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Description: The imaginary unit is the square root of negative 1. A lot of people like to call this the "definition" of _i , but the definition of sqrt df-sqrt has already been crafted with _i being mentioned explicitly, and in any case it doesn't make too much sense to define a value based on a function evaluated outside its domain. A more appropriate view is to take ax-i2m1 or i2 as the "definition", and simply postulate the existence of a number satisfying this property. This is the approach we take here. (Contributed by Mario Carneiro, 10-Jul-2013)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | sqrtm1 | ⊢ i = ( √ ‘ - 1 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 1re | ⊢ 1 ∈ ℝ | |
| 2 | 0le1 | ⊢ 0 ≤ 1 | |
| 3 | sqrtneg | ⊢ ( ( 1 ∈ ℝ ∧ 0 ≤ 1 ) → ( √ ‘ - 1 ) = ( i · ( √ ‘ 1 ) ) ) | |
| 4 | 1 2 3 | mp2an | ⊢ ( √ ‘ - 1 ) = ( i · ( √ ‘ 1 ) ) |
| 5 | sqrt1 | ⊢ ( √ ‘ 1 ) = 1 | |
| 6 | 5 | oveq2i | ⊢ ( i · ( √ ‘ 1 ) ) = ( i · 1 ) |
| 7 | ax-icn | ⊢ i ∈ ℂ | |
| 8 | 7 | mulridi | ⊢ ( i · 1 ) = i |
| 9 | 4 6 8 | 3eqtrri | ⊢ i = ( √ ‘ - 1 ) |