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Theorem bcval 11269
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 11270 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
StepHypRef Expression
1 oveq2 5786 . . . 4  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
21eleq2d 2323 . . 3  |-  ( n  =  N  ->  (
k  e.  ( 0 ... n )  <->  k  e.  ( 0 ... N
) ) )
3 fveq2 5444 . . . 4  |-  ( n  =  N  ->  ( ! `  n )  =  ( ! `  N ) )
4 oveq1 5785 . . . . . 6  |-  ( n  =  N  ->  (
n  -  k )  =  ( N  -  k ) )
54fveq2d 5448 . . . . 5  |-  ( n  =  N  ->  ( ! `  ( n  -  k ) )  =  ( ! `  ( N  -  k
) ) )
65oveq1d 5793 . . . 4  |-  ( n  =  N  ->  (
( ! `  (
n  -  k ) )  x.  ( ! `
 k ) )  =  ( ( ! `
 ( N  -  k ) )  x.  ( ! `  k
) ) )
73, 6oveq12d 5796 . . 3  |-  ( n  =  N  ->  (
( ! `  n
)  /  ( ( ! `  ( n  -  k ) )  x.  ( ! `  k ) ) )  =  ( ( ! `
 N )  / 
( ( ! `  ( N  -  k
) )  x.  ( ! `  k )
) ) )
8 eqidd 2257 . . 3  |-  ( n  =  N  ->  0  =  0 )
92, 7, 8ifbieq12d 3547 . 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 2316 . . 3  |-  ( k  =  K  ->  (
k  e.  ( 0 ... N )  <->  K  e.  ( 0 ... N
) ) )
11 oveq2 5786 . . . . . 6  |-  ( k  =  K  ->  ( N  -  k )  =  ( N  -  K ) )
1211fveq2d 5448 . . . . 5  |-  ( k  =  K  ->  ( ! `  ( N  -  k ) )  =  ( ! `  ( N  -  K
) ) )
13 fveq2 5444 . . . . 5  |-  ( k  =  K  ->  ( ! `  k )  =  ( ! `  K ) )
1412, 13oveq12d 5796 . . . 4  |-  ( k  =  K  ->  (
( ! `  ( N  -  k )
)  x.  ( ! `
 k ) )  =  ( ( ! `
 ( N  -  K ) )  x.  ( ! `  K
) ) )
1514oveq2d 5794 . . 3  |-  ( k  =  K  ->  (
( ! `  N
)  /  ( ( ! `  ( N  -  k ) )  x.  ( ! `  k ) ) )  =  ( ( ! `
 N )  / 
( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
) ) )
16 eqidd 2257 . . 3  |-  ( k  =  K  ->  0  =  0 )
1710, 15, 16ifbieq12d 3547 . 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 11268 . 2  |-  _C  =  ( n  e.  NN0 ,  k  e.  ZZ  |->  if ( k  e.  ( 0 ... n ) ,  ( ( ! `
 n )  / 
( ( ! `  ( n  -  k
) )  x.  ( ! `  k )
) ) ,  0 ) )
19 ovex 5803 . . 3  |-  ( ( ! `  N )  /  ( ( ! `
 ( N  -  K ) )  x.  ( ! `  K
) ) )  e. 
_V
20 c0ex 8786 . . 3  |-  0  e.  _V
2119, 20ifex 3583 . 2  |-  if ( K  e.  ( 0 ... N ) ,  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) ,  0 )  e.  _V
229, 17, 18, 21ovmpt2 5903 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 6    /\ wa 360    = wceq 1619    e. wcel 1621   ifcif 3525   ` cfv 4659  (class class class)co 5778   0cc0 8691    x. cmul 8696    - cmin 8991    / cdiv 9377   NN0cn0 9918   ZZcz 9977   ...cfz 10734   !cfa 11240    _C cbc 11267
This theorem is referenced by:  bcval2  11270  bcval3  11271
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-mulcl 8753  ax-i2m1 8759
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-bc 11268
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