Metamath Proof Explorer


Theorem div13d

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

Ref Expression
Hypotheses div1d.1 ( 𝜑𝐴 ∈ ℂ )
divcld.2 ( 𝜑𝐵 ∈ ℂ )
divmuld.3 ( 𝜑𝐶 ∈ ℂ )
divmuld.4 ( 𝜑𝐵 ≠ 0 )
Assertion div13d ( 𝜑 → ( ( 𝐴 / 𝐵 ) · 𝐶 ) = ( ( 𝐶 / 𝐵 ) · 𝐴 ) )

Proof

Step Hyp Ref Expression
1 div1d.1 ( 𝜑𝐴 ∈ ℂ )
2 divcld.2 ( 𝜑𝐵 ∈ ℂ )
3 divmuld.3 ( 𝜑𝐶 ∈ ℂ )
4 divmuld.4 ( 𝜑𝐵 ≠ 0 )
5 div13 ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 / 𝐵 ) · 𝐶 ) = ( ( 𝐶 / 𝐵 ) · 𝐴 ) )
6 1 2 4 3 5 syl121anc ( 𝜑 → ( ( 𝐴 / 𝐵 ) · 𝐶 ) = ( ( 𝐶 / 𝐵 ) · 𝐴 ) )