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Metamath Proof Explorer

Theorem seqdistr

Description: The distributive property for series. (Contributed by Mario Carneiro, 28-Jul-2013) (Revised by Mario Carneiro, 27-May-2014)

Ref Expression
Hypotheses seqdistr.1 
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
seqdistr.2 
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( C T ( x .+ y ) ) = ( ( C T x ) .+ ( C T y ) ) )
seqdistr.3 
|- ( ph -> N e. ( ZZ>= ` M ) )
seqdistr.4 
|- ( ( ph /\ x e. ( M ... N ) ) -> ( G ` x ) e. S )
seqdistr.5 
|- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = ( C T ( G ` x ) ) )
Assertion seqdistr 
|- ( ph -> ( seq M ( .+ , F ) ` N ) = ( C T ( seq M ( .+ , G ) ` N ) ) )

Proof

Step Hyp Ref Expression
1 seqdistr.1 
 |-  ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
2 seqdistr.2 
 |-  ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( C T ( x .+ y ) ) = ( ( C T x ) .+ ( C T y ) ) )
3 seqdistr.3 
 |-  ( ph -> N e. ( ZZ>= ` M ) )
4 seqdistr.4 
 |-  ( ( ph /\ x e. ( M ... N ) ) -> ( G ` x ) e. S )
5 seqdistr.5 
 |-  ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = ( C T ( G ` x ) ) )
6 oveq2 
 |-  ( z = ( x .+ y ) -> ( C T z ) = ( C T ( x .+ y ) ) )
7 eqid 
 |-  ( z e. S |-> ( C T z ) ) = ( z e. S |-> ( C T z ) )
8 ovex 
 |-  ( C T ( x .+ y ) ) e. _V
9 6 7 8 fvmpt 
 |-  ( ( x .+ y ) e. S -> ( ( z e. S |-> ( C T z ) ) ` ( x .+ y ) ) = ( C T ( x .+ y ) ) )
10 1 9 syl 
 |-  ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( ( z e. S |-> ( C T z ) ) ` ( x .+ y ) ) = ( C T ( x .+ y ) ) )
11 oveq2 
 |-  ( z = x -> ( C T z ) = ( C T x ) )
12 ovex 
 |-  ( C T x ) e. _V
13 11 7 12 fvmpt 
 |-  ( x e. S -> ( ( z e. S |-> ( C T z ) ) ` x ) = ( C T x ) )
14 13 ad2antrl 
 |-  ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( ( z e. S |-> ( C T z ) ) ` x ) = ( C T x ) )
15 oveq2 
 |-  ( z = y -> ( C T z ) = ( C T y ) )
16 ovex 
 |-  ( C T y ) e. _V
17 15 7 16 fvmpt 
 |-  ( y e. S -> ( ( z e. S |-> ( C T z ) ) ` y ) = ( C T y ) )
18 17 ad2antll 
 |-  ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( ( z e. S |-> ( C T z ) ) ` y ) = ( C T y ) )
19 14 18 oveq12d 
 |-  ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( ( ( z e. S |-> ( C T z ) ) ` x ) .+ ( ( z e. S |-> ( C T z ) ) ` y ) ) = ( ( C T x ) .+ ( C T y ) ) )
20 2 10 19 3eqtr4d 
 |-  ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( ( z e. S |-> ( C T z ) ) ` ( x .+ y ) ) = ( ( ( z e. S |-> ( C T z ) ) ` x ) .+ ( ( z e. S |-> ( C T z ) ) ` y ) ) )
21 oveq2 
 |-  ( z = ( G ` x ) -> ( C T z ) = ( C T ( G ` x ) ) )
22 ovex 
 |-  ( C T ( G ` x ) ) e. _V
23 21 7 22 fvmpt 
 |-  ( ( G ` x ) e. S -> ( ( z e. S |-> ( C T z ) ) ` ( G ` x ) ) = ( C T ( G ` x ) ) )
24 4 23 syl 
 |-  ( ( ph /\ x e. ( M ... N ) ) -> ( ( z e. S |-> ( C T z ) ) ` ( G ` x ) ) = ( C T ( G ` x ) ) )
25 24 5 eqtr4d 
 |-  ( ( ph /\ x e. ( M ... N ) ) -> ( ( z e. S |-> ( C T z ) ) ` ( G ` x ) ) = ( F ` x ) )
26 1 4 3 20 25 seqhomo 
 |-  ( ph -> ( ( z e. S |-> ( C T z ) ) ` ( seq M ( .+ , G ) ` N ) ) = ( seq M ( .+ , F ) ` N ) )
27 3 4 1 seqcl 
 |-  ( ph -> ( seq M ( .+ , G ) ` N ) e. S )
28 oveq2 
 |-  ( z = ( seq M ( .+ , G ) ` N ) -> ( C T z ) = ( C T ( seq M ( .+ , G ) ` N ) ) )
29 ovex 
 |-  ( C T ( seq M ( .+ , G ) ` N ) ) e. _V
30 28 7 29 fvmpt 
 |-  ( ( seq M ( .+ , G ) ` N ) e. S -> ( ( z e. S |-> ( C T z ) ) ` ( seq M ( .+ , G ) ` N ) ) = ( C T ( seq M ( .+ , G ) ` N ) ) )
31 27 30 syl 
 |-  ( ph -> ( ( z e. S |-> ( C T z ) ) ` ( seq M ( .+ , G ) ` N ) ) = ( C T ( seq M ( .+ , G ) ` N ) ) )
32 26 31 eqtr3d 
 |-  ( ph -> ( seq M ( .+ , F ) ` N ) = ( C T ( seq M ( .+ , G ) ` N ) ) )