Theorem bcp1nk

Step	Hyp	Ref	Expression
1		elfzel1 9798	. . . . . 6 |- ( K e. ( 0 ... N ) -> 0 e. ZZ )
2		elfzel2 9797	. . . . . 6 |- ( K e. ( 0 ... N ) -> N e. ZZ )
3		elfzelz 9799	. . . . . 6 |- ( K e. ( 0 ... N ) -> K e. ZZ )
4		1zzd 9074	. . . . . 6 |- ( K e. ( 0 ... N ) -> 1 e. ZZ )
5		fzaddel 9832	. . . . . 6 |- ( ( ( 0 e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ 1 e. ZZ ) ) -> ( K e. ( 0 ... N ) <-> ( K + 1 ) e. ( ( 0 + 1 ) ... ( N + 1 ) ) ) )
6	1, 2, 3, 4, 5	syl22anc 1217	. . . . 5 |- ( K e. ( 0 ... N ) -> ( K e. ( 0 ... N ) <-> ( K + 1 ) e. ( ( 0 + 1 ) ... ( N + 1 ) ) ) )
7	6	ibi 175	. . . 4 |- ( K e. ( 0 ... N ) -> ( K + 1 ) e. ( ( 0 + 1 ) ... ( N + 1 ) ) )
8		1e0p1 9216	. . . . 5 |- 1 = ( 0 + 1 )
9	8	oveq1i 5777	. . . 4 |- ( 1 ... ( N + 1 ) ) = ( ( 0 + 1 ) ... ( N + 1 ) )
10	7, 9	eleqtrrdi 2231	. . 3 |- ( K e. ( 0 ... N ) -> ( K + 1 ) e. ( 1 ... ( N + 1 ) ) )
11		bcm1k 10499	. . 3 |- ( ( K + 1 ) e. ( 1 ... ( N + 1 ) ) -> ( ( N + 1 ) _C ( K + 1 ) ) = ( ( ( N + 1 ) _C ( ( K + 1 ) - 1 ) ) x. ( ( ( N + 1 ) - ( ( K + 1 ) - 1 ) ) / ( K + 1 ) ) ) )
12	10, 11	syl 14	. 2 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C ( K + 1 ) ) = ( ( ( N + 1 ) _C ( ( K + 1 ) - 1 ) ) x. ( ( ( N + 1 ) - ( ( K + 1 ) - 1 ) ) / ( K + 1 ) ) ) )
13	3	zcnd 9167	. . . . . . 7 |- ( K e. ( 0 ... N ) -> K e. CC )
14		ax-1cn 7706	. . . . . . 7 |- 1 e. CC
15		pncan 7961	. . . . . . 7 |- ( ( K e. CC /\ 1 e. CC ) -> ( ( K + 1 ) - 1 ) = K )
16	13, 14, 15	sylancl 409	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( K + 1 ) - 1 ) = K )
17	16	oveq2d 5783	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C ( ( K + 1 ) - 1 ) ) = ( ( N + 1 ) _C K ) )
18		bcp1n 10500	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C K ) = ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )
19	17, 18	eqtrd 2170	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C ( ( K + 1 ) - 1 ) ) = ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )
20	16	oveq2d 5783	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - ( ( K + 1 ) - 1 ) ) = ( ( N + 1 ) - K ) )
21	20	oveq1d 5782	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) - ( ( K + 1 ) - 1 ) ) / ( K + 1 ) ) = ( ( ( N + 1 ) - K ) / ( K + 1 ) ) )
22	19, 21	oveq12d 5785	. . 3 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) _C ( ( K + 1 ) - 1 ) ) x. ( ( ( N + 1 ) - ( ( K + 1 ) - 1 ) ) / ( K + 1 ) ) ) = ( ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) x. ( ( ( N + 1 ) - K ) / ( K + 1 ) ) ) )
23		bcrpcl 10492	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( N _C K ) e. RR+ )
24	23	rpcnd 9478	. . . . 5 |- ( K e. ( 0 ... N ) -> ( N _C K ) e. CC )
25	2	peano2zd 9169	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. ZZ )
26	25	zred 9166	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. RR )
27	3	zred 9166	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K e. RR )
28	2	zred 9166	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> N e. RR )
29		elfzle2 9801	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K <_ N )
30	28	ltp1d 8681	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> N < ( N + 1 ) )
31	27, 28, 26, 29, 30	lelttrd 7880	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> K < ( N + 1 ) )
32		znnsub 9098	. . . . . . . . 9 |- ( ( K e. ZZ /\ ( N + 1 ) e. ZZ ) -> ( K < ( N + 1 ) <-> ( ( N + 1 ) - K ) e. NN ) )
33	3, 25, 32	syl2anc 408	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( K < ( N + 1 ) <-> ( ( N + 1 ) - K ) e. NN ) )
34	31, 33	mpbid 146	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. NN )
35	26, 34	nndivred 8763	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) / ( ( N + 1 ) - K ) ) e. RR )
36	35	recnd 7787	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) / ( ( N + 1 ) - K ) ) e. CC )
37	34	nnred 8726	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. RR )
38		elfznn0 9887	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> K e. NN0 )
39		nn0p1nn 9009	. . . . . . . 8 |- ( K e. NN0 -> ( K + 1 ) e. NN )
40	38, 39	syl 14	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( K + 1 ) e. NN )
41	37, 40	nndivred 8763	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) - K ) / ( K + 1 ) ) e. RR )
42	41	recnd 7787	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) - K ) / ( K + 1 ) ) e. CC )
43	24, 36, 42	mulassd 7782	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) x. ( ( ( N + 1 ) - K ) / ( K + 1 ) ) ) = ( ( N _C K ) x. ( ( ( N + 1 ) / ( ( N + 1 ) - K ) ) x. ( ( ( N + 1 ) - K ) / ( K + 1 ) ) ) ) )
44	25	zcnd 9167	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. CC )
45	34	nncnd 8727	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. CC )
46	40	nncnd 8727	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( K + 1 ) e. CC )
47	34	nnap0d 8759	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) # 0 )
48	40	nnap0d 8759	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( K + 1 ) # 0 )
49	44, 45, 46, 47, 48	dmdcanap2d 8574	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) / ( ( N + 1 ) - K ) ) x. ( ( ( N + 1 ) - K ) / ( K + 1 ) ) ) = ( ( N + 1 ) / ( K + 1 ) ) )
50	49	oveq2d 5783	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( N _C K ) x. ( ( ( N + 1 ) / ( ( N + 1 ) - K ) ) x. ( ( ( N + 1 ) - K ) / ( K + 1 ) ) ) ) = ( ( N _C K ) x. ( ( N + 1 ) / ( K + 1 ) ) ) )
51	43, 50	eqtrd 2170	. . 3 |- ( K e. ( 0 ... N ) -> ( ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) x. ( ( ( N + 1 ) - K ) / ( K + 1 ) ) ) = ( ( N _C K ) x. ( ( N + 1 ) / ( K + 1 ) ) ) )
52	22, 51	eqtrd 2170	. 2 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) _C ( ( K + 1 ) - 1 ) ) x. ( ( ( N + 1 ) - ( ( K + 1 ) - 1 ) ) / ( K + 1 ) ) ) = ( ( N _C K ) x. ( ( N + 1 ) / ( K + 1 ) ) ) )
53	12, 52	eqtrd 2170	1 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C ( K + 1 ) ) = ( ( N _C K ) x. ( ( N + 1 ) / ( K + 1 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   class class class wbr 3924  (class class class)co 5767   CCcc 7611   0cc0 7613   1c1 7614   + caddc 7616   x. cmul 7618   < clt 7793   - cmin 7926   / cdiv 8425   NNcn 8713   NN0cn0 8970   ZZcz 9047   ...cfz 9783   _C cbc 10486

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-fz 9784  df-seqfrec 10212  df-fac 10465  df-bc 10487

This theorem is referenced by: (None)