Theorem bcval 10527

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 10528 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 3484	. . . . 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 519	. . . . . . 7 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> N e. NN0 )
4	3	faccld 10514	. . . . . 6 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` N ) e. NN )
5	4	nnzd 9196	. . . . 5 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` N ) e. ZZ )
6		fznn0sub 9868	. . . . . . . 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 10514	. . . . . 6 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` ( N - K ) ) e. NN )
9		elfznn0 9925	. . . . . . . 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 10514	. . . . . 6 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` K ) e. NN )
12	8, 11	nnmulcld 8793	. . . . 5 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN )
13		znq 9443	. . . . 5 |- ( ( ( ! ` N ) e. ZZ /\ ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN ) -> ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) e. QQ )
14	5, 12, 13	syl2anc 409	. . . 4 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) e. QQ )
15	2, 14	eqeltrd 2217	. . 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 3487	. . . . 5 |- ( -. K e. ( 0 ... N ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) = 0 )
17		0z 9089	. . . . . 6 |- 0 e. ZZ
18		zq 9445	. . . . . 6 |- ( 0 e. ZZ -> 0 e. QQ )
19	17, 18	ax-mp 5	. . . . 5 |- 0 e. QQ
20	16, 19	eqeltrdi 2231	. . . 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 9090	. . . . 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 9195	. . . . 5 |- ( ( N e. NN0 /\ K e. ZZ ) -> N e. ZZ )
26		fzdcel 9851	. . . . 5 |- ( ( K e. ZZ /\ 0 e. ZZ /\ N e. ZZ ) -> DECID K e. ( 0 ... N ) )
27	22, 23, 25, 26	syl3anc 1217	. . . 4 |- ( ( N e. NN0 /\ K e. ZZ ) -> DECID K e. ( 0 ... N ) )
28		exmiddc 822	. . . 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 788	. 2 |- ( ( N e. NN0 /\ K e. ZZ ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) e. QQ )
31		oveq2 5790	. . . . 5 |- ( n = N -> ( 0 ... n ) = ( 0 ... N ) )
32	31	eleq2d 2210	. . . 4 |- ( n = N -> ( k e. ( 0 ... n ) <-> k e. ( 0 ... N ) ) )
33		fveq2 5429	. . . . 5 |- ( n = N -> ( ! ` n ) = ( ! ` N ) )
34		oveq1 5789	. . . . . . 7 |- ( n = N -> ( n - k ) = ( N - k ) )
35	34	fveq2d 5433	. . . . . 6 |- ( n = N -> ( ! ` ( n - k ) ) = ( ! ` ( N - k ) ) )
36	35	oveq1d 5797	. . . . 5 |- ( n = N -> ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) = ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) )
37	33, 36	oveq12d 5800	. . . 4 |- ( n = N -> ( ( ! ` n ) / ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) ) = ( ( ! ` N ) / ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) ) )
38	32, 37	ifbieq1d 3499	. . 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 2203	. . . 4 |- ( k = K -> ( k e. ( 0 ... N ) <-> K e. ( 0 ... N ) ) )
40		oveq2 5790	. . . . . . 7 |- ( k = K -> ( N - k ) = ( N - K ) )
41	40	fveq2d 5433	. . . . . 6 |- ( k = K -> ( ! ` ( N - k ) ) = ( ! ` ( N - K ) ) )
42		fveq2 5429	. . . . . 6 |- ( k = K -> ( ! ` k ) = ( ! ` K ) )
43	41, 42	oveq12d 5800	. . . . 5 |- ( k = K -> ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) = ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) )
44	43	oveq2d 5798	. . . 4 |- ( k = K -> ( ( ! ` N ) / ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
45	39, 44	ifbieq1d 3499	. . 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 10526	. . 3 |- _C = ( n e. NN0 , k e. ZZ |-> if ( k e. ( 0 ... n ) , ( ( ! ` n ) / ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) ) , 0 ) )
47	38, 45, 46	ovmpog 5913	. 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 1317	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 698  DECID wdc 820   = wceq 1332   e. wcel 1481   ifcif 3479   ` cfv 5131  (class class class)co 5782   0cc0 7644   x. cmul 7649   - cmin 7957   / cdiv 8456   NNcn 8744   NN0cn0 9001   ZZcz 9078   QQcq 9438   ...cfz 9821   !cfa 10503   _C cbc 10525

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-fz 9822  df-seqfrec 10250  df-fac 10504  df-bc 10526

This theorem is referenced by:  bcval2  10528  bcval3  10529