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Metamath Proof Explorer

Theorem divdivdivd

Description: Division of two ratios. Theorem I.15 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses div1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
divcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
divmuld.3 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
divmuldivd.4 ⊢ ( 𝜑 → 𝐷 ∈ ℂ )
divmuldivd.5 ⊢ ( 𝜑 → 𝐵 ≠ 0 )
divmuldivd.6 ⊢ ( 𝜑 → 𝐷 ≠ 0 )
divdivdivd.7 ⊢ ( 𝜑 → 𝐶 ≠ 0 )
Assertion divdivdivd ( 𝜑 → ( ( 𝐴 / 𝐵 ) / ( 𝐶 / 𝐷 ) ) = ( ( 𝐴 · 𝐷 ) / ( 𝐵 · 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 div1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 divcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 divmuld.3 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4 divmuldivd.4 ⊢ ( 𝜑 → 𝐷 ∈ ℂ )
5 divmuldivd.5 ⊢ ( 𝜑 → 𝐵 ≠ 0 )
6 divmuldivd.6 ⊢ ( 𝜑 → 𝐷 ≠ 0 )
7 divdivdivd.7 ⊢ ( 𝜑 → 𝐶 ≠ 0 )
8 2 5 jca ⊢ ( 𝜑 → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) )
9 3 7 jca ⊢ ( 𝜑 → ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) )
10 4 6 jca ⊢ ( 𝜑 → ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) )
11 divdivdiv ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) ∧ ( ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) ) → ( ( 𝐴 / 𝐵 ) / ( 𝐶 / 𝐷 ) ) = ( ( 𝐴 · 𝐷 ) / ( 𝐵 · 𝐶 ) ) )
12 1 8 9 10 11 syl22anc ⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) / ( 𝐶 / 𝐷 ) ) = ( ( 𝐴 · 𝐷 ) / ( 𝐵 · 𝐶 ) ) )