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Mirrors > Home > MPE Home > Th. List > df-frlm | Structured version Visualization version GIF version |
Description: The 𝑖-dimensional free module over a ring 𝑟 is the product of 𝑖-many copies of the ring with componentwise addition and multiplication. If 𝑖 is infinite, the allowed vectors are restricted to those with finitely many nonzero coordinates; this ensures that the resulting module is actually spanned by its unit vectors. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
df-frlm | ⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cfrlm 19909 | . 2 class freeLMod | |
2 | vr | . . 3 setvar 𝑟 | |
3 | vi | . . 3 setvar 𝑖 | |
4 | cvv 3173 | . . 3 class V | |
5 | 2 | cv 1474 | . . . 4 class 𝑟 |
6 | 3 | cv 1474 | . . . . 5 class 𝑖 |
7 | crglmod 18990 | . . . . . . 7 class ringLMod | |
8 | 5, 7 | cfv 5804 | . . . . . 6 class (ringLMod‘𝑟) |
9 | 8 | csn 4125 | . . . . 5 class {(ringLMod‘𝑟)} |
10 | 6, 9 | cxp 5036 | . . . 4 class (𝑖 × {(ringLMod‘𝑟)}) |
11 | cdsmm 19894 | . . . 4 class ⊕m | |
12 | 5, 10, 11 | co 6549 | . . 3 class (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})) |
13 | 2, 3, 4, 4, 12 | cmpt2 6551 | . 2 class (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)}))) |
14 | 1, 13 | wceq 1475 | 1 wff freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)}))) |
Colors of variables: wff setvar class |
This definition is referenced by: frlmval 19911 frlmrcl 19920 |
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