Definition df-rgmod 19945

Description: Any ring can be regarded as a left algebra over itself. The function ringLMod associates with any ring the left algebra consisting in the ring itself regarded as a left algebra over itself. It has an inner product which is simply the ring product. (Contributed by Stefan O'Rear, 6-Dec-2014.)

Assertion

Ref	Expression
df-rgmod	⊢ ringLMod = (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤)))

Detailed syntax breakdown of Definition df-rgmod

Step	Hyp	Ref	Expression
1		crglmod 19941	. 2 class ringLMod
2		vw	. . 3 setvar 𝑤
3		cvv 3494	. . 3 class V
4	2	cv 1536	. . . . 5 class 𝑤
5		cbs 16483	. . . . 5 class Base
6	4, 5	cfv 6355	. . . 4 class (Base‘𝑤)
7		csra 19940	. . . . 5 class subringAlg
8	4, 7	cfv 6355	. . . 4 class (subringAlg ‘𝑤)
9	6, 8	cfv 6355	. . 3 class ((subringAlg ‘𝑤)‘(Base‘𝑤))
10	2, 3, 9	cmpt 5146	. 2 class (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤)))
11	1, 10	wceq 1537	1 wff ringLMod = (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤)))

This definition is referenced by:  rlmfn  19962  rlmval  19963