Step |
Hyp |
Ref |
Expression |
1 |
|
cnre |
|- ( A e. CC -> E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) ) |
2 |
|
0cnd |
|- ( ( x e. RR /\ y e. RR ) -> 0 e. CC ) |
3 |
|
recn |
|- ( x e. RR -> x e. CC ) |
4 |
3
|
adantr |
|- ( ( x e. RR /\ y e. RR ) -> x e. CC ) |
5 |
|
ax-icn |
|- _i e. CC |
6 |
5
|
a1i |
|- ( ( x e. RR /\ y e. RR ) -> _i e. CC ) |
7 |
|
recn |
|- ( y e. RR -> y e. CC ) |
8 |
7
|
adantl |
|- ( ( x e. RR /\ y e. RR ) -> y e. CC ) |
9 |
6 8
|
mulcld |
|- ( ( x e. RR /\ y e. RR ) -> ( _i x. y ) e. CC ) |
10 |
2 4 9
|
adddid |
|- ( ( x e. RR /\ y e. RR ) -> ( 0 x. ( x + ( _i x. y ) ) ) = ( ( 0 x. x ) + ( 0 x. ( _i x. y ) ) ) ) |
11 |
|
remul02 |
|- ( x e. RR -> ( 0 x. x ) = 0 ) |
12 |
11
|
adantr |
|- ( ( x e. RR /\ y e. RR ) -> ( 0 x. x ) = 0 ) |
13 |
|
sn-0tie0 |
|- ( 0 x. _i ) = 0 |
14 |
13
|
oveq1i |
|- ( ( 0 x. _i ) x. y ) = ( 0 x. y ) |
15 |
2 6 8
|
mulassd |
|- ( ( x e. RR /\ y e. RR ) -> ( ( 0 x. _i ) x. y ) = ( 0 x. ( _i x. y ) ) ) |
16 |
|
remul02 |
|- ( y e. RR -> ( 0 x. y ) = 0 ) |
17 |
16
|
adantl |
|- ( ( x e. RR /\ y e. RR ) -> ( 0 x. y ) = 0 ) |
18 |
14 15 17
|
3eqtr3a |
|- ( ( x e. RR /\ y e. RR ) -> ( 0 x. ( _i x. y ) ) = 0 ) |
19 |
12 18
|
oveq12d |
|- ( ( x e. RR /\ y e. RR ) -> ( ( 0 x. x ) + ( 0 x. ( _i x. y ) ) ) = ( 0 + 0 ) ) |
20 |
|
sn-00id |
|- ( 0 + 0 ) = 0 |
21 |
19 20
|
eqtrdi |
|- ( ( x e. RR /\ y e. RR ) -> ( ( 0 x. x ) + ( 0 x. ( _i x. y ) ) ) = 0 ) |
22 |
10 21
|
eqtrd |
|- ( ( x e. RR /\ y e. RR ) -> ( 0 x. ( x + ( _i x. y ) ) ) = 0 ) |
23 |
|
oveq2 |
|- ( A = ( x + ( _i x. y ) ) -> ( 0 x. A ) = ( 0 x. ( x + ( _i x. y ) ) ) ) |
24 |
23
|
eqeq1d |
|- ( A = ( x + ( _i x. y ) ) -> ( ( 0 x. A ) = 0 <-> ( 0 x. ( x + ( _i x. y ) ) ) = 0 ) ) |
25 |
22 24
|
syl5ibrcom |
|- ( ( x e. RR /\ y e. RR ) -> ( A = ( x + ( _i x. y ) ) -> ( 0 x. A ) = 0 ) ) |
26 |
25
|
rexlimivv |
|- ( E. x e. RR E. y e. RR A = ( x + ( _i x. y ) ) -> ( 0 x. A ) = 0 ) |
27 |
1 26
|
syl |
|- ( A e. CC -> ( 0 x. A ) = 0 ) |