Theorem bcp1n 13103

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 12434	. . . . 5 |- ( K e. ( 0 ... N ) -> N e. NN0 )
2		facp1 13065	. . . . 5 |- ( N e. NN0 -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )
3	1, 2	syl 17	. . . 4 |- ( K e. ( 0 ... N ) -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) )
4		fznn0sub 12373	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( N - K ) e. NN0 )
5		facp1 13065	. . . . . . . 8 |- ( ( N - K ) e. NN0 -> ( ! ` ( ( N - K ) + 1 ) ) = ( ( ! ` ( N - K ) ) x. ( ( N - K ) + 1 ) ) )
6	4, 5	syl 17	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ! ` ( ( N - K ) + 1 ) ) = ( ( ! ` ( N - K ) ) x. ( ( N - K ) + 1 ) ) )
7	1	nn0cnd 11353	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> N e. CC )
8		1cnd 10056	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> 1 e. CC )
9		elfznn0 12433	. . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> K e. NN0 )
10	9	nn0cnd 11353	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K e. CC )
11	7, 8, 10	addsubd 10413	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) = ( ( N - K ) + 1 ) )
12	11	fveq2d 6195	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ! ` ( ( N + 1 ) - K ) ) = ( ! ` ( ( N - K ) + 1 ) ) )
13	11	oveq2d 6666	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) = ( ( ! ` ( N - K ) ) x. ( ( N - K ) + 1 ) ) )
14	6, 12, 13	3eqtr4d 2666	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ! ` ( ( N + 1 ) - K ) ) = ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) )
15	14	oveq1d 6665	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) = ( ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) )
16	4	faccld 13071	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. NN )
17	16	nncnd 11036	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. CC )
18		nn0p1nn 11332	. . . . . . . . 9 |- ( ( N - K ) e. NN0 -> ( ( N - K ) + 1 ) e. NN )
19	4, 18	syl 17	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ( N - K ) + 1 ) e. NN )
20	11, 19	eqeltrd 2701	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. NN )
21	20	nncnd 11036	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. CC )
22	9	faccld 13071	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. NN )
23	22	nncnd 11036	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. CC )
24	17, 21, 23	mul32d 10246	. . . . 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 2656	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) = ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) x. ( ( N + 1 ) - K ) ) )
26	3, 25	oveq12d 6668	. . 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 13071	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ! ` N ) e. NN )
28	27	nncnd 11036	. . . 4 |- ( K e. ( 0 ... N ) -> ( ! ` N ) e. CC )
29		nn0p1nn 11332	. . . . . 6 |- ( N e. NN0 -> ( N + 1 ) e. NN )
30	1, 29	syl 17	. . . . 5 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. NN )
31	30	nncnd 11036	. . . 4 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. CC )
32	16, 22	nnmulcld 11068	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN )
33		nncn 11028	. . . . . 6 |- ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. CC )
34		nnne0 11053	. . . . . 6 |- ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) =/= 0 )
35	33, 34	jca 554	. . . . 5 |- ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN -> ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. CC /\ ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) =/= 0 ) )
36	32, 35	syl 17	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. CC /\ ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) =/= 0 ) )
37	20	nnne0d 11065	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) =/= 0 )
38	21, 37	jca 554	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) - K ) e. CC /\ ( ( N + 1 ) - K ) =/= 0 ) )
39		divmuldiv 10725	. . . 4 |- ( ( ( ( ! ` N ) e. CC /\ ( N + 1 ) e. CC ) /\ ( ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. CC /\ ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) =/= 0 ) /\ ( ( ( N + 1 ) - K ) e. CC /\ ( ( N + 1 ) - K ) =/= 0 ) ) ) -> ( ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) = ( ( ( ! ` N ) x. ( N + 1 ) ) / ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) x. ( ( N + 1 ) - K ) ) ) )
40	28, 31, 36, 38, 39	syl22anc 1327	. . 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 ) ) ) )
41	26, 40	eqtr4d 2659	. 2 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N + 1 ) ) / ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) ) = ( ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )
42		fzelp1 12393	. . 3 |- ( K e. ( 0 ... N ) -> K e. ( 0 ... ( N + 1 ) ) )
43		bcval2 13092	. . 3 |- ( K e. ( 0 ... ( N + 1 ) ) -> ( ( N + 1 ) _C K ) = ( ( ! ` ( N + 1 ) ) / ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) ) )
44	42, 43	syl 17	. 2 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C K ) = ( ( ! ` ( N + 1 ) ) / ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) ) )
45		bcval2 13092	. . 3 |- ( K e. ( 0 ... N ) -> ( N _C K ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
46	45	oveq1d 6665	. 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 ) ) ) )
47	41, 44, 46	3eqtr4d 2666	1 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C K ) = ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   ...cfz 12326   !cfa 13060   _C cbc 13089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-fac 13061  df-bc 13090

This theorem is referenced by:  bcp1nk  13104  bcpasc  13108  bcp1ctr  25004  bcm1n  29554  bcm1nt  31623