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Metamath Proof Explorer

Theorem divcan3zi

Description: A cancellation law for division. (Eliminates a hypothesis of divcan3i with the weak deduction theorem.) (Contributed by NM, 3-Feb-2004)

	Ref	Expression
Hypotheses	divclz.1	⊢ 𝐴 ∈ ℂ
	divclz.2	⊢ 𝐵 ∈ ℂ
Assertion	divcan3zi	⊢ ( 𝐵 ≠ 0 → ( ( 𝐵 · 𝐴 ) / 𝐵 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		divclz.1	⊢ 𝐴 ∈ ℂ
2		divclz.2	⊢ 𝐵 ∈ ℂ
3		divcan3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐵 · 𝐴 ) / 𝐵 ) = 𝐴 )
4	1 2 3	mp3an12	⊢ ( 𝐵 ≠ 0 → ( ( 𝐵 · 𝐴 ) / 𝐵 ) = 𝐴 )