Theorem bcval 10220

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 10221 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 3404	. . . . 5 |- ( K e. ( 0 ... N ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
2	1	adantl 272	. . . 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 497	. . . . . . 7 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> N e. NN0 )
4	3	faccld 10207	. . . . . 6 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` N ) e. NN )
5	4	nnzd 8930	. . . . 5 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` N ) e. ZZ )
6		fznn0sub 9534	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> ( N - K ) e. NN0 )
7	6	adantl 272	. . . . . . 7 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( N - K ) e. NN0 )
8	7	faccld 10207	. . . . . 6 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` ( N - K ) ) e. NN )
9		elfznn0 9591	. . . . . . . 8 |- ( K e. ( 0 ... N ) -> K e. NN0 )
10	9	adantl 272	. . . . . . 7 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> K e. NN0 )
11	10	faccld 10207	. . . . . 6 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` K ) e. NN )
12	8, 11	nnmulcld 8534	. . . . 5 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN )
13		znq 9172	. . . . 5 |- ( ( ( ! ` N ) e. ZZ /\ ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN ) -> ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) e. QQ )
14	5, 12, 13	syl2anc 404	. . . 4 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) e. QQ )
15	2, 14	eqeltrd 2165	. . 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 3407	. . . . 5 |- ( -. K e. ( 0 ... N ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) = 0 )
17		0z 8824	. . . . . 6 |- 0 e. ZZ
18		zq 9174	. . . . . 6 |- ( 0 e. ZZ -> 0 e. QQ )
19	17, 18	ax-mp 7	. . . . 5 |- 0 e. QQ
20	16, 19	syl6eqel 2179	. . . 4 |- ( -. K e. ( 0 ... N ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) e. QQ )
21	20	adantl 272	. . 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 8825	. . . . 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 8929	. . . . 5 |- ( ( N e. NN0 /\ K e. ZZ ) -> N e. ZZ )
26		fzdcel 9517	. . . . 5 |- ( ( K e. ZZ /\ 0 e. ZZ /\ N e. ZZ ) -> DECID K e. ( 0 ... N ) )
27	22, 23, 25, 26	syl3anc 1175	. . . 4 |- ( ( N e. NN0 /\ K e. ZZ ) -> DECID K e. ( 0 ... N ) )
28		exmiddc 783	. . . 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 748	. 2 |- ( ( N e. NN0 /\ K e. ZZ ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) e. QQ )
31		oveq2 5676	. . . . 5 |- ( n = N -> ( 0 ... n ) = ( 0 ... N ) )
32	31	eleq2d 2158	. . . 4 |- ( n = N -> ( k e. ( 0 ... n ) <-> k e. ( 0 ... N ) ) )
33		fveq2 5320	. . . . 5 |- ( n = N -> ( ! ` n ) = ( ! ` N ) )
34		oveq1 5675	. . . . . . 7 |- ( n = N -> ( n - k ) = ( N - k ) )
35	34	fveq2d 5324	. . . . . 6 |- ( n = N -> ( ! ` ( n - k ) ) = ( ! ` ( N - k ) ) )
36	35	oveq1d 5683	. . . . 5 |- ( n = N -> ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) = ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) )
37	33, 36	oveq12d 5686	. . . 4 |- ( n = N -> ( ( ! ` n ) / ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) ) = ( ( ! ` N ) / ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) ) )
38	32, 37	ifbieq1d 3419	. . 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 2151	. . . 4 |- ( k = K -> ( k e. ( 0 ... N ) <-> K e. ( 0 ... N ) ) )
40		oveq2 5676	. . . . . . 7 |- ( k = K -> ( N - k ) = ( N - K ) )
41	40	fveq2d 5324	. . . . . 6 |- ( k = K -> ( ! ` ( N - k ) ) = ( ! ` ( N - K ) ) )
42		fveq2 5320	. . . . . 6 |- ( k = K -> ( ! ` k ) = ( ! ` K ) )
43	41, 42	oveq12d 5686	. . . . 5 |- ( k = K -> ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) = ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) )
44	43	oveq2d 5684	. . . 4 |- ( k = K -> ( ( ! ` N ) / ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
45	39, 44	ifbieq1d 3419	. . 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 10219	. . 3 |- _C = ( n e. NN0 , k e. ZZ |-> if ( k e. ( 0 ... n ) , ( ( ! ` n ) / ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) ) , 0 ) )
47	38, 45, 46	ovmpt2g 5795	. 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 1275	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 665  DECID wdc 781   = wceq 1290   e. wcel 1439   ifcif 3399   ` cfv 5030  (class class class)co 5668   0cc0 7413   x. cmul 7418   - cmin 7716   / cdiv 8202   NNcn 8485   NN0cn0 8736   ZZcz 8813   QQcq 9167   ...cfz 9487   !cfa 10196   _C cbc 10218

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3962  ax-sep 3965  ax-nul 3973  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-iinf 4418  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-1re 7502  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-mulrcl 7507  ax-addcom 7508  ax-mulcom 7509  ax-addass 7510  ax-mulass 7511  ax-distr 7512  ax-i2m1 7513  ax-0lt1 7514  ax-1rid 7515  ax-0id 7516  ax-rnegex 7517  ax-precex 7518  ax-cnre 7519  ax-pre-ltirr 7520  ax-pre-ltwlin 7521  ax-pre-lttrn 7522  ax-pre-apti 7523  ax-pre-ltadd 7524  ax-pre-mulgt0 7525  ax-pre-mulext 7526

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-if 3400  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-tr 3945  df-id 4131  df-po 4134  df-iso 4135  df-iord 4204  df-on 4206  df-ilim 4207  df-suc 4209  df-iom 4421  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-riota 5624  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-1st 5927  df-2nd 5928  df-recs 6086  df-frec 6172  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591  df-sub 7718  df-neg 7719  df-reap 8115  df-ap 8122  df-div 8203  df-inn 8486  df-n0 8737  df-z 8814  df-uz 9083  df-q 9168  df-fz 9488  df-iseq 9916  df-fac 10197  df-bc 10219

This theorem is referenced by:  bcval2  10221  bcval3  10222