This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Theorem 0cn
Description: Zero is a complex number. See also 0cnALT . (Contributed by NM, 19-Feb-2005)
|
|
Ref |
Expression |
|
Assertion |
0cn |
⊢ 0 ∈ ℂ |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ax-i2m1 |
⊢ ( ( i · i ) + 1 ) = 0 |
| 2 |
|
ax-icn |
⊢ i ∈ ℂ |
| 3 |
|
mulcl |
⊢ ( ( i ∈ ℂ ∧ i ∈ ℂ ) → ( i · i ) ∈ ℂ ) |
| 4 |
2 2 3
|
mp2an |
⊢ ( i · i ) ∈ ℂ |
| 5 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 6 |
|
addcl |
⊢ ( ( ( i · i ) ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( i · i ) + 1 ) ∈ ℂ ) |
| 7 |
4 5 6
|
mp2an |
⊢ ( ( i · i ) + 1 ) ∈ ℂ |
| 8 |
1 7
|
eqeltrri |
⊢ 0 ∈ ℂ |