let K be Ring; :: thesis: for G being VectorSpace-Sequence of K
for r being Element of K
for v being Element of product (carr G)
for i being Element of dom (carr G) holds
( ([:(multop G):] . (r,v)) . i = the lmult of (G . i) . (r,(v . i)) & ( for vi being Vector of (G . i) st vi = v . i holds
([:(multop G):] . (r,v)) . i = r * vi ) )
let G be VectorSpace-Sequence of K; :: thesis: for r being Element of K
for v being Element of product (carr G)
for i being Element of dom (carr G) holds
( ([:(multop G):] . (r,v)) . i = the lmult of (G . i) . (r,(v . i)) & ( for vi being Vector of (G . i) st vi = v . i holds
([:(multop G):] . (r,v)) . i = r * vi ) )
let r be Element of K; :: thesis: for v being Element of product (carr G)
for i being Element of dom (carr G) holds
( ([:(multop G):] . (r,v)) . i = the lmult of (G . i) . (r,(v . i)) & ( for vi being Vector of (G . i) st vi = v . i holds
([:(multop G):] . (r,v)) . i = r * vi ) )
let v be Element of product (carr G); :: thesis: for i being Element of dom (carr G) holds
( ([:(multop G):] . (r,v)) . i = the lmult of (G . i) . (r,(v . i)) & ( for vi being Vector of (G . i) st vi = v . i holds
([:(multop G):] . (r,v)) . i = r * vi ) )
let i be Element of dom (carr G); :: thesis: ( ([:(multop G):] . (r,v)) . i = the lmult of (G . i) . (r,(v . i)) & ( for vi being Vector of (G . i) st vi = v . i holds
([:(multop G):] . (r,v)) . i = r * vi ) )
([:(multop G):] . (r,v)) . i = ((multop G) . i) . (r,(v . i)) by PRVECT_2:def 2;
hence ( ([:(multop G):] . (r,v)) . i = the lmult of (G . i) . (r,(v . i)) & ( for vi being Vector of (G . i) st vi = v . i holds
([:(multop G):] . (r,v)) . i = r * vi ) ) by Def8; :: thesis: verum