Theorem divcan2

Description: A cancellation law for division. (Contributed by NM, 3-Feb-2004) (Revised by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	divcan2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 · ( 𝐴 / 𝐵 ) ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( 𝐴 / 𝐵 ) = ( 𝐴 / 𝐵 )
2		simp1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → 𝐴 ∈ ℂ )
3		divcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 / 𝐵 ) ∈ ℂ )
4		3simpc	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) )
5		divmul	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐴 / 𝐵 ) ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 / 𝐵 ) = ( 𝐴 / 𝐵 ) ↔ ( 𝐵 · ( 𝐴 / 𝐵 ) ) = 𝐴 ) )
6	2 3 4 5	syl3anc	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = ( 𝐴 / 𝐵 ) ↔ ( 𝐵 · ( 𝐴 / 𝐵 ) ) = 𝐴 ) )
7	1 6	mpbii	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 · ( 𝐴 / 𝐵 ) ) = 𝐴 )