Metamath Proof Explorer
Description: A number is equal to the reciprocal of its reciprocal. Theorem I.10
of Apostol p. 18. (Contributed by NM, 9-Feb-1995)
|
|
Ref |
Expression |
|
Hypotheses |
divclz.1 |
⊢ 𝐴 ∈ ℂ |
|
|
reccl.2 |
⊢ 𝐴 ≠ 0 |
|
Assertion |
recreci |
⊢ ( 1 / ( 1 / 𝐴 ) ) = 𝐴 |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
divclz.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
reccl.2 |
⊢ 𝐴 ≠ 0 |
3 |
|
recrec |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / ( 1 / 𝐴 ) ) = 𝐴 ) |
4 |
1 2 3
|
mp2an |
⊢ ( 1 / ( 1 / 𝐴 ) ) = 𝐴 |