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Metamath Proof Explorer

Theorem divdivdivi

Description: Division of two ratios. Theorem I.15 of Apostol p. 18. (Contributed by NM, 22-Feb-1995)

Ref Expression
Hypotheses divclz.1 ⊢ 𝐴 ∈ ℂ
divclz.2 ⊢ 𝐵 ∈ ℂ
divmulz.3 ⊢ 𝐶 ∈ ℂ
divmuldiv.4 ⊢ 𝐷 ∈ ℂ
divmuldiv.5 ⊢ 𝐵 ≠ 0
divmuldiv.6 ⊢ 𝐷 ≠ 0
divdivdiv.7 ⊢ 𝐶 ≠ 0
Assertion divdivdivi ( ( 𝐴 / 𝐵 ) / ( 𝐶 / 𝐷 ) ) = ( ( 𝐴 · 𝐷 ) / ( 𝐵 · 𝐶 ) )

Proof

Step Hyp Ref Expression
1 divclz.1 ⊢ 𝐴 ∈ ℂ
2 divclz.2 ⊢ 𝐵 ∈ ℂ
3 divmulz.3 ⊢ 𝐶 ∈ ℂ
4 divmuldiv.4 ⊢ 𝐷 ∈ ℂ
5 divmuldiv.5 ⊢ 𝐵 ≠ 0
6 divmuldiv.6 ⊢ 𝐷 ≠ 0
7 divdivdiv.7 ⊢ 𝐶 ≠ 0
8 2 5 pm3.2i ⊢ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 )
9 3 7 pm3.2i ⊢ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 )
10 4 6 pm3.2i ⊢ ( 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 mp4an ⊢ ( ( 𝐴 / 𝐵 ) / ( 𝐶 / 𝐷 ) ) = ( ( 𝐴 · 𝐷 ) / ( 𝐵 · 𝐶 ) )