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Metamath Proof Explorer
Description: A number divided by itself is one. (Contributed by NM, 1-Aug-2004)
(Proof shortened by SN, 9-Jul-2025)
|
|
Ref |
Expression |
|
Assertion |
divid |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝐴 / 𝐴 ) = 1 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
eqid |
⊢ 𝐴 = 𝐴 |
| 2 |
|
diveq1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( 𝐴 / 𝐴 ) = 1 ↔ 𝐴 = 𝐴 ) ) |
| 3 |
1 2
|
mpbiri |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝐴 / 𝐴 ) = 1 ) |
| 4 |
3
|
3anidm12 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝐴 / 𝐴 ) = 1 ) |