Metamath Proof Explorer


Theorem mulcan2d

Description: Cancellation law for multiplication. Theorem I.7 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses mulcand.1 ( 𝜑𝐴 ∈ ℂ )
mulcand.2 ( 𝜑𝐵 ∈ ℂ )
mulcand.3 ( 𝜑𝐶 ∈ ℂ )
mulcand.4 ( 𝜑𝐶 ≠ 0 )
Assertion mulcan2d ( 𝜑 → ( ( 𝐴 · 𝐶 ) = ( 𝐵 · 𝐶 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 mulcand.1 ( 𝜑𝐴 ∈ ℂ )
2 mulcand.2 ( 𝜑𝐵 ∈ ℂ )
3 mulcand.3 ( 𝜑𝐶 ∈ ℂ )
4 mulcand.4 ( 𝜑𝐶 ≠ 0 )
5 1 3 mulcomd ( 𝜑 → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
6 2 3 mulcomd ( 𝜑 → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
7 5 6 eqeq12d ( 𝜑 → ( ( 𝐴 · 𝐶 ) = ( 𝐵 · 𝐶 ) ↔ ( 𝐶 · 𝐴 ) = ( 𝐶 · 𝐵 ) ) )
8 1 2 3 4 mulcand ( 𝜑 → ( ( 𝐶 · 𝐴 ) = ( 𝐶 · 𝐵 ) ↔ 𝐴 = 𝐵 ) )
9 7 8 bitrd ( 𝜑 → ( ( 𝐴 · 𝐶 ) = ( 𝐵 · 𝐶 ) ↔ 𝐴 = 𝐵 ) )