Theorem mulcani

Description: Cancellation law for multiplication. Theorem I.7 of Apostol p. 18. (Contributed by NM, 26-Jan-1995)

	Ref	Expression
Hypotheses	mulcan.1	⊢ 𝐴 ∈ ℂ
	mulcan.2	⊢ 𝐵 ∈ ℂ
	mulcan.3	⊢ 𝐶 ∈ ℂ
	mulcan.4	⊢ 𝐶 ≠ 0
Assertion	mulcani	⊢ ( ( 𝐶 · 𝐴 ) = ( 𝐶 · 𝐵 ) ↔ 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		mulcan.1	⊢ 𝐴 ∈ ℂ
2		mulcan.2	⊢ 𝐵 ∈ ℂ
3		mulcan.3	⊢ 𝐶 ∈ ℂ
4		mulcan.4	⊢ 𝐶 ≠ 0
5	3 4	pm3.2i	⊢ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 )
6		mulcan	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐶 · 𝐴 ) = ( 𝐶 · 𝐵 ) ↔ 𝐴 = 𝐵 ) )
7	1 2 5 6	mp3an	⊢ ( ( 𝐶 · 𝐴 ) = ( 𝐶 · 𝐵 ) ↔ 𝐴 = 𝐵 )