Step |
Hyp |
Ref |
Expression |
1 |
|
cxpne0 |
|- ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( A ^c B ) =/= 0 ) |
2 |
1
|
3com23 |
|- ( ( A e. CC /\ B e. CC /\ A =/= 0 ) -> ( A ^c B ) =/= 0 ) |
3 |
2
|
3expia |
|- ( ( A e. CC /\ B e. CC ) -> ( A =/= 0 -> ( A ^c B ) =/= 0 ) ) |
4 |
3
|
necon4d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^c B ) = 0 -> A = 0 ) ) |
5 |
|
ax-1ne0 |
|- 1 =/= 0 |
6 |
|
cxp0 |
|- ( A e. CC -> ( A ^c 0 ) = 1 ) |
7 |
6
|
neeq1d |
|- ( A e. CC -> ( ( A ^c 0 ) =/= 0 <-> 1 =/= 0 ) ) |
8 |
5 7
|
mpbiri |
|- ( A e. CC -> ( A ^c 0 ) =/= 0 ) |
9 |
8
|
adantr |
|- ( ( A e. CC /\ B e. CC ) -> ( A ^c 0 ) =/= 0 ) |
10 |
|
oveq2 |
|- ( B = 0 -> ( A ^c B ) = ( A ^c 0 ) ) |
11 |
10
|
neeq1d |
|- ( B = 0 -> ( ( A ^c B ) =/= 0 <-> ( A ^c 0 ) =/= 0 ) ) |
12 |
9 11
|
syl5ibrcom |
|- ( ( A e. CC /\ B e. CC ) -> ( B = 0 -> ( A ^c B ) =/= 0 ) ) |
13 |
12
|
necon2d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^c B ) = 0 -> B =/= 0 ) ) |
14 |
4 13
|
jcad |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^c B ) = 0 -> ( A = 0 /\ B =/= 0 ) ) ) |
15 |
|
0cxp |
|- ( ( B e. CC /\ B =/= 0 ) -> ( 0 ^c B ) = 0 ) |
16 |
|
oveq1 |
|- ( A = 0 -> ( A ^c B ) = ( 0 ^c B ) ) |
17 |
16
|
eqeq1d |
|- ( A = 0 -> ( ( A ^c B ) = 0 <-> ( 0 ^c B ) = 0 ) ) |
18 |
15 17
|
syl5ibrcom |
|- ( ( B e. CC /\ B =/= 0 ) -> ( A = 0 -> ( A ^c B ) = 0 ) ) |
19 |
18
|
expimpd |
|- ( B e. CC -> ( ( B =/= 0 /\ A = 0 ) -> ( A ^c B ) = 0 ) ) |
20 |
19
|
ancomsd |
|- ( B e. CC -> ( ( A = 0 /\ B =/= 0 ) -> ( A ^c B ) = 0 ) ) |
21 |
20
|
adantl |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A = 0 /\ B =/= 0 ) -> ( A ^c B ) = 0 ) ) |
22 |
14 21
|
impbid |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^c B ) = 0 <-> ( A = 0 /\ B =/= 0 ) ) ) |