Description: The quotient of two complex numbers is one iff they are equal. Deduction form of diveq1 . Generalization of diveq1d . (Contributed by David Moews, 28-Feb-2017)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | div1d.1 | |- ( ph -> A e. CC ) |
|
| divcld.2 | |- ( ph -> B e. CC ) |
||
| divcld.3 | |- ( ph -> B =/= 0 ) |
||
| Assertion | diveq1ad | |- ( ph -> ( ( A / B ) = 1 <-> A = B ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | div1d.1 | |- ( ph -> A e. CC ) |
|
| 2 | divcld.2 | |- ( ph -> B e. CC ) |
|
| 3 | divcld.3 | |- ( ph -> B =/= 0 ) |
|
| 4 | diveq1 | |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( ( A / B ) = 1 <-> A = B ) ) |
|
| 5 | 1 2 3 4 | syl3anc | |- ( ph -> ( ( A / B ) = 1 <-> A = B ) ) |