Theorem div32

Description: A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	div32	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 / 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐶 / 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		div23	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 · 𝐶 ) / 𝐵 ) = ( ( 𝐴 / 𝐵 ) · 𝐶 ) )
2		divass	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 · 𝐶 ) / 𝐵 ) = ( 𝐴 · ( 𝐶 / 𝐵 ) ) )
3	1 2	eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 / 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐶 / 𝐵 ) ) )
4	3	3com23	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 / 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐶 / 𝐵 ) ) )