# Metamath Proof Explorer

## Theorem dmdcan

Description: Cancellation law for division and multiplication. (Contributed by Scott Fenton, 7-Jun-2013) (Proof shortened by Fan Zheng, 3-Jul-2016)

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

### Proof

Step Hyp Ref Expression
1 simp1l ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → 𝐴 ∈ ℂ )
2 simp3 ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → 𝐶 ∈ ℂ )
3 simp1r ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → 𝐴 ≠ 0 )
4 divcl ( ( 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝐶 / 𝐴 ) ∈ ℂ )
5 2 1 3 4 syl3anc ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( 𝐶 / 𝐴 ) ∈ ℂ )
6 simp2l ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → 𝐵 ∈ ℂ )
7 simp2r ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → 𝐵 ≠ 0 )
8 div23 ( ( 𝐴 ∈ ℂ ∧ ( 𝐶 / 𝐴 ) ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 · ( 𝐶 / 𝐴 ) ) / 𝐵 ) = ( ( 𝐴 / 𝐵 ) · ( 𝐶 / 𝐴 ) ) )
9 1 5 6 7 8 syl112anc ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · ( 𝐶 / 𝐴 ) ) / 𝐵 ) = ( ( 𝐴 / 𝐵 ) · ( 𝐶 / 𝐴 ) ) )
10 divcan2 ( ( 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝐴 · ( 𝐶 / 𝐴 ) ) = 𝐶 )
11 2 1 3 10 syl3anc ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · ( 𝐶 / 𝐴 ) ) = 𝐶 )
12 11 oveq1d ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · ( 𝐶 / 𝐴 ) ) / 𝐵 ) = ( 𝐶 / 𝐵 ) )
13 9 12 eqtr3d ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 / 𝐵 ) · ( 𝐶 / 𝐴 ) ) = ( 𝐶 / 𝐵 ) )