Step |
Hyp |
Ref |
Expression |
1 |
|
9cn |
|- 9 e. CC |
2 |
|
10nn0 |
|- ; 1 0 e. NN0 |
3 |
2
|
nn0cni |
|- ; 1 0 e. CC |
4 |
|
ax-1cn |
|- 1 e. CC |
5 |
3 4
|
mulcli |
|- ( ; 1 0 x. 1 ) e. CC |
6 |
1 5 4
|
adddii |
|- ( 9 x. ( ( ; 1 0 x. 1 ) + 1 ) ) = ( ( 9 x. ( ; 1 0 x. 1 ) ) + ( 9 x. 1 ) ) |
7 |
3
|
mulid1i |
|- ( ; 1 0 x. 1 ) = ; 1 0 |
8 |
7
|
oveq2i |
|- ( 9 x. ( ; 1 0 x. 1 ) ) = ( 9 x. ; 1 0 ) |
9 |
1 3
|
mulcomi |
|- ( 9 x. ; 1 0 ) = ( ; 1 0 x. 9 ) |
10 |
8 9
|
eqtri |
|- ( 9 x. ( ; 1 0 x. 1 ) ) = ( ; 1 0 x. 9 ) |
11 |
1
|
mulid1i |
|- ( 9 x. 1 ) = 9 |
12 |
10 11
|
oveq12i |
|- ( ( 9 x. ( ; 1 0 x. 1 ) ) + ( 9 x. 1 ) ) = ( ( ; 1 0 x. 9 ) + 9 ) |
13 |
6 12
|
eqtri |
|- ( 9 x. ( ( ; 1 0 x. 1 ) + 1 ) ) = ( ( ; 1 0 x. 9 ) + 9 ) |
14 |
|
dfdec10 |
|- ; 1 1 = ( ( ; 1 0 x. 1 ) + 1 ) |
15 |
14
|
oveq2i |
|- ( 9 x. ; 1 1 ) = ( 9 x. ( ( ; 1 0 x. 1 ) + 1 ) ) |
16 |
|
dfdec10 |
|- ; 9 9 = ( ( ; 1 0 x. 9 ) + 9 ) |
17 |
13 15 16
|
3eqtr4i |
|- ( 9 x. ; 1 1 ) = ; 9 9 |