Metamath Proof Explorer
Description: If two complex numbers are unequal, their quotient is not one.
Contrapositive of diveq1d . (Contributed by David Moews, 28-Feb-2017)
|
|
Ref |
Expression |
|
Hypotheses |
div1d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
divcld.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
|
|
divcld.3 |
⊢ ( 𝜑 → 𝐵 ≠ 0 ) |
|
|
divne1d.4 |
⊢ ( 𝜑 → 𝐴 ≠ 𝐵 ) |
|
Assertion |
divne1d |
⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ≠ 1 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
div1d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
divcld.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
3 |
|
divcld.3 |
⊢ ( 𝜑 → 𝐵 ≠ 0 ) |
4 |
|
divne1d.4 |
⊢ ( 𝜑 → 𝐴 ≠ 𝐵 ) |
5 |
1 2 3
|
diveq1ad |
⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) = 1 ↔ 𝐴 = 𝐵 ) ) |
6 |
5
|
necon3bid |
⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) ≠ 1 ↔ 𝐴 ≠ 𝐵 ) ) |
7 |
4 6
|
mpbird |
⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ≠ 1 ) |