Metamath Proof Explorer


Theorem dmmcand

Description: Cancellation law for division and multiplication. (Contributed by Glauco Siliprandi, 5-Apr-2020)

Ref Expression
Hypotheses dmmcand.a ( 𝜑𝐴 ∈ ℂ )
dmmcand.b ( 𝜑𝐵 ∈ ℂ )
dmmcand.c ( 𝜑𝐶 ∈ ℂ )
dmmcand.bn0 ( 𝜑𝐵 ≠ 0 )
Assertion dmmcand ( 𝜑 → ( ( 𝐴 / 𝐵 ) · ( 𝐵 · 𝐶 ) ) = ( 𝐴 · 𝐶 ) )

Proof

Step Hyp Ref Expression
1 dmmcand.a ( 𝜑𝐴 ∈ ℂ )
2 dmmcand.b ( 𝜑𝐵 ∈ ℂ )
3 dmmcand.c ( 𝜑𝐶 ∈ ℂ )
4 dmmcand.bn0 ( 𝜑𝐵 ≠ 0 )
5 2 3 mulcld ( 𝜑 → ( 𝐵 · 𝐶 ) ∈ ℂ )
6 1 2 5 4 div32d ( 𝜑 → ( ( 𝐴 / 𝐵 ) · ( 𝐵 · 𝐶 ) ) = ( 𝐴 · ( ( 𝐵 · 𝐶 ) / 𝐵 ) ) )
7 3 2 4 divcan3d ( 𝜑 → ( ( 𝐵 · 𝐶 ) / 𝐵 ) = 𝐶 )
8 7 oveq2d ( 𝜑 → ( 𝐴 · ( ( 𝐵 · 𝐶 ) / 𝐵 ) ) = ( 𝐴 · 𝐶 ) )
9 eqidd ( 𝜑 → ( 𝐴 · 𝐶 ) = ( 𝐴 · 𝐶 ) )
10 6 8 9 3eqtrd ( 𝜑 → ( ( 𝐴 / 𝐵 ) · ( 𝐵 · 𝐶 ) ) = ( 𝐴 · 𝐶 ) )