Theorem div13

Description: A commutative/associative law for division. (Contributed by NM, 22-Apr-2005)

	Ref	Expression
Assertion	div13	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 / 𝐵 ) · 𝐶 ) = ( ( 𝐶 / 𝐵 ) · 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		mulcom	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
2	1	oveq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐶 ) / 𝐵 ) = ( ( 𝐶 · 𝐴 ) / 𝐵 ) )
3	2	3adant2	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐶 ) / 𝐵 ) = ( ( 𝐶 · 𝐴 ) / 𝐵 ) )
4		div23	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 · 𝐶 ) / 𝐵 ) = ( ( 𝐴 / 𝐵 ) · 𝐶 ) )
5	4	3com23	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐶 ) / 𝐵 ) = ( ( 𝐴 / 𝐵 ) · 𝐶 ) )
6		div23	⊢ ( ( 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐶 · 𝐴 ) / 𝐵 ) = ( ( 𝐶 / 𝐵 ) · 𝐴 ) )
7	6	3coml	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝐶 · 𝐴 ) / 𝐵 ) = ( ( 𝐶 / 𝐵 ) · 𝐴 ) )
8	3 5 7	3eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 / 𝐵 ) · 𝐶 ) = ( ( 𝐶 / 𝐵 ) · 𝐴 ) )