Metamath Proof Explorer


Theorem addcand

Description: Cancellation law for addition. Theorem I.1 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses muld.1 ( 𝜑𝐴 ∈ ℂ )
addcomd.2 ( 𝜑𝐵 ∈ ℂ )
addcand.3 ( 𝜑𝐶 ∈ ℂ )
Assertion addcand ( 𝜑 → ( ( 𝐴 + 𝐵 ) = ( 𝐴 + 𝐶 ) ↔ 𝐵 = 𝐶 ) )

Proof

Step Hyp Ref Expression
1 muld.1 ( 𝜑𝐴 ∈ ℂ )
2 addcomd.2 ( 𝜑𝐵 ∈ ℂ )
3 addcand.3 ( 𝜑𝐶 ∈ ℂ )
4 addcan ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) = ( 𝐴 + 𝐶 ) ↔ 𝐵 = 𝐶 ) )
5 1 2 3 4 syl3anc ( 𝜑 → ( ( 𝐴 + 𝐵 ) = ( 𝐴 + 𝐶 ) ↔ 𝐵 = 𝐶 ) )