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Metamath Proof Explorer
Description: Division into zero is zero. (Contributed by NM, 14-Mar-2005) (Proof
shortened by SN, 9-Jul-2025)
|
|
Ref |
Expression |
|
Assertion |
div0 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 0 / 𝐴 ) = 0 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
0cn |
⊢ 0 ∈ ℂ |
| 2 |
|
eqid |
⊢ 0 = 0 |
| 3 |
|
diveq0 |
⊢ ( ( 0 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( 0 / 𝐴 ) = 0 ↔ 0 = 0 ) ) |
| 4 |
2 3
|
mpbiri |
⊢ ( ( 0 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 0 / 𝐴 ) = 0 ) |
| 5 |
1 4
|
mp3an1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 0 / 𝐴 ) = 0 ) |