Theorem bccval 38537

Description: Value of the generalized binomial coefficient, C choose K. (Contributed by Steve Rodriguez, 22-Apr-2020.)

Hypotheses

Ref	Expression
bccval.c	|- ( ph -> C e. CC )
bccval.k	|- ( ph -> K e. NN0 )

Assertion

Ref	Expression
bccval	|- ( ph -> ( CC𝑐 K ) = ( ( C FallFac K ) / ( ! ` K ) ) )

Proof of Theorem bccval

Dummy variables k c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bcc 38536	. . 3 |- C𝑐 = ( c e. CC , k e. NN0 |-> ( ( c FallFac k ) / ( ! ` k ) ) )
2	1	a1i 11	. 2 |- ( ph -> C𝑐 = ( c e. CC , k e. NN0 |-> ( ( c FallFac k ) / ( ! ` k ) ) ) )
3		simprl 794	. . . 4 |- ( ( ph /\ ( c = C /\ k = K ) ) -> c = C )
4		simprr 796	. . . 4 |- ( ( ph /\ ( c = C /\ k = K ) ) -> k = K )
5	3, 4	oveq12d 6668	. . 3 |- ( ( ph /\ ( c = C /\ k = K ) ) -> ( c FallFac k ) = ( C FallFac K ) )
6	4	fveq2d 6195	. . 3 |- ( ( ph /\ ( c = C /\ k = K ) ) -> ( ! ` k ) = ( ! ` K ) )
7	5, 6	oveq12d 6668	. 2 |- ( ( ph /\ ( c = C /\ k = K ) ) -> ( ( c FallFac k ) / ( ! ` k ) ) = ( ( C FallFac K ) / ( ! ` K ) ) )
8		bccval.c	. 2 |- ( ph -> C e. CC )
9		bccval.k	. 2 |- ( ph -> K e. NN0 )
10		ovexd 6680	. 2 |- ( ph -> ( ( C FallFac K ) / ( ! ` K ) ) e. _V )
11	2, 7, 8, 9, 10	ovmpt2d 6788	1 |- ( ph -> ( CC𝑐 K ) = ( ( C FallFac K ) / ( ! ` K ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   / cdiv 10684   NN0cn0 11292   !cfa 13060   FallFac cfallfac 14735  C_𝑐cbcc 38535

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

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-bcc 38536

This theorem is referenced by:  bcccl  38538  bcc0  38539  bccp1k  38540  bccn0  38542  bccbc  38544  binomcxplemwb  38547