Metamath Proof Explorer


Theorem isummulc2

Description: An infinite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005) (Revised by Mario Carneiro, 23-Apr-2014)

Ref Expression
Hypotheses isumcl.1 Z = M
isumcl.2 φ M
isumcl.3 φ k Z F k = A
isumcl.4 φ k Z A
isumcl.5 φ seq M + F dom
summulc.6 φ B
Assertion isummulc2 φ B k Z A = k Z B A

Proof

Step Hyp Ref Expression
1 isumcl.1 Z = M
2 isumcl.2 φ M
3 isumcl.3 φ k Z F k = A
4 isumcl.4 φ k Z A
5 isumcl.5 φ seq M + F dom
6 summulc.6 φ B
7 eqidd φ m Z k Z B A m = k Z B A m
8 6 adantr φ k Z B
9 8 4 mulcld φ k Z B A
10 9 fmpttd φ k Z B A : Z
11 10 ffvelrnda φ m Z k Z B A m
12 1 2 3 4 5 isumclim2 φ seq M + F k Z A
13 3 4 eqeltrd φ k Z F k
14 13 ralrimiva φ k Z F k
15 fveq2 k = m F k = F m
16 15 eleq1d k = m F k F m
17 16 rspccva k Z F k m Z F m
18 14 17 sylan φ m Z F m
19 simpr φ k Z k Z
20 ovex B A V
21 eqid k Z B A = k Z B A
22 21 fvmpt2 k Z B A V k Z B A k = B A
23 19 20 22 sylancl φ k Z k Z B A k = B A
24 3 oveq2d φ k Z B F k = B A
25 23 24 eqtr4d φ k Z k Z B A k = B F k
26 25 ralrimiva φ k Z k Z B A k = B F k
27 nffvmpt1 _ k k Z B A m
28 27 nfeq1 k k Z B A m = B F m
29 fveq2 k = m k Z B A k = k Z B A m
30 15 oveq2d k = m B F k = B F m
31 29 30 eqeq12d k = m k Z B A k = B F k k Z B A m = B F m
32 28 31 rspc m Z k Z k Z B A k = B F k k Z B A m = B F m
33 26 32 mpan9 φ m Z k Z B A m = B F m
34 1 2 6 12 18 33 isermulc2 φ seq M + k Z B A B k Z A
35 1 2 7 11 34 isumclim φ m Z k Z B A m = B k Z A
36 sumfc m Z k Z B A m = k Z B A
37 35 36 eqtr3di φ B k Z A = k Z B A