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Metamath Proof Explorer

Theorem divreczi

Description: Relationship between division and reciprocal. Theorem I.9 of Apostol p. 18. (Contributed by NM, 11-Oct-1999)

Ref Expression
Hypotheses divclz.1 ⊢ A ∈ ℂ
divclz.2 ⊢ B ∈ ℂ
Assertion divreczi ⊢ B ≠ 0 → A B = A ⁢ 1 B

Proof

Step Hyp Ref Expression
1 divclz.1 ⊢ A ∈ ℂ
2 divclz.2 ⊢ B ∈ ℂ
3 divrec ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ B ≠ 0 → A B = A ⁢ 1 B
4 1 2 3 mp3an12 ⊢ B ≠ 0 → A B = A ⁢ 1 B