This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dmdcan2d

Description: Cancellation law for division and multiplication. (Contributed by David Moews, 28-Feb-2017)

Ref Expression
Hypotheses div1d.1 ⊢ φ → A ∈ ℂ
divcld.2 ⊢ φ → B ∈ ℂ
divmuld.3 ⊢ φ → C ∈ ℂ
divmuld.4 ⊢ φ → B ≠ 0
divdiv23d.5 ⊢ φ → C ≠ 0
Assertion dmdcan2d ⊢ φ → A B ⁢ B C = A C

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 divdiv23d.5 ⊢ φ → C ≠ 0
6 1 2 4 divcld ⊢ φ → A B ∈ ℂ
7 2 3 5 divcld ⊢ φ → B C ∈ ℂ
8 6 7 mulcomd ⊢ φ → A B ⁢ B C = B C ⁢ A B
9 1 2 3 4 5 dmdcand ⊢ φ → B C ⁢ A B = A C
10 8 9 eqtrd ⊢ φ → A B ⁢ B C = A C