Metamath Proof Explorer


Theorem fprodclf

Description: Closure of a finite product of complex numbers. A version of fprodcl using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020)

Ref Expression
Hypotheses fprodclf.kph
|- F/ k ph
fprodclf.a
|- ( ph -> A e. Fin )
fprodclf.b
|- ( ( ph /\ k e. A ) -> B e. CC )
Assertion fprodclf
|- ( ph -> prod_ k e. A B e. CC )

Proof

Step Hyp Ref Expression
1 fprodclf.kph
 |-  F/ k ph
2 fprodclf.a
 |-  ( ph -> A e. Fin )
3 fprodclf.b
 |-  ( ( ph /\ k e. A ) -> B e. CC )
4 ssidd
 |-  ( ph -> CC C_ CC )
5 mulcl
 |-  ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
6 5 adantl
 |-  ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
7 1cnd
 |-  ( ph -> 1 e. CC )
8 1 4 6 2 3 7 fprodcllemf
 |-  ( ph -> prod_ k e. A B e. CC )