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Metamath Proof Explorer
Description: A number is zero iff its square is zero (where square is represented
using multiplication). (Contributed by Mario Carneiro, 27-May-2016)
|
|
Ref |
Expression |
|
Hypothesis |
msq0d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
Assertion |
msq0d |
⊢ ( 𝜑 → ( ( 𝐴 · 𝐴 ) = 0 ↔ 𝐴 = 0 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
msq0d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
| 2 |
1 1
|
mul0ord |
⊢ ( 𝜑 → ( ( 𝐴 · 𝐴 ) = 0 ↔ ( 𝐴 = 0 ∨ 𝐴 = 0 ) ) ) |
| 3 |
|
oridm |
⊢ ( ( 𝐴 = 0 ∨ 𝐴 = 0 ) ↔ 𝐴 = 0 ) |
| 4 |
2 3
|
bitrdi |
⊢ ( 𝜑 → ( ( 𝐴 · 𝐴 ) = 0 ↔ 𝐴 = 0 ) ) |