Step |
Hyp |
Ref |
Expression |
1 |
|
0dp2dp.a |
|- A e. NN0 |
2 |
|
0dp2dp.b |
|- B e. RR+ |
3 |
|
0p1e1 |
|- ( 0 + 1 ) = 1 |
4 |
|
0z |
|- 0 e. ZZ |
5 |
|
1z |
|- 1 e. ZZ |
6 |
1 2 3 4 5
|
dpexpp1 |
|- ( ( A . B ) x. ( ; 1 0 ^ 0 ) ) = ( ( 0 . _ A B ) x. ( ; 1 0 ^ 1 ) ) |
7 |
|
10nn0 |
|- ; 1 0 e. NN0 |
8 |
7
|
nn0cni |
|- ; 1 0 e. CC |
9 |
|
exp0 |
|- ( ; 1 0 e. CC -> ( ; 1 0 ^ 0 ) = 1 ) |
10 |
8 9
|
ax-mp |
|- ( ; 1 0 ^ 0 ) = 1 |
11 |
10
|
oveq2i |
|- ( ( A . B ) x. ( ; 1 0 ^ 0 ) ) = ( ( A . B ) x. 1 ) |
12 |
|
exp1 |
|- ( ; 1 0 e. CC -> ( ; 1 0 ^ 1 ) = ; 1 0 ) |
13 |
8 12
|
ax-mp |
|- ( ; 1 0 ^ 1 ) = ; 1 0 |
14 |
13
|
oveq2i |
|- ( ( 0 . _ A B ) x. ( ; 1 0 ^ 1 ) ) = ( ( 0 . _ A B ) x. ; 1 0 ) |
15 |
6 11 14
|
3eqtr3ri |
|- ( ( 0 . _ A B ) x. ; 1 0 ) = ( ( A . B ) x. 1 ) |
16 |
1 2
|
rpdpcl |
|- ( A . B ) e. RR+ |
17 |
|
rpcn |
|- ( ( A . B ) e. RR+ -> ( A . B ) e. CC ) |
18 |
16 17
|
ax-mp |
|- ( A . B ) e. CC |
19 |
|
mulid1 |
|- ( ( A . B ) e. CC -> ( ( A . B ) x. 1 ) = ( A . B ) ) |
20 |
18 19
|
ax-mp |
|- ( ( A . B ) x. 1 ) = ( A . B ) |
21 |
15 20
|
eqtri |
|- ( ( 0 . _ A B ) x. ; 1 0 ) = ( A . B ) |