Definition df-frlm 22053

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 22038 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 ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})))

Distinct variable group:   𝑖,𝑟

Detailed syntax breakdown of Definition df-frlm

Step	Hyp	Ref	Expression
1		cfrlm 22052	. 2 class freeLMod
2		vr	. . 3 setvar 𝑟
3		vi	. . 3 setvar 𝑖
4		cvv 3451	. . 3 class V
5	2	cv 1569	. . . 4 class 𝑟
6	3	cv 1569	. . . . 5 class 𝑖
7		crglmod 21447	. . . . . . 7 class ringLMod
8	5, 7	cfv 6538	. . . . . 6 class (ringLMod‘𝑟)
9	8	csn 4584	. . . . 5 class {(ringLMod‘𝑟)}
10	6, 9	cxp 5649	. . . 4 class (𝑖 × {(ringLMod‘𝑟)})
11		cdsmm 22037	. . . 4 class ⊕m
12	5, 10, 11	co 7420	. . 3 class (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)}))
13	2, 3, 4, 4, 12	cmpo 7422	. 2 class (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})))
14	1, 13	wceq 1570	1 wff freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})))

This definition is used by:  frlmval  22054  frlmrcl  22063  reldmfrlm  43564