Theorem bcp1n 10943

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 10272	. . . . 5 |- ( K e. ( 0 ... N ) -> N e. NN0 )
2		facp1 10912	. . . . 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 10214	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( N - K ) e. NN0 )
5		facp1 10912	. . . . . . . 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 9385	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> N e. CC )
8		1cnd 8123	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> 1 e. CC )
9		elfznn0 10271	. . . . . . . . . 10 |- ( K e. ( 0 ... N ) -> K e. NN0 )
10	9	nn0cnd 9385	. . . . . . . . 9 |- ( K e. ( 0 ... N ) -> K e. CC )
11	7, 8, 10	addsubd 8439	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) = ( ( N - K ) + 1 ) )
12	11	fveq2d 5603	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ! ` ( ( N + 1 ) - K ) ) = ( ! ` ( ( N - K ) + 1 ) ) )
13	11	oveq2d 5983	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) = ( ( ! ` ( N - K ) ) x. ( ( N - K ) + 1 ) ) )
14	6, 12, 13	3eqtr4d 2250	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ! ` ( ( N + 1 ) - K ) ) = ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) )
15	14	oveq1d 5982	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) = ( ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) )
16	4	faccld 10918	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. NN )
17	16	nncnd 9085	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. CC )
18		nn0p1nn 9369	. . . . . . . . 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 2284	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. NN )
21	20	nncnd 9085	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. CC )
22	9	faccld 10918	. . . . . . 7 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. NN )
23	22	nncnd 9085	. . . . . 6 |- ( K e. ( 0 ... N ) -> ( ! ` K ) e. CC )
24	17, 21, 23	mul32d 8260	. . . . 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 2240	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) = ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) x. ( ( N + 1 ) - K ) ) )
26	3, 25	oveq12d 5985	. . 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 10918	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ! ` N ) e. NN )
28	27	nncnd 9085	. . . 4 |- ( K e. ( 0 ... N ) -> ( ! ` N ) e. CC )
29	16, 22	nnmulcld 9120	. . . . 5 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN )
30	29	nncnd 9085	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. CC )
31		nn0p1nn 9369	. . . . . 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 9085	. . . 4 |- ( K e. ( 0 ... N ) -> ( N + 1 ) e. CC )
34	29	nnap0d 9117	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) # 0 )
35	20	nnap0d 9117	. . . 4 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) # 0 )
36	28, 30, 33, 21, 34, 35	divmuldivapd 8940	. . 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 2243	. 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 10231	. . 3 |- ( K e. ( 0 ... N ) -> K e. ( 0 ... ( N + 1 ) ) )
39		bcval2 10932	. . 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 10932	. . 3 |- ( K e. ( 0 ... N ) -> ( N _C K ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
42	41	oveq1d 5982	. 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 2250	1 |- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C K ) = ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   ` cfv 5290  (class class class)co 5967   0cc0 7960   1c1 7961   + caddc 7963   x. cmul 7965   - cmin 8278   / cdiv 8780   NNcn 9071   NN0cn0 9330   ...cfz 10165   !cfa 10907   _C cbc 10929

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-fz 10166  df-seqfrec 10630  df-fac 10908  df-bc 10930

This theorem is referenced by:  bcp1nk  10944  bcpasc  10948