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Metamath Proof Explorer

Theorem div32d

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

Ref Expression
Hypotheses div1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
divcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
divmuld.3 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
divmuld.4 ⊢ ( 𝜑 → 𝐵 ≠ 0 )
Assertion div32d ( 𝜑 → ( ( 𝐴 / 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐶 / 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 div1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 divcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 divmuld.3 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4 divmuld.4 ⊢ ( 𝜑 → 𝐵 ≠ 0 )
5 div32 ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 / 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐶 / 𝐵 ) ) )
6 1 2 4 3 5 syl121anc ⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐶 / 𝐵 ) ) )