Description: A factor of a nonzero complex number is nonzero. Partial converse of mulne0d and consequence of mulne0bd . (Contributed by David Moews, 28-Feb-2017)
Ref | Expression | ||
---|---|---|---|
Hypotheses | mulne0bad.1 | |- ( ph -> A e. CC ) |
|
mulne0bad.2 | |- ( ph -> B e. CC ) |
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mulne0bad.3 | |- ( ph -> ( A x. B ) =/= 0 ) |
||
Assertion | mulne0bbd | |- ( ph -> B =/= 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulne0bad.1 | |- ( ph -> A e. CC ) |
|
2 | mulne0bad.2 | |- ( ph -> B e. CC ) |
|
3 | mulne0bad.3 | |- ( ph -> ( A x. B ) =/= 0 ) |
|
4 | 1 2 | mulne0bd | |- ( ph -> ( ( A =/= 0 /\ B =/= 0 ) <-> ( A x. B ) =/= 0 ) ) |
5 | 3 4 | mpbird | |- ( ph -> ( A =/= 0 /\ B =/= 0 ) ) |
6 | 5 | simprd | |- ( ph -> B =/= 0 ) |