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 ) |
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 ) |