Theorem divmul13i

Description: Swap denominators of two ratios. (Contributed by NM, 6-Aug-1999)

	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	divmul13i	⊢ ( ( 𝐴 / 𝐵 ) · ( 𝐶 / 𝐷 ) ) = ( ( 𝐶 / 𝐵 ) · ( 𝐴 / 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	3 1	mulcomi	⊢ ( 𝐶 · 𝐴 ) = ( 𝐴 · 𝐶 )
8	7	oveq1i	⊢ ( ( 𝐶 · 𝐴 ) / ( 𝐵 · 𝐷 ) ) = ( ( 𝐴 · 𝐶 ) / ( 𝐵 · 𝐷 ) )
9	3 2 1 4 5 6	divmuldivi	⊢ ( ( 𝐶 / 𝐵 ) · ( 𝐴 / 𝐷 ) ) = ( ( 𝐶 · 𝐴 ) / ( 𝐵 · 𝐷 ) )
10	1 2 3 4 5 6	divmuldivi	⊢ ( ( 𝐴 / 𝐵 ) · ( 𝐶 / 𝐷 ) ) = ( ( 𝐴 · 𝐶 ) / ( 𝐵 · 𝐷 ) )
11	8 9 10	3eqtr4ri	⊢ ( ( 𝐴 / 𝐵 ) · ( 𝐶 / 𝐷 ) ) = ( ( 𝐶 / 𝐵 ) · ( 𝐴 / 𝐷 ) )