Description: A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by Mario Carneiro, 27-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypothesis | msq0d.1 | |- ( ph -> A e. CC ) |
|
Assertion | msq0d | |- ( ph -> ( ( A x. A ) = 0 <-> A = 0 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | msq0d.1 | |- ( ph -> A e. CC ) |
|
2 | mul0or | |- ( ( A e. CC /\ A e. CC ) -> ( ( A x. A ) = 0 <-> ( A = 0 \/ A = 0 ) ) ) |
|
3 | 1 1 2 | syl2anc | |- ( ph -> ( ( A x. A ) = 0 <-> ( A = 0 \/ A = 0 ) ) ) |
4 | oridm | |- ( ( A = 0 \/ A = 0 ) <-> A = 0 ) |
|
5 | 3 4 | bitrdi | |- ( ph -> ( ( A x. A ) = 0 <-> A = 0 ) ) |