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Definition df-mpl 19128
Description: Define the subalgebra of the power series algebra generated by the variables; this is the polynomial algebra (the set of power series with finite degree). (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.)
Assertion
Ref Expression
df-mpl mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑖 mPwSer 𝑟) / 𝑤(𝑤s {𝑓 ∈ (Base‘𝑤) ∣ 𝑓 finSupp (0g𝑟)}))
Distinct variable group:   𝑓,𝑖,𝑟,𝑤

Detailed syntax breakdown of Definition df-mpl
StepHypRef Expression
1 cmpl 19123 . 2 class mPoly
2 vi . . 3 setvar 𝑖
3 vr . . 3 setvar 𝑟
4 cvv 3173 . . 3 class V
5 vw . . . 4 setvar 𝑤
62cv 1474 . . . . 5 class 𝑖
73cv 1474 . . . . 5 class 𝑟
8 cmps 19121 . . . . 5 class mPwSer
96, 7, 8co 6527 . . . 4 class (𝑖 mPwSer 𝑟)
105cv 1474 . . . . 5 class 𝑤
11 vf . . . . . . . 8 setvar 𝑓
1211cv 1474 . . . . . . 7 class 𝑓
13 c0g 15872 . . . . . . . 8 class 0g
147, 13cfv 5790 . . . . . . 7 class (0g𝑟)
15 cfsupp 8136 . . . . . . 7 class finSupp
1612, 14, 15wbr 4578 . . . . . 6 wff 𝑓 finSupp (0g𝑟)
17 cbs 15644 . . . . . . 7 class Base
1810, 17cfv 5790 . . . . . 6 class (Base‘𝑤)
1916, 11, 18crab 2900 . . . . 5 class {𝑓 ∈ (Base‘𝑤) ∣ 𝑓 finSupp (0g𝑟)}
20 cress 15645 . . . . 5 class s
2110, 19, 20co 6527 . . . 4 class (𝑤s {𝑓 ∈ (Base‘𝑤) ∣ 𝑓 finSupp (0g𝑟)})
225, 9, 21csb 3499 . . 3 class (𝑖 mPwSer 𝑟) / 𝑤(𝑤s {𝑓 ∈ (Base‘𝑤) ∣ 𝑓 finSupp (0g𝑟)})
232, 3, 4, 4, 22cmpt2 6529 . 2 class (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑖 mPwSer 𝑟) / 𝑤(𝑤s {𝑓 ∈ (Base‘𝑤) ∣ 𝑓 finSupp (0g𝑟)}))
241, 23wceq 1475 1 wff mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑖 mPwSer 𝑟) / 𝑤(𝑤s {𝑓 ∈ (Base‘𝑤) ∣ 𝑓 finSupp (0g𝑟)}))
Colors of variables: wff setvar class
This definition is referenced by:  reldmmpl  19197  mplval  19198
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