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Metamath Proof Explorer

Theorem divne1d

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