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Theorem bcval3 11313
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 11311 . . 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 3573 . . 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 2316 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 1628    e. wcel 1688   ifcif 3566   ` cfv 5221  (class class class)co 5819   0cc0 8732    x. cmul 8737    - cmin 9032    / cdiv 9418   NN0cn0 9960   ZZcz 10019   ...cfz 10776   !cfa 11282    _C cbc 11309
This theorem is referenced by:  bcval4  11314  bccmpl  11316  bcval5  11324  bcpasc  11327  bccl  11328  hashbc  11385  binomlem  12281
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-mulcl 8794  ax-i2m1 8800
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-bc 11310
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