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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.
StepHypRef 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 )
) ) )
21adantl 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 )
43faccld 10207 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  K  e.  ZZ )  /\  K  e.  ( 0 ... N ) )  ->  ( ! `  N )  e.  NN )
54nnzd 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 )
76adantl 272 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  K  e.  ZZ )  /\  K  e.  ( 0 ... N ) )  ->  ( N  -  K )  e.  NN0 )
87faccld 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 )
109adantl 272 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  K  e.  ZZ )  /\  K  e.  ( 0 ... N ) )  ->  K  e.  NN0 )
1110faccld 10207 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  K  e.  ZZ )  /\  K  e.  ( 0 ... N ) )  ->  ( ! `  K )  e.  NN )
128, 11nnmulcld 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 )
145, 12, 13syl2anc 404 . . . 4  |-  ( ( ( N  e.  NN0  /\  K  e.  ZZ )  /\  K  e.  ( 0 ... N ) )  ->  ( ( ! `  N )  /  ( ( ! `
 ( N  -  K ) )  x.  ( ! `  K
) ) )  e.  QQ )
152, 14eqeltrd 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 )
1917, 18ax-mp 7 . . . . 5  |-  0  e.  QQ
2016, 19syl6eqel 2179 . . . 4  |-  ( -.  K  e.  ( 0 ... N )  ->  if ( K  e.  ( 0 ... N ) ,  ( ( ! `
 N )  / 
( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
) ) ,  0 )  e.  QQ )
2120adantl 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 )
2524nn0zd 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 ) )
2722, 23, 25, 26syl3anc 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
) ) )
2927, 28syl 14 . . 3  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( K  e.  ( 0 ... N )  \/  -.  K  e.  ( 0 ... N
) ) )
3015, 21, 29mpjaodan 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
) )
3231eleq2d 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 ) )
3534fveq2d 5324 . . . . . 6  |-  ( n  =  N  ->  ( ! `  ( n  -  k ) )  =  ( ! `  ( N  -  k
) ) )
3635oveq1d 5683 . . . . 5  |-  ( n  =  N  ->  (
( ! `  (
n  -  k ) )  x.  ( ! `
 k ) )  =  ( ( ! `
 ( N  -  k ) )  x.  ( ! `  k
) ) )
3733, 36oveq12d 5686 . . . 4  |-  ( n  =  N  ->  (
( ! `  n
)  /  ( ( ! `  ( n  -  k ) )  x.  ( ! `  k ) ) )  =  ( ( ! `
 N )  / 
( ( ! `  ( N  -  k
) )  x.  ( ! `  k )
) ) )
3832, 37ifbieq1d 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 ) )
4140fveq2d 5324 . . . . . 6  |-  ( k  =  K  ->  ( ! `  ( N  -  k ) )  =  ( ! `  ( N  -  K
) ) )
42 fveq2 5320 . . . . . 6  |-  ( k  =  K  ->  ( ! `  k )  =  ( ! `  K ) )
4341, 42oveq12d 5686 . . . . 5  |-  ( k  =  K  ->  (
( ! `  ( N  -  k )
)  x.  ( ! `
 k ) )  =  ( ( ! `
 ( N  -  K ) )  x.  ( ! `  K
) ) )
4443oveq2d 5684 . . . 4  |-  ( k  =  K  ->  (
( ! `  N
)  /  ( ( ! `  ( N  -  k ) )  x.  ( ! `  k ) ) )  =  ( ( ! `
 N )  / 
( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
) ) )
4539, 44ifbieq1d 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 ) )
4738, 45, 46ovmpt2g 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 ) )
4830, 47mpd3an3 1275 1  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  _C  K
)  =  if ( K  e.  ( 0 ... N ) ,  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) ,  0 ) )
Colors of variables: wff set class
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
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