Theorem bcp1nk 10873

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 10118	. . . . . 6 |- ( K e. ( 0 ... N ) -> 0 e. ZZ )
2		elfzel2 10117	. . . . . 6 |- ( K e. ( 0 ... N ) -> N e. ZZ )
3		elfzelz 10119	. . . . . 6 |- ( K e. ( 0 ... N ) -> K e. ZZ )
4		1zzd 9372	. . . . . 6 |- ( K e. ( 0 ... N ) -> 1 e. ZZ )
5		fzaddel 10153	. . . . . 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 1250	. . . . 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 9517	. . . . 5 |- 1 = ( 0 + 1 )
9	8	oveq1i 5935	. . . 4 |- ( 1 ... ( N + 1 ) ) = ( ( 0 + 1 ) ... ( N + 1 ) )
10	7, 9	eleqtrrdi 2290	. . 3 |- ( K e. ( 0 ... N ) -> ( K + 1 ) e. ( 1 ... ( N + 1 ) ) )
11		bcm1k 10871	. . 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 9468	. . . . . . 7 |- ( K e. ( 0 ... N ) -> K e. CC )
14		ax-1cn 7991	. . . . . . 7 |- 1 e. CC
15		pncan 8251	. . . . . . 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 5941	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C ( ( K + 1 ) - 1 ) ) = ( ( N + 1 ) _C K ) )
18		bcp1n 10872	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C K ) = ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )
19	17, 18	eqtrd 2229	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C ( ( K + 1 ) - 1 ) ) = ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )
20	16	oveq2d 5941	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - ( ( K + 1 ) - 1 ) ) = ( ( N + 1 ) - K ) )
21	20	oveq1d 5940	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) - ( ( K + 1 ) - 1 ) ) / ( K + 1 ) ) = ( ( ( N + 1 ) - K ) / ( K + 1 ) ) )
22	19, 21	oveq12d 5943	. . 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 10864	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( N _C K ) e. RR+ )
24	23	rpcnd 9792	. . . . 5 |- ( K e. ( 0 ... N ) -> ( N _C K ) e. CC )
25	2	peano2zd 9470	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. ZZ )
26	25	zred 9467	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. RR )
27	3	zred 9467	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K e. RR )
28	2	zred 9467	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> N e. RR )
29		elfzle2 10122	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K <_ N )
30	28	ltp1d 8976	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> N < ( N + 1 ) )
31	27, 28, 26, 29, 30	lelttrd 8170	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> K < ( N + 1 ) )
32		znnsub 9396	. . . . . . . . 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 9059	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) / ( ( N + 1 ) - K ) ) e. RR )
36	35	recnd 8074	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) / ( ( N + 1 ) - K ) ) e. CC )
37	34	nnred 9022	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. RR )
38		elfznn0 10208	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> K e. NN0 )
39		nn0p1nn 9307	. . . . . . . 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 9059	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) - K ) / ( K + 1 ) ) e. RR )
42	41	recnd 8074	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) - K ) / ( K + 1 ) ) e. CC )
43	24, 36, 42	mulassd 8069	. . . 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 9468	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. CC )
45	34	nncnd 9023	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. CC )
46	40	nncnd 9023	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( K + 1 ) e. CC )
47	34	nnap0d 9055	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) # 0 )
48	40	nnap0d 9055	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( K + 1 ) # 0 )
49	44, 45, 46, 47, 48	dmdcanap2d 8867	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) / ( ( N + 1 ) - K ) ) x. ( ( ( N + 1 ) - K ) / ( K + 1 ) ) ) = ( ( N + 1 ) / ( K + 1 ) ) )
50	49	oveq2d 5941	. . . 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 2229	. . 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 2229	. 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 2229	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 1364   e. wcel 2167   class class class wbr 4034  (class class class)co 5925   CCcc 7896   0cc0 7898   1c1 7899   + caddc 7901   x. cmul 7903   < clt 8080   - cmin 8216   / cdiv 8718   NNcn 9009   NN0cn0 9268   ZZcz 9345   ...cfz 10102   _C cbc 10858

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-mulrcl 7997  ax-addcom 7998  ax-mulcom 7999  ax-addass 8000  ax-mulass 8001  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-1rid 8005  ax-0id 8006  ax-rnegex 8007  ax-precex 8008  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-apti 8013  ax-pre-ltadd 8014  ax-pre-mulgt0 8015  ax-pre-mulext 8016

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-reap 8621  df-ap 8628  df-div 8719  df-inn 9010  df-n0 9269  df-z 9346  df-uz 9621  df-q 9713  df-rp 9748  df-fz 10103  df-seqfrec 10559  df-fac 10837  df-bc 10859

This theorem is referenced by: (None)