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Metamath Proof Explorer

Theorem isummulc1

Description: An infinite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005) (Revised by Mario Carneiro, 23-Apr-2014)

Ref Expression
Hypotheses isumcl.1 
|- Z = ( ZZ>= ` M )
isumcl.2 
|- ( ph -> M e. ZZ )
isumcl.3 
|- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
isumcl.4 
|- ( ( ph /\ k e. Z ) -> A e. CC )
isumcl.5 
|- ( ph -> seq M ( + , F ) e. dom ~~> )
summulc.6 
|- ( ph -> B e. CC )
Assertion isummulc1 
|- ( ph -> ( sum_ k e. Z A x. B ) = sum_ k e. Z ( A x. B ) )

Proof

Step Hyp Ref Expression
1 isumcl.1 
 |-  Z = ( ZZ>= ` M )
2 isumcl.2 
 |-  ( ph -> M e. ZZ )
3 isumcl.3 
 |-  ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
4 isumcl.4 
 |-  ( ( ph /\ k e. Z ) -> A e. CC )
5 isumcl.5 
 |-  ( ph -> seq M ( + , F ) e. dom ~~> )
6 summulc.6 
 |-  ( ph -> B e. CC )
7 1 2 3 4 5 6 isummulc2 
 |-  ( ph -> ( B x. sum_ k e. Z A ) = sum_ k e. Z ( B x. A ) )
8 1 2 3 4 5 isumcl 
 |-  ( ph -> sum_ k e. Z A e. CC )
9 8 6 mulcomd 
 |-  ( ph -> ( sum_ k e. Z A x. B ) = ( B x. sum_ k e. Z A ) )
10 6 adantr 
 |-  ( ( ph /\ k e. Z ) -> B e. CC )
11 4 10 mulcomd 
 |-  ( ( ph /\ k e. Z ) -> ( A x. B ) = ( B x. A ) )
12 11 sumeq2dv 
 |-  ( ph -> sum_ k e. Z ( A x. B ) = sum_ k e. Z ( B x. A ) )
13 7 9 12 3eqtr4d 
 |-  ( ph -> ( sum_ k e. Z A x. B ) = sum_ k e. Z ( A x. B ) )