Theorem bcp1nk 11181

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 10409	. . . . . 6 |- ( K e. ( 0 ... N ) -> 0 e. ZZ )
2		elfzel2 10408	. . . . . 6 |- ( K e. ( 0 ... N ) -> N e. ZZ )
3		elfzelz 10410	. . . . . 6 |- ( K e. ( 0 ... N ) -> K e. ZZ )
4		1zzd 9653	. . . . . 6 |- ( K e. ( 0 ... N ) -> 1 e. ZZ )
5		fzaddel 10446	. . . . . 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 1279	. . . . 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 9800	. . . . 5 |- 1 = ( 0 + 1 )
9	8	oveq1i 6088	. . . 4 |- ( 1 ... ( N + 1 ) ) = ( ( 0 + 1 ) ... ( N + 1 ) )
10	7, 9	eleqtrrdi 2332	. . 3 |- ( K e. ( 0 ... N ) -> ( K + 1 ) e. ( 1 ... ( N + 1 ) ) )
11		bcm1k 11179	. . 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 9751	. . . . . . 7 |- ( K e. ( 0 ... N ) -> K e. CC )
14		ax-1cn 8265	. . . . . . 7 |- 1 e. CC
15		pncan 8525	. . . . . . 7 |- ( ( K e. CC /\ 1 e. CC ) -> ( ( K + 1 ) - 1 ) = K )
16	13, 14, 15	sylancl 417	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( K + 1 ) - 1 ) = K )
17	16	oveq2d 6094	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C ( ( K + 1 ) - 1 ) ) = ( ( N + 1 ) _C K ) )
18		bcp1n 11180	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C K ) = ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )
19	17, 18	eqtrd 2271	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C ( ( K + 1 ) - 1 ) ) = ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )
20	16	oveq2d 6094	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - ( ( K + 1 ) - 1 ) ) = ( ( N + 1 ) - K ) )
21	20	oveq1d 6093	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) - ( ( K + 1 ) - 1 ) ) / ( K + 1 ) ) = ( ( ( N + 1 ) - K ) / ( K + 1 ) ) )
22	19, 21	oveq12d 6096	. . 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 11172	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( N _C K ) e. RR+ )
24	23	rpcnd 10081	. . . . 5 |- ( K e. ( 0 ... N ) -> ( N _C K ) e. CC )
25	2	peano2zd 9753	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. ZZ )
26	25	zred 9750	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. RR )
27	3	zred 9750	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K e. RR )
28	2	zred 9750	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> N e. RR )
29		elfzle2 10414	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K <_ N )
30	28	ltp1d 9253	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> N < ( N + 1 ) )
31	27, 28, 26, 29, 30	lelttrd 8444	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> K < ( N + 1 ) )
32		znnsub 9678	. . . . . . . . 9 |- ( ( K e. ZZ /\ ( N + 1 ) e. ZZ ) -> ( K < ( N + 1 ) <-> ( ( N + 1 ) - K ) e. NN ) )
33	3, 25, 32	syl2anc 415	. . . . . . . 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 9336	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) / ( ( N + 1 ) - K ) ) e. RR )
36	35	recnd 8347	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) / ( ( N + 1 ) - K ) ) e. CC )
37	34	nnred 9299	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. RR )
38		elfznn0 10502	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> K e. NN0 )
39		nn0p1nn 9584	. . . . . . . 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 9336	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) - K ) / ( K + 1 ) ) e. RR )
42	41	recnd 8347	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) - K ) / ( K + 1 ) ) e. CC )
43	24, 36, 42	mulassd 8342	. . . 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 9751	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. CC )
45	34	nncnd 9300	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. CC )
46	40	nncnd 9300	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( K + 1 ) e. CC )
47	34	nnap0d 9332	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) # 0 )
48	40	nnap0d 9332	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( K + 1 ) # 0 )
49	44, 45, 46, 47, 48	dmdcanap2d 9144	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) / ( ( N + 1 ) - K ) ) x. ( ( ( N + 1 ) - K ) / ( K + 1 ) ) ) = ( ( N + 1 ) / ( K + 1 ) ) )
50	49	oveq2d 6094	. . . 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 2271	. . 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 2271	. 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 2271	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 1402   e. wcel 2209   class class class wbr 4128  (class class class)co 6078   CCcc 8170   0cc0 8172   1c1 8173   + caddc 8175   x. cmul 8177   < clt 8353   - cmin 8490   / cdiv 8995   NNcn 9286   NN0cn0 9545   ZZcz 9626   ...cfz 10393   _C cbc 11166

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-mulrcl 8271  ax-addcom 8272  ax-mulcom 8273  ax-addass 8274  ax-mulass 8275  ax-distr 8276  ax-i2m1 8277  ax-0lt1 8278  ax-1rid 8279  ax-0id 8280  ax-rnegex 8281  ax-precex 8282  ax-cnre 8283  ax-pre-ltirr 8284  ax-pre-ltwlin 8285  ax-pre-lttrn 8286  ax-pre-apti 8287  ax-pre-ltadd 8288  ax-pre-mulgt0 8289  ax-pre-mulext 8290

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-recs 6569  df-frec 6655  df-pnf 8355  df-mnf 8356  df-xr 8357  df-ltxr 8358  df-le 8359  df-sub 8492  df-neg 8493  df-reap 8896  df-ap 8903  df-div 8996  df-inn 9287  df-n0 9546  df-z 9627  df-uz 9904  df-q 10002  df-rp 10037  df-fz 10394  df-seqfrec 10866  df-fac 11145  df-bc 11167

This theorem is referenced by: (None)