Theorem bcval 13091

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 13092 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.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . 4 |- ( n = N -> ( 0 ... n ) = ( 0 ... N ) )
2	1	eleq2d 2687	. . 3 |- ( n = N -> ( k e. ( 0 ... n ) <-> k e. ( 0 ... N ) ) )
3		fveq2 6191	. . . 4 |- ( n = N -> ( ! ` n ) = ( ! ` N ) )
4		oveq1 6657	. . . . . 6 |- ( n = N -> ( n - k ) = ( N - k ) )
5	4	fveq2d 6195	. . . . 5 |- ( n = N -> ( ! ` ( n - k ) ) = ( ! ` ( N - k ) ) )
6	5	oveq1d 6665	. . . 4 |- ( n = N -> ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) = ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) )
7	3, 6	oveq12d 6668	. . 3 |- ( n = N -> ( ( ! ` n ) / ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) ) = ( ( ! ` N ) / ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) ) )
8	2, 7	ifbieq1d 4109	. 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 ) )
9		eleq1 2689	. . 3 |- ( k = K -> ( k e. ( 0 ... N ) <-> K e. ( 0 ... N ) ) )
10		oveq2 6658	. . . . . 6 |- ( k = K -> ( N - k ) = ( N - K ) )
11	10	fveq2d 6195	. . . . 5 |- ( k = K -> ( ! ` ( N - k ) ) = ( ! ` ( N - K ) ) )
12		fveq2 6191	. . . . 5 |- ( k = K -> ( ! ` k ) = ( ! ` K ) )
13	11, 12	oveq12d 6668	. . . 4 |- ( k = K -> ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) = ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) )
14	13	oveq2d 6666	. . 3 |- ( k = K -> ( ( ! ` N ) / ( ( ! ` ( N - k ) ) x. ( ! ` k ) ) ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) )
15	9, 14	ifbieq1d 4109	. 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 ) )
16		df-bc 13090	. 2 |- _C = ( n e. NN0 , k e. ZZ |-> if ( k e. ( 0 ... n ) , ( ( ! ` n ) / ( ( ! ` ( n - k ) ) x. ( ! ` k ) ) ) , 0 ) )
17		ovex 6678	. . 3 |- ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) e. _V
18		c0ex 10034	. . 3 |- 0 e. _V
19	17, 18	ifex 4156	. 2 |- if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) e. _V
20	8, 15, 16, 19	ovmpt2 6796	1 |- ( ( N e. NN0 /\ K e. ZZ ) -> ( N _C K ) = if ( K e. ( 0 ... N ) , ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) , 0 ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   ` cfv 5888  (class class class)co 6650   0cc0 9936   x. cmul 9941   - cmin 10266   / cdiv 10684   NN0cn0 11292   ZZcz 11377   ...cfz 12326   !cfa 13060   _C cbc 13089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-bc 13090

This theorem is referenced by:  bcval2  13092  bcval3  13093  bcneg1  31622  bccolsum  31625  fwddifnp1  32272