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Metamath Proof Explorer

Theorem rec11i

Description: Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999)

Ref Expression
Hypotheses divclz.1 ⊢ A ∈ ℂ
divclz.2 ⊢ B ∈ ℂ
Assertion rec11i ⊢ A ≠ 0 ∧ B ≠ 0 → 1 A = 1 B ↔ A = B

Proof

Step Hyp Ref Expression
1 divclz.1 ⊢ A ∈ ℂ
2 divclz.2 ⊢ B ∈ ℂ
3 rec11 ⊢ A ∈ ℂ ∧ A ≠ 0 ∧ B ∈ ℂ ∧ B ≠ 0 → 1 A = 1 B ↔ A = B
4 3 an4s ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ A ≠ 0 ∧ B ≠ 0 → 1 A = 1 B ↔ A = B
5 1 2 4 mpanl12 ⊢ A ≠ 0 ∧ B ≠ 0 → 1 A = 1 B ↔ A = B