Theorem bcp1n 10507

Description: The proportion of one binomial coefficient to another with N increased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)

Assertion

Ref	Expression
bcp1n	|- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C K ) = ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )

Proof of Theorem bcp1n

Step	Hyp	Ref	Expression
1		elfz3nn0 9895	. . . . 5 |- ( K e. ( 0 ... N ) -> N e. NN0 )
2		facp1 10476	. . . . 5 |- ( N e. NN0 -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )
3	1, 2	syl 14	. . . 4 |- ( K e. ( 0 ... N ) -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )
4		fznn0sub 9837	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( N - K ) e. NN0 )
5		facp1 10476	. . . . . . . 8 |- ( ( N - K ) e. NN0 -> ( ! ` ( ( N - K ) + 1 ) ) = ( ( ! ` ( N - K ) ) x. ( ( N - K ) + 1 ) ) )
6	4, 5	syl 14	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ! ` ( ( N - K ) + 1 ) ) = ( ( ! ` ( N - K ) ) x. ( ( N - K ) + 1 ) ) )
7	1	nn0cnd 9032	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> N e. CC )
8		1cnd 7782	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> 1 e. CC )
9		elfznn0 9894	. . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> K e. NN0 )
10	9	nn0cnd 9032	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K e. CC )
11	7, 8, 10	addsubd 8094	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) = ( ( N - K ) + 1 ) )
12	11	fveq2d 5425	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ! ` ( ( N + 1 ) - K ) ) = ( ! ` ( ( N - K ) + 1 ) ) )
13	11	oveq2d 5790	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) = ( ( ! ` ( N - K ) ) x. ( ( N - K ) + 1 ) ) )
14	6, 12, 13	3eqtr4d 2182	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ! ` ( ( N + 1 ) - K ) ) = ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) )
15	14	oveq1d 5789	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) = ( ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) )
16	4	faccld 10482	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. NN )
17	16	nncnd 8734	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. CC )
18		nn0p1nn 9016	. . . . . . . . 9 |- ( ( N - K ) e. NN0 -> ( ( N - K ) + 1 ) e. NN )
19	4, 18	syl 14	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ( N - K ) + 1 ) e. NN )
20	11, 19	eqeltrd 2216	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. NN )
21	20	nncnd 8734	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. CC )
22	9	faccld 10482	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. NN )
23	22	nncnd 8734	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. CC )
24	17, 21, 23	mul32d 7915	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) = ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) x. ( ( N + 1 ) - K ) ) )
25	15, 24	eqtrd 2172	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) = ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) x. ( ( N + 1 ) - K ) ) )
26	3, 25	oveq12d 5792	. . 3 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N + 1 ) ) / ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) ) = ( ( ( ! ` N ) x. ( N + 1 ) ) / ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) x. ( ( N + 1 ) - K ) ) ) )
27	1	faccld 10482	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ! ` N ) e. NN )
28	27	nncnd 8734	. . . 4 |- ( K e. ( 0 ... N ) -> ( ! ` N ) e. CC )
29	16, 22	nnmulcld 8769	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN )
30	29	nncnd 8734	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. CC )
31		nn0p1nn 9016	. . . . . 6 |- ( N e. NN0 -> ( N + 1 ) e. NN )
32	1, 31	syl 14	. . . . 5 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. NN )
33	32	nncnd 8734	. . . 4 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. CC )
34	29	nnap0d 8766	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) # 0 )
35	20	nnap0d 8766	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) # 0 )
36	28, 30, 33, 21, 34, 35	divmuldivapd 8592	. . 3 |- ( K e. ( 0 ... N ) -> ( ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) = ( ( ( ! ` N ) x. ( N + 1 ) ) / ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) x. ( ( N + 1 ) - K ) ) ) )
37	26, 36	eqtr4d 2175	. 2 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N + 1 ) ) / ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) ) = ( ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )
38		fzelp1 9854	. . 3 |- ( K e. ( 0 ... N ) -> K e. ( 0 ... ( N + 1 ) ) )
39		bcval2 10496	. . 3 |- ( K e. ( 0 ... ( N + 1 ) ) -> ( ( N + 1 ) _C K ) = ( ( ! ` ( N + 1 ) ) / ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) ) )
40	38, 39	syl 14	. 2 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C K ) = ( ( ! ` ( N + 1 ) ) / ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) ) )
41		bcval2 10496	. . 3 |- ( K e. ( 0 ... N ) -> ( N _C K ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
42	41	oveq1d 5789	. 2 |- ( K e. ( 0 ... N ) -> ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) = ( ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )
43	37, 40, 42	3eqtr4d 2182	1 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C K ) = ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   ` cfv 5123  (class class class)co 5774   0cc0 7620   1c1 7621   + caddc 7623   x. cmul 7625   - cmin 7933   / cdiv 8432   NNcn 8720   NN0cn0 8977   ...cfz 9790   !cfa 10471   _C cbc 10493

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 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738

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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-fz 9791  df-seqfrec 10219  df-fac 10472  df-bc 10494

This theorem is referenced by:  bcp1nk  10508  bcpasc  10512