Description: The multiplication operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 6-Oct-2015) (Revised by Thierry Arnoux, 17-Dec-2017) Revise df-cnfld . (Revised by GG, 27-Apr-2025)
Ref | Expression | ||
---|---|---|---|
Assertion | cnfldmul | |- x. = ( .r ` CCfld ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-mulf | |- x. : ( CC X. CC ) --> CC |
|
2 | ffn | |- ( x. : ( CC X. CC ) --> CC -> x. Fn ( CC X. CC ) ) |
|
3 | 1 2 | ax-mp | |- x. Fn ( CC X. CC ) |
4 | fnov | |- ( x. Fn ( CC X. CC ) <-> x. = ( x e. CC , y e. CC |-> ( x x. y ) ) ) |
|
5 | 3 4 | mpbi | |- x. = ( x e. CC , y e. CC |-> ( x x. y ) ) |
6 | mpocnfldmul | |- ( x e. CC , y e. CC |-> ( x x. y ) ) = ( .r ` CCfld ) |
|
7 | 5 6 | eqtri | |- x. = ( .r ` CCfld ) |