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Theorem bcval 11583
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 11584 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 6080 . . . 4  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
21eleq2d 2502 . . 3  |-  ( n  =  N  ->  (
k  e.  ( 0 ... n )  <->  k  e.  ( 0 ... N
) ) )
3 fveq2 5719 . . . 4  |-  ( n  =  N  ->  ( ! `  n )  =  ( ! `  N ) )
4 oveq1 6079 . . . . . 6  |-  ( n  =  N  ->  (
n  -  k )  =  ( N  -  k ) )
54fveq2d 5723 . . . . 5  |-  ( n  =  N  ->  ( ! `  ( n  -  k ) )  =  ( ! `  ( N  -  k
) ) )
65oveq1d 6087 . . . 4  |-  ( n  =  N  ->  (
( ! `  (
n  -  k ) )  x.  ( ! `
 k ) )  =  ( ( ! `
 ( N  -  k ) )  x.  ( ! `  k
) ) )
73, 6oveq12d 6090 . . 3  |-  ( n  =  N  ->  (
( ! `  n
)  /  ( ( ! `  ( n  -  k ) )  x.  ( ! `  k ) ) )  =  ( ( ! `
 N )  / 
( ( ! `  ( N  -  k
) )  x.  ( ! `  k )
) ) )
8 eqidd 2436 . . 3  |-  ( n  =  N  ->  0  =  0 )
92, 7, 8ifbieq12d 3753 . 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 2495 . . 3  |-  ( k  =  K  ->  (
k  e.  ( 0 ... N )  <->  K  e.  ( 0 ... N
) ) )
11 oveq2 6080 . . . . . 6  |-  ( k  =  K  ->  ( N  -  k )  =  ( N  -  K ) )
1211fveq2d 5723 . . . . 5  |-  ( k  =  K  ->  ( ! `  ( N  -  k ) )  =  ( ! `  ( N  -  K
) ) )
13 fveq2 5719 . . . . 5  |-  ( k  =  K  ->  ( ! `  k )  =  ( ! `  K ) )
1412, 13oveq12d 6090 . . . 4  |-  ( k  =  K  ->  (
( ! `  ( N  -  k )
)  x.  ( ! `
 k ) )  =  ( ( ! `
 ( N  -  K ) )  x.  ( ! `  K
) ) )
1514oveq2d 6088 . . 3  |-  ( k  =  K  ->  (
( ! `  N
)  /  ( ( ! `  ( N  -  k ) )  x.  ( ! `  k ) ) )  =  ( ( ! `
 N )  / 
( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
) ) )
16 eqidd 2436 . . 3  |-  ( k  =  K  ->  0  =  0 )
1710, 15, 16ifbieq12d 3753 . 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 11582 . 2  |-  _C  =  ( n  e.  NN0 ,  k  e.  ZZ  |->  if ( k  e.  ( 0 ... n ) ,  ( ( ! `
 n )  / 
( ( ! `  ( n  -  k
) )  x.  ( ! `  k )
) ) ,  0 ) )
19 ovex 6097 . . 3  |-  ( ( ! `  N )  /  ( ( ! `
 ( N  -  K ) )  x.  ( ! `  K
) ) )  e. 
_V
20 c0ex 9074 . . 3  |-  0  e.  _V
2119, 20ifex 3789 . 2  |-  if ( K  e.  ( 0 ... N ) ,  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) ,  0 )  e.  _V
229, 17, 18, 21ovmpt2 6200 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 1652    e. wcel 1725   ifcif 3731   ` cfv 5445  (class class class)co 6072   0cc0 8979    x. cmul 8984    - cmin 9280    / cdiv 9666   NN0cn0 10210   ZZcz 10271   ...cfz 11032   !cfa 11554    _C cbc 11581
This theorem is referenced by:  bcval2  11584  bcval3  11585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-mulcl 9041  ax-i2m1 9047
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-iota 5409  df-fun 5447  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-bc 11582
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