Theorem bcp1n 10682

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 10058	. . . . 5 |- ( K e. ( 0 ... N ) -> N e. NN0 )
2		facp1 10651	. . . . 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 10000	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( N - K ) e. NN0 )
5		facp1 10651	. . . . . . . 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 9177	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> N e. CC )
8		1cnd 7923	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> 1 e. CC )
9		elfznn0 10057	. . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> K e. NN0 )
10	9	nn0cnd 9177	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K e. CC )
11	7, 8, 10	addsubd 8238	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) = ( ( N - K ) + 1 ) )
12	11	fveq2d 5498	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ! ` ( ( N + 1 ) - K ) ) = ( ! ` ( ( N - K ) + 1 ) ) )
13	11	oveq2d 5866	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) = ( ( ! ` ( N - K ) ) x. ( ( N - K ) + 1 ) ) )
14	6, 12, 13	3eqtr4d 2213	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ! ` ( ( N + 1 ) - K ) ) = ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) )
15	14	oveq1d 5865	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) = ( ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) )
16	4	faccld 10657	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. NN )
17	16	nncnd 8879	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. CC )
18		nn0p1nn 9161	. . . . . . . . 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 2247	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. NN )
21	20	nncnd 8879	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. CC )
22	9	faccld 10657	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. NN )
23	22	nncnd 8879	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. CC )
24	17, 21, 23	mul32d 8059	. . . . 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 2203	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) = ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) x. ( ( N + 1 ) - K ) ) )
26	3, 25	oveq12d 5868	. . 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 10657	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ! ` N ) e. NN )
28	27	nncnd 8879	. . . 4 |- ( K e. ( 0 ... N ) -> ( ! ` N ) e. CC )
29	16, 22	nnmulcld 8914	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN )
30	29	nncnd 8879	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. CC )
31		nn0p1nn 9161	. . . . . 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 8879	. . . 4 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. CC )
34	29	nnap0d 8911	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) # 0 )
35	20	nnap0d 8911	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) # 0 )
36	28, 30, 33, 21, 34, 35	divmuldivapd 8736	. . 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 2206	. 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 10017	. . 3 |- ( K e. ( 0 ... N ) -> K e. ( 0 ... ( N + 1 ) ) )
39		bcval2 10671	. . 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 10671	. . 3 |- ( K e. ( 0 ... N ) -> ( N _C K ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
42	41	oveq1d 5865	. 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 2213	1 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C K ) = ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   ` cfv 5196  (class class class)co 5850   0cc0 7761   1c1 7762   + caddc 7764   x. cmul 7766   - cmin 8077   / cdiv 8576   NNcn 8865   NN0cn0 9122   ...cfz 9952   !cfa 10646   _C cbc 10668

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-n0 9123  df-z 9200  df-uz 9475  df-q 9566  df-fz 9953  df-seqfrec 10389  df-fac 10647  df-bc 10669

This theorem is referenced by:  bcp1nk  10683  bcpasc  10687