Metamath Proof Explorer


Theorem divmuleqd

Description: Cross-multiply in an equality of ratios. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses div1d.1 ( 𝜑𝐴 ∈ ℂ )
divcld.2 ( 𝜑𝐵 ∈ ℂ )
divmuld.3 ( 𝜑𝐶 ∈ ℂ )
divmuldivd.4 ( 𝜑𝐷 ∈ ℂ )
divmuldivd.5 ( 𝜑𝐵 ≠ 0 )
divmuldivd.6 ( 𝜑𝐷 ≠ 0 )
Assertion divmuleqd ( 𝜑 → ( ( 𝐴 / 𝐵 ) = ( 𝐶 / 𝐷 ) ↔ ( 𝐴 · 𝐷 ) = ( 𝐶 · 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 div1d.1 ( 𝜑𝐴 ∈ ℂ )
2 divcld.2 ( 𝜑𝐵 ∈ ℂ )
3 divmuld.3 ( 𝜑𝐶 ∈ ℂ )
4 divmuldivd.4 ( 𝜑𝐷 ∈ ℂ )
5 divmuldivd.5 ( 𝜑𝐵 ≠ 0 )
6 divmuldivd.6 ( 𝜑𝐷 ≠ 0 )
7 2 5 jca ( 𝜑 → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) )
8 4 6 jca ( 𝜑 → ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) )
9 divmuleq ( ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ∧ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) ) → ( ( 𝐴 / 𝐵 ) = ( 𝐶 / 𝐷 ) ↔ ( 𝐴 · 𝐷 ) = ( 𝐶 · 𝐵 ) ) )
10 1 3 7 8 9 syl22anc ( 𝜑 → ( ( 𝐴 / 𝐵 ) = ( 𝐶 / 𝐷 ) ↔ ( 𝐴 · 𝐷 ) = ( 𝐶 · 𝐵 ) ) )