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Metamath Proof Explorer

Theorem div12

Description: A commutative/associative law for division. (Contributed by NM, 30-Apr-2005)

Ref Expression
Assertion div12 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 · ( 𝐵 / 𝐶 ) ) = ( 𝐵 · ( 𝐴 / 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 divcl ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) → ( 𝐵 / 𝐶 ) ∈ ℂ )
2 1 3expb ⊢ ( ( 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐵 / 𝐶 ) ∈ ℂ )
3 mulcom ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 / 𝐶 ) ∈ ℂ ) → ( 𝐴 · ( 𝐵 / 𝐶 ) ) = ( ( 𝐵 / 𝐶 ) · 𝐴 ) )
4 2 3 sylan2 ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) ) → ( 𝐴 · ( 𝐵 / 𝐶 ) ) = ( ( 𝐵 / 𝐶 ) · 𝐴 ) )
5 4 3impb ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 · ( 𝐵 / 𝐶 ) ) = ( ( 𝐵 / 𝐶 ) · 𝐴 ) )
6 div13 ⊢ ( ( 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ∧ 𝐴 ∈ ℂ ) → ( ( 𝐵 / 𝐶 ) · 𝐴 ) = ( ( 𝐴 / 𝐶 ) · 𝐵 ) )
7 6 3comr ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐵 / 𝐶 ) · 𝐴 ) = ( ( 𝐴 / 𝐶 ) · 𝐵 ) )
8 divcl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) → ( 𝐴 / 𝐶 ) ∈ ℂ )
9 8 3expb ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 / 𝐶 ) ∈ ℂ )
10 mulcom ⊢ ( ( ( 𝐴 / 𝐶 ) ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 / 𝐶 ) · 𝐵 ) = ( 𝐵 · ( 𝐴 / 𝐶 ) ) )
11 9 10 stoic3 ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 / 𝐶 ) · 𝐵 ) = ( 𝐵 · ( 𝐴 / 𝐶 ) ) )
12 11 3com23 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) · 𝐵 ) = ( 𝐵 · ( 𝐴 / 𝐶 ) ) )
13 5 7 12 3eqtrd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 · ( 𝐵 / 𝐶 ) ) = ( 𝐵 · ( 𝐴 / 𝐶 ) ) )