Description: Define inversion, which maps a nonzero extended complex number or element of the complex projective line (Riemann sphere) to its inverse. Beware of the overloading: the equality ( invc0 ) = infty is to be understood in the complex projective line, but 0 as an extended complex number does not have an inverse, which we can state as ( invc0 ) e/ CCbar . Note that this definition relies on df-bj-mulc , which does not bypass ordinary complex multiplication, but defines extended complex multiplication on top of it. Therefore, we could have used directly / instead of ( iota_ ... .cc ... ) . (Contributed by BJ, 22-Jun-2019)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | df-bj-invc | |- invc = ( x e. ( CCbar u. CChat ) |-> if ( x = 0 , infty , if ( x e. CC , ( iota_ y e. CC ( x .cc y ) = 1 ) , 0 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | cinvc | |- invc |
|
| 1 | vx | |- x |
|
| 2 | cccbar | |- CCbar |
|
| 3 | ccchat | |- CChat |
|
| 4 | 2 3 | cun | |- ( CCbar u. CChat ) |
| 5 | 1 | cv | |- x |
| 6 | cc0 | |- 0 |
|
| 7 | 5 6 | wceq | |- x = 0 |
| 8 | cinfty | |- infty |
|
| 9 | cc | |- CC |
|
| 10 | 5 9 | wcel | |- x e. CC |
| 11 | vy | |- y |
|
| 12 | cmulc | |- .cc |
|
| 13 | 11 | cv | |- y |
| 14 | 5 13 12 | co | |- ( x .cc y ) |
| 15 | c1 | |- 1 |
|
| 16 | 14 15 | wceq | |- ( x .cc y ) = 1 |
| 17 | 16 11 9 | crio | |- ( iota_ y e. CC ( x .cc y ) = 1 ) |
| 18 | 10 17 6 | cif | |- if ( x e. CC , ( iota_ y e. CC ( x .cc y ) = 1 ) , 0 ) |
| 19 | 7 8 18 | cif | |- if ( x = 0 , infty , if ( x e. CC , ( iota_ y e. CC ( x .cc y ) = 1 ) , 0 ) ) |
| 20 | 1 4 19 | cmpt | |- ( x e. ( CCbar u. CChat ) |-> if ( x = 0 , infty , if ( x e. CC , ( iota_ y e. CC ( x .cc y ) = 1 ) , 0 ) ) ) |
| 21 | 0 20 | wceq | |- invc = ( x e. ( CCbar u. CChat ) |-> if ( x = 0 , infty , if ( x e. CC , ( iota_ y e. CC ( x .cc y ) = 1 ) , 0 ) ) ) |