Theorem bcp1nk 10979

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 10216	. . . . . 6 |- ( K e. ( 0 ... N ) -> 0 e. ZZ )
2		elfzel2 10215	. . . . . 6 |- ( K e. ( 0 ... N ) -> N e. ZZ )
3		elfzelz 10217	. . . . . 6 |- ( K e. ( 0 ... N ) -> K e. ZZ )
4		1zzd 9469	. . . . . 6 |- ( K e. ( 0 ... N ) -> 1 e. ZZ )
5		fzaddel 10251	. . . . . 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 1272	. . . . 5 |- ( K e. ( 0 ... N ) -> ( K e. ( 0 ... N ) <-> ( K + 1 ) e. ( ( 0 + 1 ) ... ( N + 1 ) ) ) )
7	6	ibi 176	. . . 4 |- ( K e. ( 0 ... N ) -> ( K + 1 ) e. ( ( 0 + 1 ) ... ( N + 1 ) ) )
8		1e0p1 9615	. . . . 5 |- 1 = ( 0 + 1 )
9	8	oveq1i 6010	. . . 4 |- ( 1 ... ( N + 1 ) ) = ( ( 0 + 1 ) ... ( N + 1 ) )
10	7, 9	eleqtrrdi 2323	. . 3 |- ( K e. ( 0 ... N ) -> ( K + 1 ) e. ( 1 ... ( N + 1 ) ) )
11		bcm1k 10977	. . 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 9566	. . . . . . 7 |- ( K e. ( 0 ... N ) -> K e. CC )
14		ax-1cn 8088	. . . . . . 7 |- 1 e. CC
15		pncan 8348	. . . . . . 7 |- ( ( K e. CC /\ 1 e. CC ) -> ( ( K + 1 ) - 1 ) = K )
16	13, 14, 15	sylancl 413	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( K + 1 ) - 1 ) = K )
17	16	oveq2d 6016	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C ( ( K + 1 ) - 1 ) ) = ( ( N + 1 ) _C K ) )
18		bcp1n 10978	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C K ) = ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )
19	17, 18	eqtrd 2262	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C ( ( K + 1 ) - 1 ) ) = ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )
20	16	oveq2d 6016	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - ( ( K + 1 ) - 1 ) ) = ( ( N + 1 ) - K ) )
21	20	oveq1d 6015	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) - ( ( K + 1 ) - 1 ) ) / ( K + 1 ) ) = ( ( ( N + 1 ) - K ) / ( K + 1 ) ) )
22	19, 21	oveq12d 6018	. . 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 10970	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( N _C K ) e. RR+ )
24	23	rpcnd 9890	. . . . 5 |- ( K e. ( 0 ... N ) -> ( N _C K ) e. CC )
25	2	peano2zd 9568	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. ZZ )
26	25	zred 9565	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. RR )
27	3	zred 9565	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K e. RR )
28	2	zred 9565	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> N e. RR )
29		elfzle2 10220	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K <_ N )
30	28	ltp1d 9073	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> N < ( N + 1 ) )
31	27, 28, 26, 29, 30	lelttrd 8267	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> K < ( N + 1 ) )
32		znnsub 9494	. . . . . . . . 9 |- ( ( K e. ZZ /\ ( N + 1 ) e. ZZ ) -> ( K < ( N + 1 ) <-> ( ( N + 1 ) - K ) e. NN ) )
33	3, 25, 32	syl2anc 411	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( K < ( N + 1 ) <-> ( ( N + 1 ) - K ) e. NN ) )
34	31, 33	mpbid 147	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. NN )
35	26, 34	nndivred 9156	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) / ( ( N + 1 ) - K ) ) e. RR )
36	35	recnd 8171	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) / ( ( N + 1 ) - K ) ) e. CC )
37	34	nnred 9119	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. RR )
38		elfznn0 10306	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> K e. NN0 )
39		nn0p1nn 9404	. . . . . . . 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 9156	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) - K ) / ( K + 1 ) ) e. RR )
42	41	recnd 8171	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) - K ) / ( K + 1 ) ) e. CC )
43	24, 36, 42	mulassd 8166	. . . 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 9566	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. CC )
45	34	nncnd 9120	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. CC )
46	40	nncnd 9120	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( K + 1 ) e. CC )
47	34	nnap0d 9152	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) # 0 )
48	40	nnap0d 9152	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( K + 1 ) # 0 )
49	44, 45, 46, 47, 48	dmdcanap2d 8964	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) / ( ( N + 1 ) - K ) ) x. ( ( ( N + 1 ) - K ) / ( K + 1 ) ) ) = ( ( N + 1 ) / ( K + 1 ) ) )
50	49	oveq2d 6016	. . . 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 2262	. . 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 2262	. 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 2262	1 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C ( K + 1 ) ) = ( ( N _C K ) x. ( ( N + 1 ) / ( K + 1 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   class class class wbr 4082  (class class class)co 6000   CCcc 7993   0cc0 7995   1c1 7996   + caddc 7998   x. cmul 8000   < clt 8177   - cmin 8313   / cdiv 8815   NNcn 9106   NN0cn0 9365   ZZcz 9442   ...cfz 10200   _C cbc 10964

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-fz 10201  df-seqfrec 10665  df-fac 10943  df-bc 10965

This theorem is referenced by: (None)