Metamath Proof Explorer

Theorem divmuldivi

Description: Multiplication of two ratios. Theorem I.14 of Apostol p. 18. (Contributed by NM, 16-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
Assertion divmuldivi ( ( 𝐴 / 𝐵 ) · ( 𝐶 / 𝐷 ) ) = ( ( 𝐴 · 𝐶 ) / ( 𝐵 · 𝐷 ) )

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 2 5 pm3.2i ⊢ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 )
8 4 6 pm3.2i ⊢ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 )
9 divmuldiv ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ∧ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) ) → ( ( 𝐴 / 𝐵 ) · ( 𝐶 / 𝐷 ) ) = ( ( 𝐴 · 𝐶 ) / ( 𝐵 · 𝐷 ) ) )
10 1 3 7 8 9 mp4an ⊢ ( ( 𝐴 / 𝐵 ) · ( 𝐶 / 𝐷 ) ) = ( ( 𝐴 · 𝐶 ) / ( 𝐵 · 𝐷 ) )