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Definition df-frlm 20416
Description: Define the function associating with a ring and a set the direct sum indexed by that set of copies of that ring regarded as a left module over itself. Recall from df-dsmm 20401 that an element of a direct sum has finitely many nonzero coordinates. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
df-frlm freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟m (𝑖 × {(ringLMod‘𝑟)})))
Distinct variable group:   𝑖,𝑟

Detailed syntax breakdown of Definition df-frlm
StepHypRef Expression
1 cfrlm 20415 . 2 class freeLMod
2 vr . . 3 setvar 𝑟
3 vi . . 3 setvar 𝑖
4 cvv 3385 . . 3 class V
52cv 1652 . . . 4 class 𝑟
63cv 1652 . . . . 5 class 𝑖
7 crglmod 19492 . . . . . . 7 class ringLMod
85, 7cfv 6101 . . . . . 6 class (ringLMod‘𝑟)
98csn 4368 . . . . 5 class {(ringLMod‘𝑟)}
106, 9cxp 5310 . . . 4 class (𝑖 × {(ringLMod‘𝑟)})
11 cdsmm 20400 . . . 4 class m
125, 10, 11co 6878 . . 3 class (𝑟m (𝑖 × {(ringLMod‘𝑟)}))
132, 3, 4, 4, 12cmpt2 6880 . 2 class (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟m (𝑖 × {(ringLMod‘𝑟)})))
141, 13wceq 1653 1 wff freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟m (𝑖 × {(ringLMod‘𝑟)})))
Colors of variables: wff setvar class
This definition is referenced by:  frlmval  20417  frlmrcl  20426
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