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Mirrors > Home > MPE Home > Th. List > df-rgmod | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
df-rgmod | ⊢ ringLMod = (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crglmod 20072 | . 2 class ringLMod | |
2 | vw | . . 3 setvar 𝑤 | |
3 | cvv 3400 | . . 3 class V | |
4 | 2 | cv 1541 | . . . . 5 class 𝑤 |
5 | cbs 16598 | . . . . 5 class Base | |
6 | 4, 5 | cfv 6349 | . . . 4 class (Base‘𝑤) |
7 | csra 20071 | . . . . 5 class subringAlg | |
8 | 4, 7 | cfv 6349 | . . . 4 class (subringAlg ‘𝑤) |
9 | 6, 8 | cfv 6349 | . . 3 class ((subringAlg ‘𝑤)‘(Base‘𝑤)) |
10 | 2, 3, 9 | cmpt 5120 | . 2 class (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤))) |
11 | 1, 10 | wceq 1542 | 1 wff ringLMod = (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤))) |
Colors of variables: wff setvar class |
This definition is referenced by: rlmfn 20093 rlmval 20094 |
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