Description: The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017)
Ref | Expression | ||
---|---|---|---|
Hypotheses | mulcncf.1 | |- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) ) |
|
mulcncf.2 | |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) ) |
||
Assertion | mulcncf | |- ( ph -> ( x e. X |-> ( A x. B ) ) e. ( X -cn-> CC ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulcncf.1 | |- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) ) |
|
2 | mulcncf.2 | |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) ) |
|
3 | eqid | |- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
|
4 | 3 | mulcn | |- x. e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) ) |
5 | 4 | a1i | |- ( ph -> x. e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) ) ) |
6 | 3 5 1 2 | cncfmpt2f | |- ( ph -> ( x e. X |-> ( A x. B ) ) e. ( X -cn-> CC ) ) |