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Metamath Proof Explorer

Theorem div23

Description: A commutative/associative law for division. (Contributed by NM, 2-Aug-2004)

Ref Expression
Assertion div23 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 · 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) · 𝐵 ) )

Proof

Step Hyp Ref Expression
1 mulcom ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
2 1 oveq1d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) / 𝐶 ) = ( ( 𝐵 · 𝐴 ) / 𝐶 ) )
3 2 3adant3 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 · 𝐵 ) / 𝐶 ) = ( ( 𝐵 · 𝐴 ) / 𝐶 ) )
4 divass ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐵 · 𝐴 ) / 𝐶 ) = ( 𝐵 · ( 𝐴 / 𝐶 ) ) )
5 4 3com12 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐵 · 𝐴 ) / 𝐶 ) = ( 𝐵 · ( 𝐴 / 𝐶 ) ) )
6 simp2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → 𝐵 ∈ ℂ )
7 divcl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) → ( 𝐴 / 𝐶 ) ∈ ℂ )
8 7 3expb ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 / 𝐶 ) ∈ ℂ )
9 8 3adant2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 / 𝐶 ) ∈ ℂ )
10 6 9 mulcomd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐵 · ( 𝐴 / 𝐶 ) ) = ( ( 𝐴 / 𝐶 ) · 𝐵 ) )
11 3 5 10 3eqtrd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 · 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) · 𝐵 ) )