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Metamath Proof Explorer

Theorem divreci

Description: Relationship between division and reciprocal. Theorem I.9 of Apostol p. 18. (Contributed by NM, 9-Feb-1995)

Ref Expression
Hypotheses divclz.1 ⊢ A ∈ ℂ
divclz.2 ⊢ B ∈ ℂ
divcl.3 ⊢ B ≠ 0
Assertion divreci ⊢ A B = A ⁢ 1 B

Proof

Step Hyp Ref Expression
1 divclz.1 ⊢ A ∈ ℂ
2 divclz.2 ⊢ B ∈ ℂ
3 divcl.3 ⊢ B ≠ 0
4 1 2 divreczi ⊢ B ≠ 0 → A B = A ⁢ 1 B
5 3 4 ax-mp ⊢ A B = A ⁢ 1 B