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Metamath Proof Explorer

Theorem recidd

Description: Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses div1d.1 ⊢ φ → A ∈ ℂ
reccld.2 ⊢ φ → A ≠ 0
Assertion recidd ⊢ φ → A ⁢ 1 A = 1

Proof

Step Hyp Ref Expression
1 div1d.1 ⊢ φ → A ∈ ℂ
2 reccld.2 ⊢ φ → A ≠ 0
3 recid ⊢ A ∈ ℂ ∧ A ≠ 0 → A ⁢ 1 A = 1
4 1 2 3 syl2anc ⊢ φ → A ⁢ 1 A = 1