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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 )