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Metamath Proof Explorer

Theorem divmul13

Description: Swap the denominators in the product of two ratios. (Contributed by NM, 3-May-2005)

Ref Expression
Assertion divmul13 ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) ) → ( ( 𝐴 / 𝐶 ) · ( 𝐵 / 𝐷 ) ) = ( ( 𝐵 / 𝐶 ) · ( 𝐴 / 𝐷 ) ) )

Proof

Step Hyp Ref Expression
1 mulcom ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
2 1 adantr ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
3 2 oveq1d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) ) → ( ( 𝐴 · 𝐵 ) / ( 𝐶 · 𝐷 ) ) = ( ( 𝐵 · 𝐴 ) / ( 𝐶 · 𝐷 ) ) )
4 divmuldiv ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) ) → ( ( 𝐴 / 𝐶 ) · ( 𝐵 / 𝐷 ) ) = ( ( 𝐴 · 𝐵 ) / ( 𝐶 · 𝐷 ) ) )
5 divmuldiv ⊢ ( ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) ∧ ( ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) ) → ( ( 𝐵 / 𝐶 ) · ( 𝐴 / 𝐷 ) ) = ( ( 𝐵 · 𝐴 ) / ( 𝐶 · 𝐷 ) ) )
6 5 ancom1s ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) ) → ( ( 𝐵 / 𝐶 ) · ( 𝐴 / 𝐷 ) ) = ( ( 𝐵 · 𝐴 ) / ( 𝐶 · 𝐷 ) ) )
7 3 4 6 3eqtr4d ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) ) → ( ( 𝐴 / 𝐶 ) · ( 𝐵 / 𝐷 ) ) = ( ( 𝐵 / 𝐶 ) · ( 𝐴 / 𝐷 ) ) )