Step |
Hyp |
Ref |
Expression |
1 |
|
1p1times |
|- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( A + A ) ) |
2 |
|
ax-1cn |
|- 1 e. CC |
3 |
2 2
|
addcli |
|- ( 1 + 1 ) e. CC |
4 |
3
|
mul01i |
|- ( ( 1 + 1 ) x. 0 ) = 0 |
5 |
|
negid |
|- ( A e. CC -> ( A + -u A ) = 0 ) |
6 |
4 5
|
eqtr4id |
|- ( A e. CC -> ( ( 1 + 1 ) x. 0 ) = ( A + -u A ) ) |
7 |
1 6
|
eqeq12d |
|- ( A e. CC -> ( ( ( 1 + 1 ) x. A ) = ( ( 1 + 1 ) x. 0 ) <-> ( A + A ) = ( A + -u A ) ) ) |
8 |
|
id |
|- ( A e. CC -> A e. CC ) |
9 |
|
0cnd |
|- ( A e. CC -> 0 e. CC ) |
10 |
3
|
a1i |
|- ( A e. CC -> ( 1 + 1 ) e. CC ) |
11 |
|
1re |
|- 1 e. RR |
12 |
11 11
|
readdcli |
|- ( 1 + 1 ) e. RR |
13 |
|
0lt1 |
|- 0 < 1 |
14 |
11 11 13 13
|
addgt0ii |
|- 0 < ( 1 + 1 ) |
15 |
12 14
|
gt0ne0ii |
|- ( 1 + 1 ) =/= 0 |
16 |
15
|
a1i |
|- ( A e. CC -> ( 1 + 1 ) =/= 0 ) |
17 |
8 9 10 16
|
mulcand |
|- ( A e. CC -> ( ( ( 1 + 1 ) x. A ) = ( ( 1 + 1 ) x. 0 ) <-> A = 0 ) ) |
18 |
|
negcl |
|- ( A e. CC -> -u A e. CC ) |
19 |
8 8 18
|
addcand |
|- ( A e. CC -> ( ( A + A ) = ( A + -u A ) <-> A = -u A ) ) |
20 |
7 17 19
|
3bitr3rd |
|- ( A e. CC -> ( A = -u A <-> A = 0 ) ) |