Theorem seradd

Description: The sum of two infinite series. (Contributed by NM, 17-Mar-2005) (Revised by Mario Carneiro, 26-May-2014)

	Ref	Expression
Hypotheses	seradd.1	|- ( ph -> N e. ( ZZ>= ` M ) )
	seradd.2	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. CC )
	seradd.3	|- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) e. CC )
	seradd.4	|- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( ( F ` k ) + ( G ` k ) ) )
Assertion	seradd	|- ( ph -> ( seq M ( + , H ) ` N ) = ( ( seq M ( + , F ) ` N ) + ( seq M ( + , G ) ` N ) ) )

Proof

Step	Hyp	Ref	Expression
1		seradd.1	|- ( ph -> N e. ( ZZ>= ` M ) )
2		seradd.2	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. CC )
3		seradd.3	|- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) e. CC )
4		seradd.4	|- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( ( F ` k ) + ( G ` k ) ) )
5		addcl	|- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
6	5	adantl	|- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
7		addcom	|- ( ( x e. CC /\ y e. CC ) -> ( x + y ) = ( y + x ) )
8	7	adantl	|- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) = ( y + x ) )
9		addass	|- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( ( x + y ) + z ) = ( x + ( y + z ) ) )
10	9	adantl	|- ( ( ph /\ ( x e. CC /\ y e. CC /\ z e. CC ) ) -> ( ( x + y ) + z ) = ( x + ( y + z ) ) )
11	6 8 10 1 2 3 4	seqcaopr	|- ( ph -> ( seq M ( + , H ) ` N ) = ( ( seq M ( + , F ) ` N ) + ( seq M ( + , G ) ` N ) ) )