Theorem bcval 10926

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 10927 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 3580	. . . . 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 10913	. . . . . 6 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` N ) e. NN )
5	4	nnzd 9524	. . . . 5 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` N ) e. ZZ )
6		fznn0sub 10209	. . . . . . . 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 10913	. . . . . 6 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` ( N - K ) ) e. NN )
9		elfznn0 10266	. . . . . . . 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 10913	. . . . . 6 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ! ` K ) e. NN )
12	8, 11	nnmulcld 9115	. . . . 5 |- ( ( ( N e. NN0 /\ K e. ZZ ) /\ K e. ( 0 ... N ) ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN )
13		znq 9775	. . . . 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 2283	. . 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 3583	. . . . 5 |- ( -. K e. ( 0 ... N ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) = 0 )
17		0z 9413	. . . . . 6 |- 0 e. ZZ
18		zq 9777	. . . . . 6 |- ( 0 e. ZZ -> 0 e. QQ )
19	17, 18	ax-mp 5	. . . . 5 |- 0 e. QQ
20	16, 19	eqeltrdi 2297	. . . 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 9414	. . . . 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 9523	. . . . 5 |- ( ( N e. NN0 /\ K e. ZZ ) -> N e. ZZ )
26		fzdcel 10192	. . . . 5 |- ( ( K e. ZZ /\ 0 e. ZZ /\ N e. ZZ ) -> DECID K e. ( 0 ... N ) )
27	22, 23, 25, 26	syl3anc 1250	. . . 4 |- ( ( N e. NN0 /\ K e. ZZ ) -> DECID K e. ( 0 ... N ) )
28		exmiddc 838	. . . 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 800	. 2 |- ( ( N e. NN0 /\ K e. ZZ ) -> if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) e. QQ )
31		oveq2 5970	. . . . 5 |- ( n = N -> ( 0 ... n ) = ( 0 ... N ) )
32	31	eleq2d 2276	. . . 4 |- ( n = N -> ( k e. ( 0 ... n ) <-> k e. ( 0 ... N ) ) )
33		fveq2 5594	. . . . 5 |- ( n = N -> ( ! ` n ) = ( ! ` N ) )
34		oveq1 5969	. . . . . . 7 |- ( n = N -> ( n - k ) = ( N - k ) )
35	34	fveq2d 5598	. . . . . 6 |- ( n = N -> ( ! ` ( n - k ) ) = ( ! ` ( N - k ) ) )
36	35	oveq1d 5977	. . . . 5 |- ( n = N -> ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) = ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) )
37	33, 36	oveq12d 5980	. . . 4 |- ( n = N -> ( ( ! ` n ) / ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) ) = ( ( ! ` N ) / ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) ) )
38	32, 37	ifbieq1d 3598	. . 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 2269	. . . 4 |- ( k = K -> ( k e. ( 0 ... N ) <-> K e. ( 0 ... N ) ) )
40		oveq2 5970	. . . . . . 7 |- ( k = K -> ( N - k ) = ( N - K ) )
41	40	fveq2d 5598	. . . . . 6 |- ( k = K -> ( ! ` ( N - k ) ) = ( ! ` ( N - K ) ) )
42		fveq2 5594	. . . . . 6 |- ( k = K -> ( ! ` k ) = ( ! ` K ) )
43	41, 42	oveq12d 5980	. . . . 5 |- ( k = K -> ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) = ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) )
44	43	oveq2d 5978	. . . 4 |- ( k = K -> ( ( ! ` N ) / ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
45	39, 44	ifbieq1d 3598	. . 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 10925	. . 3 |- _C = ( n e. NN0 , k e. ZZ |-> if ( k e. ( 0 ... n ) , ( ( ! ` n ) / ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) ) , 0 ) )
47	38, 45, 46	ovmpog 6098	. 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 1351	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 710  DECID wdc 836   = wceq 1373   e. wcel 2177   ifcif 3575   ` cfv 5285  (class class class)co 5962   0cc0 7955   x. cmul 7960   - cmin 8273   / cdiv 8775   NNcn 9066   NN0cn0 9325   ZZcz 9402   QQcq 9770   ...cfz 10160   !cfa 10902   _C cbc 10924

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 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-mulrcl 8054  ax-addcom 8055  ax-mulcom 8056  ax-addass 8057  ax-mulass 8058  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-1rid 8062  ax-0id 8063  ax-rnegex 8064  ax-precex 8065  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-apti 8070  ax-pre-ltadd 8071  ax-pre-mulgt0 8072  ax-pre-mulext 8073

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 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-po 4356  df-iso 4357  df-iord 4426  df-on 4428  df-ilim 4429  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-frec 6495  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-reap 8678  df-ap 8685  df-div 8776  df-inn 9067  df-n0 9326  df-z 9403  df-uz 9679  df-q 9771  df-fz 10161  df-seqfrec 10625  df-fac 10903  df-bc 10925

This theorem is referenced by:  bcval2  10927  bcval3  10928