Metamath Proof Explorer


Theorem bccn1

Description: Generalized binomial coefficient: C choose 1 . (Contributed by Steve Rodriguez, 22-Apr-2020)

Ref Expression
Hypothesis bccn0.c
|- ( ph -> C e. CC )
Assertion bccn1
|- ( ph -> ( C _Cc 1 ) = C )

Proof

Step Hyp Ref Expression
1 bccn0.c
 |-  ( ph -> C e. CC )
2 0nn0
 |-  0 e. NN0
3 2 a1i
 |-  ( ph -> 0 e. NN0 )
4 1 3 bccp1k
 |-  ( ph -> ( C _Cc ( 0 + 1 ) ) = ( ( C _Cc 0 ) x. ( ( C - 0 ) / ( 0 + 1 ) ) ) )
5 0p1e1
 |-  ( 0 + 1 ) = 1
6 5 oveq2i
 |-  ( C _Cc ( 0 + 1 ) ) = ( C _Cc 1 )
7 6 a1i
 |-  ( ph -> ( C _Cc ( 0 + 1 ) ) = ( C _Cc 1 ) )
8 1 bccn0
 |-  ( ph -> ( C _Cc 0 ) = 1 )
9 1 subid1d
 |-  ( ph -> ( C - 0 ) = C )
10 5 a1i
 |-  ( ph -> ( 0 + 1 ) = 1 )
11 9 10 oveq12d
 |-  ( ph -> ( ( C - 0 ) / ( 0 + 1 ) ) = ( C / 1 ) )
12 1 div1d
 |-  ( ph -> ( C / 1 ) = C )
13 11 12 eqtrd
 |-  ( ph -> ( ( C - 0 ) / ( 0 + 1 ) ) = C )
14 8 13 oveq12d
 |-  ( ph -> ( ( C _Cc 0 ) x. ( ( C - 0 ) / ( 0 + 1 ) ) ) = ( 1 x. C ) )
15 4 7 14 3eqtr3d
 |-  ( ph -> ( C _Cc 1 ) = ( 1 x. C ) )
16 1 mulid2d
 |-  ( ph -> ( 1 x. C ) = C )
17 15 16 eqtrd
 |-  ( ph -> ( C _Cc 1 ) = C )