Theorem bcval 10820

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 10821 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 3562	. . . . 5 |- ( K e. ( 0 ... N ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
2	1	adantl 277	. . . 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 527	. . . . . . 7 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> N e. NN0 )
4	3	faccld 10807	. . . . . 6 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` N ) e. NN )
5	4	nnzd 9438	. . . . 5 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` N ) e. ZZ )
6		fznn0sub 10123	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( N - K ) e. NN0 )
7	6	adantl 277	. . . . . . 7 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( N - K ) e. NN0 )
8	7	faccld 10807	. . . . . 6 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` ( N - K ) ) e. NN )
9		elfznn0 10180	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> K e. NN0 )
10	9	adantl 277	. . . . . . 7 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> K e. NN0 )
11	10	faccld 10807	. . . . . 6 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` K ) e. NN )
12	8, 11	nnmulcld 9031	. . . . 5 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN )
13		znq 9689	. . . . 5 |- ( ( ( ! ` N ) e. ZZ /\ ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN ) -> ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) e. QQ )
14	5, 12, 13	syl2anc 411	. . . 4 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) e. QQ )
15	2, 14	eqeltrd 2270	. . 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 3565	. . . . 5 |- ( -. K e. ( 0 ... N ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) = 0 )
17		0z 9328	. . . . . 6 |- 0 e. ZZ
18		zq 9691	. . . . . 6 |- ( 0 e. ZZ -> 0 e. QQ )
19	17, 18	ax-mp 5	. . . . 5 |- 0 e. QQ
20	16, 19	eqeltrdi 2284	. . . 4 |- ( -. K e. ( 0 ... N ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) e. QQ )
21	20	adantl 277	. . 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 110	. . . . 5 |- ( ( N e. NN0 /\ K e. ZZ ) -> K e. ZZ )
23		0zd 9329	. . . . 5 |- ( ( N e. NN0 /\ K e. ZZ ) -> 0 e. ZZ )
24		simpl 109	. . . . . 6 |- ( ( N e. NN0 /\ K e. ZZ ) -> N e. NN0 )
25	24	nn0zd 9437	. . . . 5 |- ( ( N e. NN0 /\ K e. ZZ ) -> N e. ZZ )
26		fzdcel 10106	. . . . 5 |- ( ( K e. ZZ /\ 0 e. ZZ /\ N e. ZZ ) -> DECID K e. ( 0 ... N ) )
27	22, 23, 25, 26	syl3anc 1249	. . . 4 |- ( ( N e. NN0 /\ K e. ZZ ) -> DECID K e. ( 0 ... N ) )
28		exmiddc 837	. . . 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 799	. 2 |- ( ( N e. NN0 /\ K e. ZZ ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) e. QQ )
31		oveq2 5926	. . . . 5 |- ( n = N -> ( 0 ... n ) = ( 0 ... N ) )
32	31	eleq2d 2263	. . . 4 |- ( n = N -> ( k e. ( 0 ... n ) <-> k e. ( 0 ... N ) ) )
33		fveq2 5554	. . . . 5 |- ( n = N -> ( ! ` n ) = ( ! ` N ) )
34		oveq1 5925	. . . . . . 7 |- ( n = N -> ( n - k ) = ( N - k ) )
35	34	fveq2d 5558	. . . . . 6 |- ( n = N -> ( ! ` ( n - k ) ) = ( ! ` ( N - k ) ) )
36	35	oveq1d 5933	. . . . 5 |- ( n = N -> ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) = ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) )
37	33, 36	oveq12d 5936	. . . 4 |- ( n = N -> ( ( ! ` n ) / ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) ) = ( ( ! ` N ) / ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) ) )
38	32, 37	ifbieq1d 3579	. . 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 2256	. . . 4 |- ( k = K -> ( k e. ( 0 ... N ) <-> K e. ( 0 ... N ) ) )
40		oveq2 5926	. . . . . . 7 |- ( k = K -> ( N - k ) = ( N - K ) )
41	40	fveq2d 5558	. . . . . 6 |- ( k = K -> ( ! ` ( N - k ) ) = ( ! ` ( N - K ) ) )
42		fveq2 5554	. . . . . 6 |- ( k = K -> ( ! ` k ) = ( ! ` K ) )
43	41, 42	oveq12d 5936	. . . . 5 |- ( k = K -> ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) = ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) )
44	43	oveq2d 5934	. . . 4 |- ( k = K -> ( ( ! ` N ) / ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
45	39, 44	ifbieq1d 3579	. . 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 10819	. . 3 |- _C = ( n e. NN0 , k e. ZZ |-> if ( k e. ( 0 ... n ) , ( ( ! ` n ) / ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) ) , 0 ) )
47	38, 45, 46	ovmpog 6053	. 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 1349	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 104   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2164   ifcif 3557   ` cfv 5254  (class class class)co 5918   0cc0 7872   x. cmul 7877   - cmin 8190   / cdiv 8691   NNcn 8982   NN0cn0 9240   ZZcz 9317   QQcq 9684   ...cfz 10074   !cfa 10796   _C cbc 10818

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-fz 10075  df-seqfrec 10519  df-fac 10797  df-bc 10819

This theorem is referenced by:  bcval2  10821  bcval3  10822