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Metamath Proof Explorer

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 ) )