Theorem bcp1n 10674

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 10050	. . . . 5 |- ( K e. ( 0 ... N ) -> N e. NN0 )
2		facp1 10643	. . . . 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 9992	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( N - K ) e. NN0 )
5		facp1 10643	. . . . . . . 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 9169	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> N e. CC )
8		1cnd 7915	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> 1 e. CC )
9		elfznn0 10049	. . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> K e. NN0 )
10	9	nn0cnd 9169	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K e. CC )
11	7, 8, 10	addsubd 8230	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) = ( ( N - K ) + 1 ) )
12	11	fveq2d 5490	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ! ` ( ( N + 1 ) - K ) ) = ( ! ` ( ( N - K ) + 1 ) ) )
13	11	oveq2d 5858	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) = ( ( ! ` ( N - K ) ) x. ( ( N - K ) + 1 ) ) )
14	6, 12, 13	3eqtr4d 2208	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ! ` ( ( N + 1 ) - K ) ) = ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) )
15	14	oveq1d 5857	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) = ( ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) )
16	4	faccld 10649	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. NN )
17	16	nncnd 8871	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. CC )
18		nn0p1nn 9153	. . . . . . . . 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 2243	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. NN )
21	20	nncnd 8871	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. CC )
22	9	faccld 10649	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. NN )
23	22	nncnd 8871	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. CC )
24	17, 21, 23	mul32d 8051	. . . . 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 2198	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) = ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) x. ( ( N + 1 ) - K ) ) )
26	3, 25	oveq12d 5860	. . 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 10649	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ! ` N ) e. NN )
28	27	nncnd 8871	. . . 4 |- ( K e. ( 0 ... N ) -> ( ! ` N ) e. CC )
29	16, 22	nnmulcld 8906	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN )
30	29	nncnd 8871	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. CC )
31		nn0p1nn 9153	. . . . . 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 8871	. . . 4 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. CC )
34	29	nnap0d 8903	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) # 0 )
35	20	nnap0d 8903	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) # 0 )
36	28, 30, 33, 21, 34, 35	divmuldivapd 8728	. . 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 2201	. 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 10009	. . 3 |- ( K e. ( 0 ... N ) -> K e. ( 0 ... ( N + 1 ) ) )
39		bcval2 10663	. . 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 10663	. . 3 |- ( K e. ( 0 ... N ) -> ( N _C K ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
42	41	oveq1d 5857	. 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 2208	1 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C K ) = ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   ` cfv 5188  (class class class)co 5842   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   - cmin 8069   / cdiv 8568   NNcn 8857   NN0cn0 9114   ...cfz 9944   !cfa 10638   _C cbc 10660

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-fz 9945  df-seqfrec 10381  df-fac 10639  df-bc 10661

This theorem is referenced by:  bcp1nk  10675  bcpasc  10679