Metamath Proof Explorer


Theorem frlmfzowrd

Description: A vector of a module with indices from 0 to N - 1 is a word over the scalars of the module. (Contributed by SN, 31-Aug-2023)

Ref Expression
Hypotheses frlmfzowrd.w
|- W = ( K freeLMod ( 0 ..^ N ) )
frlmfzowrd.b
|- B = ( Base ` W )
frlmfzowrd.s
|- S = ( Base ` K )
Assertion frlmfzowrd
|- ( X e. B -> X e. Word S )

Proof

Step Hyp Ref Expression
1 frlmfzowrd.w
 |-  W = ( K freeLMod ( 0 ..^ N ) )
2 frlmfzowrd.b
 |-  B = ( Base ` W )
3 frlmfzowrd.s
 |-  S = ( Base ` K )
4 ovex
 |-  ( 0 ..^ N ) e. _V
5 1 3 2 frlmbasf
 |-  ( ( ( 0 ..^ N ) e. _V /\ X e. B ) -> X : ( 0 ..^ N ) --> S )
6 4 5 mpan
 |-  ( X e. B -> X : ( 0 ..^ N ) --> S )
7 iswrdi
 |-  ( X : ( 0 ..^ N ) --> S -> X e. Word S )
8 6 7 syl
 |-  ( X e. B -> X e. Word S )