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Theorem bcval 11550
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 11551 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 oveq2 6048 . . . 4  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
21eleq2d 2471 . . 3  |-  ( n  =  N  ->  (
k  e.  ( 0 ... n )  <->  k  e.  ( 0 ... N
) ) )
3 fveq2 5687 . . . 4  |-  ( n  =  N  ->  ( ! `  n )  =  ( ! `  N ) )
4 oveq1 6047 . . . . . 6  |-  ( n  =  N  ->  (
n  -  k )  =  ( N  -  k ) )
54fveq2d 5691 . . . . 5  |-  ( n  =  N  ->  ( ! `  ( n  -  k ) )  =  ( ! `  ( N  -  k
) ) )
65oveq1d 6055 . . . 4  |-  ( n  =  N  ->  (
( ! `  (
n  -  k ) )  x.  ( ! `
 k ) )  =  ( ( ! `
 ( N  -  k ) )  x.  ( ! `  k
) ) )
73, 6oveq12d 6058 . . 3  |-  ( n  =  N  ->  (
( ! `  n
)  /  ( ( ! `  ( n  -  k ) )  x.  ( ! `  k ) ) )  =  ( ( ! `
 N )  / 
( ( ! `  ( N  -  k
) )  x.  ( ! `  k )
) ) )
8 eqidd 2405 . . 3  |-  ( n  =  N  ->  0  =  0 )
92, 7, 8ifbieq12d 3721 . 2  |-  ( n  =  N  ->  if ( k  e.  ( 0 ... n ) ,  ( ( ! `
 n )  / 
( ( ! `  ( n  -  k
) )  x.  ( ! `  k )
) ) ,  0 )  =  if ( k  e.  ( 0 ... N ) ,  ( ( ! `  N )  /  (
( ! `  ( N  -  k )
)  x.  ( ! `
 k ) ) ) ,  0 ) )
10 eleq1 2464 . . 3  |-  ( k  =  K  ->  (
k  e.  ( 0 ... N )  <->  K  e.  ( 0 ... N
) ) )
11 oveq2 6048 . . . . . 6  |-  ( k  =  K  ->  ( N  -  k )  =  ( N  -  K ) )
1211fveq2d 5691 . . . . 5  |-  ( k  =  K  ->  ( ! `  ( N  -  k ) )  =  ( ! `  ( N  -  K
) ) )
13 fveq2 5687 . . . . 5  |-  ( k  =  K  ->  ( ! `  k )  =  ( ! `  K ) )
1412, 13oveq12d 6058 . . . 4  |-  ( k  =  K  ->  (
( ! `  ( N  -  k )
)  x.  ( ! `
 k ) )  =  ( ( ! `
 ( N  -  K ) )  x.  ( ! `  K
) ) )
1514oveq2d 6056 . . 3  |-  ( k  =  K  ->  (
( ! `  N
)  /  ( ( ! `  ( N  -  k ) )  x.  ( ! `  k ) ) )  =  ( ( ! `
 N )  / 
( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
) ) )
16 eqidd 2405 . . 3  |-  ( k  =  K  ->  0  =  0 )
1710, 15, 16ifbieq12d 3721 . 2  |-  ( k  =  K  ->  if ( k  e.  ( 0 ... N ) ,  ( ( ! `
 N )  / 
( ( ! `  ( N  -  k
) )  x.  ( ! `  k )
) ) ,  0 )  =  if ( K  e.  ( 0 ... N ) ,  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) ,  0 ) )
18 df-bc 11549 . 2  |-  _C  =  ( n  e.  NN0 ,  k  e.  ZZ  |->  if ( k  e.  ( 0 ... n ) ,  ( ( ! `
 n )  / 
( ( ! `  ( n  -  k
) )  x.  ( ! `  k )
) ) ,  0 ) )
19 ovex 6065 . . 3  |-  ( ( ! `  N )  /  ( ( ! `
 ( N  -  K ) )  x.  ( ! `  K
) ) )  e. 
_V
20 c0ex 9041 . . 3  |-  0  e.  _V
2119, 20ifex 3757 . 2  |-  if ( K  e.  ( 0 ... N ) ,  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) ,  0 )  e.  _V
229, 17, 18, 21ovmpt2 6168 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:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   ifcif 3699   ` cfv 5413  (class class class)co 6040   0cc0 8946    x. cmul 8951    - cmin 9247    / cdiv 9633   NN0cn0 10177   ZZcz 10238   ...cfz 10999   !cfa 11521    _C cbc 11548
This theorem is referenced by:  bcval2  11551  bcval3  11552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-mulcl 9008  ax-i2m1 9014
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-bc 11549
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