Theorem bcp1nk 11128

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 10361	. . . . . 6 |- ( K e. ( 0 ... N ) -> 0 e. ZZ )
2		elfzel2 10360	. . . . . 6 |- ( K e. ( 0 ... N ) -> N e. ZZ )
3		elfzelz 10362	. . . . . 6 |- ( K e. ( 0 ... N ) -> K e. ZZ )
4		1zzd 9606	. . . . . 6 |- ( K e. ( 0 ... N ) -> 1 e. ZZ )
5		fzaddel 10396	. . . . . 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 1275	. . . . 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 9753	. . . . 5 |- 1 = ( 0 + 1 )
9	8	oveq1i 6062	. . . 4 |- ( 1 ... ( N + 1 ) ) = ( ( 0 + 1 ) ... ( N + 1 ) )
10	7, 9	eleqtrrdi 2328	. . 3 |- ( K e. ( 0 ... N ) -> ( K + 1 ) e. ( 1 ... ( N + 1 ) ) )
11		bcm1k 11126	. . 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 9704	. . . . . . 7 |- ( K e. ( 0 ... N ) -> K e. CC )
14		ax-1cn 8222	. . . . . . 7 |- 1 e. CC
15		pncan 8481	. . . . . . 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 6068	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C ( ( K + 1 ) - 1 ) ) = ( ( N + 1 ) _C K ) )
18		bcp1n 11127	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C K ) = ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )
19	17, 18	eqtrd 2267	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C ( ( K + 1 ) - 1 ) ) = ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )
20	16	oveq2d 6068	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - ( ( K + 1 ) - 1 ) ) = ( ( N + 1 ) - K ) )
21	20	oveq1d 6067	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) - ( ( K + 1 ) - 1 ) ) / ( K + 1 ) ) = ( ( ( N + 1 ) - K ) / ( K + 1 ) ) )
22	19, 21	oveq12d 6070	. . 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 11119	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( N _C K ) e. RR+ )
24	23	rpcnd 10034	. . . . 5 |- ( K e. ( 0 ... N ) -> ( N _C K ) e. CC )
25	2	peano2zd 9706	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. ZZ )
26	25	zred 9703	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. RR )
27	3	zred 9703	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K e. RR )
28	2	zred 9703	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> N e. RR )
29		elfzle2 10365	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K <_ N )
30	28	ltp1d 9206	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> N < ( N + 1 ) )
31	27, 28, 26, 29, 30	lelttrd 8400	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> K < ( N + 1 ) )
32		znnsub 9631	. . . . . . . . 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 9289	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) / ( ( N + 1 ) - K ) ) e. RR )
36	35	recnd 8304	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) / ( ( N + 1 ) - K ) ) e. CC )
37	34	nnred 9252	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. RR )
38		elfznn0 10452	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> K e. NN0 )
39		nn0p1nn 9537	. . . . . . . 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 9289	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) - K ) / ( K + 1 ) ) e. RR )
42	41	recnd 8304	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) - K ) / ( K + 1 ) ) e. CC )
43	24, 36, 42	mulassd 8299	. . . 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 9704	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. CC )
45	34	nncnd 9253	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. CC )
46	40	nncnd 9253	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( K + 1 ) e. CC )
47	34	nnap0d 9285	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) # 0 )
48	40	nnap0d 9285	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( K + 1 ) # 0 )
49	44, 45, 46, 47, 48	dmdcanap2d 9097	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) / ( ( N + 1 ) - K ) ) x. ( ( ( N + 1 ) - K ) / ( K + 1 ) ) ) = ( ( N + 1 ) / ( K + 1 ) ) )
50	49	oveq2d 6068	. . . 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 2267	. . 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 2267	. 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 2267	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 1398   e. wcel 2205   class class class wbr 4111  (class class class)co 6052   CCcc 8127   0cc0 8129   1c1 8130   + caddc 8132   x. cmul 8134   < clt 8310   - cmin 8446   / cdiv 8948   NNcn 9239   NN0cn0 9498   ZZcz 9579   ...cfz 10345   _C cbc 11113

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

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-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  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-n0 9499  df-z 9580  df-uz 9857  df-q 9955  df-rp 9990  df-fz 10346  df-seqfrec 10814  df-fac 11092  df-bc 11114

This theorem is referenced by: (None)