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 freeLMod ( 0 ... 𝑁 ) )
	frlmfzwrd.b	⊢ 𝐵 = ( Base ‘ 𝑊 )
	frlmfzwrd.s	⊢ 𝑆 = ( Base ‘ 𝐾 )
Assertion	frlmfzwrd	⊢ ( 𝑋 ∈ 𝐵 → 𝑋 ∈ Word 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		frlmfzwrd.w	⊢ 𝑊 = ( 𝐾 freeLMod ( 0 ... 𝑁 ) )
2		frlmfzwrd.b	⊢ 𝐵 = ( Base ‘ 𝑊 )
3		frlmfzwrd.s	⊢ 𝑆 = ( Base ‘ 𝐾 )
4		ovex	⊢ ( 0 ... 𝑁 ) ∈ V
5	1 3 2	frlmbasf	⊢ ( ( ( 0 ... 𝑁 ) ∈ V ∧ 𝑋 ∈ 𝐵 ) → 𝑋 : ( 0 ... 𝑁 ) ⟶ 𝑆 )
6	4 5	mpan	⊢ ( 𝑋 ∈ 𝐵 → 𝑋 : ( 0 ... 𝑁 ) ⟶ 𝑆 )
7		ffz0iswrd	⊢ ( 𝑋 : ( 0 ... 𝑁 ) ⟶ 𝑆 → 𝑋 ∈ Word 𝑆 )
8	6 7	syl	⊢ ( 𝑋 ∈ 𝐵 → 𝑋 ∈ Word 𝑆 )