Theorem bcval 10488

Description: Value of the binomial coefficient, N choose K. Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when 0 <_ K <_ N does not hold. See bcval2 10489 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Assertion

Ref	Expression
bcval	|- ( ( N e. NN0 /\ K e. ZZ ) -> ( N _C K ) = if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) )

Proof of Theorem bcval

Dummy variables k n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iftrue 3474	. . . . 5 |- ( K e. ( 0 ... N ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
2	1	adantl 275	. . . 4 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
3		simpll 518	. . . . . . 7 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> N e. NN0 )
4	3	faccld 10475	. . . . . 6 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` N ) e. NN )
5	4	nnzd 9165	. . . . 5 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` N ) e. ZZ )
6		fznn0sub 9830	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( N - K ) e. NN0 )
7	6	adantl 275	. . . . . . 7 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( N - K ) e. NN0 )
8	7	faccld 10475	. . . . . 6 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` ( N - K ) ) e. NN )
9		elfznn0 9887	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> K e. NN0 )
10	9	adantl 275	. . . . . . 7 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> K e. NN0 )
11	10	faccld 10475	. . . . . 6 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` K ) e. NN )
12	8, 11	nnmulcld 8762	. . . . 5 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN )
13		znq 9409	. . . . 5 |- ( ( ( ! ` N ) e. ZZ /\ ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN ) -> ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) e. QQ )
14	5, 12, 13	syl2anc 408	. . . 4 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) e. QQ )
15	2, 14	eqeltrd 2214	. . 3 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) e. QQ )
16		iffalse 3477	. . . . 5 |- ( -. K e. ( 0 ... N ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) = 0 )
17		0z 9058	. . . . . 6 |- 0 e. ZZ
18		zq 9411	. . . . . 6 |- ( 0 e. ZZ -> 0 e. QQ )
19	17, 18	ax-mp 5	. . . . 5 |- 0 e. QQ
20	16, 19	eqeltrdi 2228	. . . 4 |- ( -. K e. ( 0 ... N ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) e. QQ )
21	20	adantl 275	. . 3 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ -. K e. ( 0 ... N ) ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) e. QQ )
22		simpr 109	. . . . 5 |- ( ( N e. NN0 /\ K e. ZZ ) -> K e. ZZ )
23		0zd 9059	. . . . 5 |- ( ( N e. NN0 /\ K e. ZZ ) -> 0 e. ZZ )
24		simpl 108	. . . . . 6 |- ( ( N e. NN0 /\ K e. ZZ ) -> N e. NN0 )
25	24	nn0zd 9164	. . . . 5 |- ( ( N e. NN0 /\ K e. ZZ ) -> N e. ZZ )
26		fzdcel 9813	. . . . 5 |- ( ( K e. ZZ /\ 0 e. ZZ /\ N e. ZZ ) -> DECID K e. ( 0 ... N ) )
27	22, 23, 25, 26	syl3anc 1216	. . . 4 |- ( ( N e. NN0 /\ K e. ZZ ) -> DECID K e. ( 0 ... N ) )
28		exmiddc 821	. . . 4 |- (DECID K e. ( 0 ... N ) -> ( K e. ( 0 ... N ) \/ -. K e. ( 0 ... N ) ) )
29	27, 28	syl 14	. . 3 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( K e. ( 0 ... N ) \/ -. K e. ( 0 ... N ) ) )
30	15, 21, 29	mpjaodan 787	. 2 |- ( ( N e. NN0 /\ K e. ZZ ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) e. QQ )
31		oveq2 5775	. . . . 5 |- ( n = N -> ( 0 ... n ) = ( 0 ... N ) )
32	31	eleq2d 2207	. . . 4 |- ( n = N -> ( k e. ( 0 ... n ) <-> k e. ( 0 ... N ) ) )
33		fveq2 5414	. . . . 5 |- ( n = N -> ( ! ` n ) = ( ! ` N ) )
34		oveq1 5774	. . . . . . 7 |- ( n = N -> ( n - k ) = ( N - k ) )
35	34	fveq2d 5418	. . . . . 6 |- ( n = N -> ( ! ` ( n - k ) ) = ( ! ` ( N - k ) ) )
36	35	oveq1d 5782	. . . . 5 |- ( n = N -> ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) = ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) )
37	33, 36	oveq12d 5785	. . . 4 |- ( n = N -> ( ( ! ` n ) / ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) ) = ( ( ! ` N ) / ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) ) )
38	32, 37	ifbieq1d 3489	. . 3 |- ( n = N -> if ( k e. ( 0 ... n ) , ( ( ! ` n ) / ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) ) , 0 ) = if ( k e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) ) , 0 ) )
39		eleq1 2200	. . . 4 |- ( k = K -> ( k e. ( 0 ... N ) <-> K e. ( 0 ... N ) ) )
40		oveq2 5775	. . . . . . 7 |- ( k = K -> ( N - k ) = ( N - K ) )
41	40	fveq2d 5418	. . . . . 6 |- ( k = K -> ( ! ` ( N - k ) ) = ( ! ` ( N - K ) ) )
42		fveq2 5414	. . . . . 6 |- ( k = K -> ( ! ` k ) = ( ! ` K ) )
43	41, 42	oveq12d 5785	. . . . 5 |- ( k = K -> ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) = ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) )
44	43	oveq2d 5783	. . . 4 |- ( k = K -> ( ( ! ` N ) / ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
45	39, 44	ifbieq1d 3489	. . 3 |- ( k = K -> if ( k e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) ) , 0 ) = if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) )
46		df-bc 10487	. . 3 |- _C = ( n e. NN0 , k e. ZZ |-> if ( k e. ( 0 ... n ) , ( ( ! ` n ) / ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) ) , 0 ) )
47	38, 45, 46	ovmpog 5898	. 2 |- ( ( N e. NN0 /\ K e. ZZ /\ if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) e. QQ ) -> ( N _C K ) = if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) )
48	30, 47	mpd3an3 1316	1 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( N _C K ) = if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 697  DECID wdc 819   = wceq 1331   e. wcel 1480   ifcif 3469   ` cfv 5118  (class class class)co 5767   0cc0 7613   x. cmul 7618   - cmin 7926   / cdiv 8425   NNcn 8713   NN0cn0 8970   ZZcz 9047   QQcq 9404   ...cfz 9783   !cfa 10464   _C cbc 10486

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 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731

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 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-fz 9784  df-seqfrec 10212  df-fac 10465  df-bc 10487

This theorem is referenced by:  bcval2  10489  bcval3  10490