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Metamath Proof Explorer
Description: Multiplication of a number and its reciprocal. (Contributed by NM, 22-Jun-2006) (Proof shortened by Mario Carneiro, 27-May-2016)
|
|
Ref |
Expression |
|
Assertion |
recid2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( 1 / 𝐴 ) · 𝐴 ) = 1 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 2 |
|
divcan1 |
⊢ ( ( 1 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( 1 / 𝐴 ) · 𝐴 ) = 1 ) |
| 3 |
1 2
|
mp3an1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( 1 / 𝐴 ) · 𝐴 ) = 1 ) |