Theorem frlmfzwrd

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

	Ref	Expression
Hypotheses	frlmfzwrd.w	$$⊢W=K\mathrm{freeLMod}\left(0\dots N\right)$$
	frlmfzwrd.b	$$⊢B={\mathrm{Base}}_W$$
	frlmfzwrd.s	$$⊢S={\mathrm{Base}}_K$$
Assertion	frlmfzwrd	$$⊢X\in B\to X\in \mathrm{Word}S$$

Proof

Step	Hyp	Ref	Expression
1		frlmfzwrd.w	$$⊢W=K\mathrm{freeLMod}\left(0\dots N\right)$$
2		frlmfzwrd.b	$$⊢B={\mathrm{Base}}_W$$
3		frlmfzwrd.s	$$⊢S={\mathrm{Base}}_K$$
4		ovex	$$⊢\left(0\dots N\right)\in \mathrm{V}$$
5	1 3 2	frlmbasf	$$⊢\left(\left(0\dots N\right)\in \mathrm{V}\wedge X\in B\right)\to X:\left(0\dots N\right)⟶S$$
6	4 5	mpan	$$⊢X\in B\to X:\left(0\dots N\right)⟶S$$
7		ffz0iswrd	$$⊢X:\left(0\dots N\right)⟶S\to X\in \mathrm{Word}S$$
8	6 7	syl	$$⊢X\in B\to X\in \mathrm{Word}S$$