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Theorem bcval3 11271
Description: Value of the binomial coefficient,  N choose  K, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
Assertion
Ref Expression
bcval3  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  0 )

Proof of Theorem bcval3
StepHypRef Expression
1 bcval 11269 . . 3  |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  _C  K
)  =  if ( K  e.  ( 0 ... N ) ,  ( ( ! `  N )  /  (
( ! `  ( N  -  K )
)  x.  ( ! `
 K ) ) ) ,  0 ) )
213adant3 980 . 2  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  if ( K  e.  ( 0 ... N ) ,  ( ( ! `
 N )  / 
( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
) ) ,  0 ) )
3 iffalse 3532 . . 3  |-  ( -.  K  e.  ( 0 ... N )  ->  if ( K  e.  ( 0 ... N ) ,  ( ( ! `
 N )  / 
( ( ! `  ( N  -  K
) )  x.  ( ! `  K )
) ) ,  0 )  =  0 )
433ad2ant3 983 . 2  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  if ( K  e.  ( 0 ... N ) ,  ( ( ! `  N
)  /  ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) ) ) ,  0 )  =  0 )
52, 4eqtrd 2288 1  |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0 ... N ) )  ->  ( N  _C  K )  =  0 )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ w3a 939    = 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:  bcval4  11272  bccmpl  11274  bcval5  11282  bcpasc  11285  bccl  11286  hashbc  11342  binomlem  12238
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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