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 ) |