Description: Reciprocal is one-to-one. (Contributed by Mario Carneiro, 27-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | div1d.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℂ ) | |
divcld.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℂ ) | ||
divne0d.3 | ⊢ ( 𝜑 → 𝐴 ≠ 0 ) | ||
divne0d.4 | ⊢ ( 𝜑 → 𝐵 ≠ 0 ) | ||
rec11d.5 | ⊢ ( 𝜑 → ( 1 / 𝐴 ) = ( 1 / 𝐵 ) ) | ||
Assertion | rec11d | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | div1d.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℂ ) | |
2 | divcld.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℂ ) | |
3 | divne0d.3 | ⊢ ( 𝜑 → 𝐴 ≠ 0 ) | |
4 | divne0d.4 | ⊢ ( 𝜑 → 𝐵 ≠ 0 ) | |
5 | rec11d.5 | ⊢ ( 𝜑 → ( 1 / 𝐴 ) = ( 1 / 𝐵 ) ) | |
6 | rec11 | ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 1 / 𝐴 ) = ( 1 / 𝐵 ) ↔ 𝐴 = 𝐵 ) ) | |
7 | 1 3 2 4 6 | syl22anc | ⊢ ( 𝜑 → ( ( 1 / 𝐴 ) = ( 1 / 𝐵 ) ↔ 𝐴 = 𝐵 ) ) |
8 | 5 7 | mpbid | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) |