Definition df-mplcoe 14470

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).

The index set (which has an element for each variable) is 𝑖, the coefficients are in ring 𝑟, and for each variable there is a "degree" such that the coefficient is zero for a term where the powers are all greater than those degrees. (Degree is in quotes because there is no guarantee that coefficients below that degree are nonzero, as we do not assume decidable equality for 𝑟). (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 7-Oct-2025.)

Assertion

Ref	Expression
df-mplcoe	⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑤⦌(𝑤 ↾s {𝑓 ∈ (Base‘𝑤) ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))}))

Distinct variable group:   𝑎,𝑏,𝑓,𝑖,𝑘,𝑟,𝑤

Detailed syntax breakdown of Definition df-mplcoe

Step	Hyp	Ref	Expression
1		cmpl 14468	. 2 class mPoly
2		vi	. . 3 setvar 𝑖
3		vr	. . 3 setvar 𝑟
4		cvv 2773	. . 3 class V
5		vw	. . . 4 setvar 𝑤
6	2	cv 1372	. . . . 5 class 𝑖
7	3	cv 1372	. . . . 5 class 𝑟
8		cmps 14467	. . . . 5 class mPwSer
9	6, 7, 8	co 5951	. . . 4 class (𝑖 mPwSer 𝑟)
10	5	cv 1372	. . . . 5 class 𝑤
11		vk	. . . . . . . . . . . . 13 setvar 𝑘
12	11	cv 1372	. . . . . . . . . . . 12 class 𝑘
13		va	. . . . . . . . . . . . 13 setvar 𝑎
14	13	cv 1372	. . . . . . . . . . . 12 class 𝑎
15	12, 14	cfv 5276	. . . . . . . . . . 11 class (𝑎‘𝑘)
16		vb	. . . . . . . . . . . . 13 setvar 𝑏
17	16	cv 1372	. . . . . . . . . . . 12 class 𝑏
18	12, 17	cfv 5276	. . . . . . . . . . 11 class (𝑏‘𝑘)
19		clt 8114	. . . . . . . . . . 11 class <
20	15, 18, 19	wbr 4047	. . . . . . . . . 10 wff (𝑎‘𝑘) < (𝑏‘𝑘)
21	20, 11, 6	wral 2485	. . . . . . . . 9 wff ∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘)
22		vf	. . . . . . . . . . . 12 setvar 𝑓
23	22	cv 1372	. . . . . . . . . . 11 class 𝑓
24	17, 23	cfv 5276	. . . . . . . . . 10 class (𝑓‘𝑏)
25		c0g 13132	. . . . . . . . . . 11 class 0g
26	7, 25	cfv 5276	. . . . . . . . . 10 class (0g‘𝑟)
27	24, 26	wceq 1373	. . . . . . . . 9 wff (𝑓‘𝑏) = (0g‘𝑟)
28	21, 27	wi 4	. . . . . . . 8 wff (∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))
29		cn0 9302	. . . . . . . . 9 class ℕ0
30		cmap 6742	. . . . . . . . 9 class ↑𝑚
31	29, 6, 30	co 5951	. . . . . . . 8 class (ℕ0 ↑𝑚 𝑖)
32	28, 16, 31	wral 2485	. . . . . . 7 wff ∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))
33	32, 13, 31	wrex 2486	. . . . . 6 wff ∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))
34		cbs 12876	. . . . . . 7 class Base
35	10, 34	cfv 5276	. . . . . 6 class (Base‘𝑤)
36	33, 22, 35	crab 2489	. . . . 5 class {𝑓 ∈ (Base‘𝑤) ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))}
37		cress 12877	. . . . 5 class ↾s
38	10, 36, 37	co 5951	. . . 4 class (𝑤 ↾s {𝑓 ∈ (Base‘𝑤) ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))})
39	5, 9, 38	csb 3094	. . 3 class ⦋(𝑖 mPwSer 𝑟) / 𝑤⦌(𝑤 ↾s {𝑓 ∈ (Base‘𝑤) ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))})
40	2, 3, 4, 4, 39	cmpo 5953	. 2 class (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑤⦌(𝑤 ↾s {𝑓 ∈ (Base‘𝑤) ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))}))
41	1, 40	wceq 1373	1 wff mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑤⦌(𝑤 ↾s {𝑓 ∈ (Base‘𝑤) ∣ ∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))}))

This definition is referenced by:  reldmmpl  14495  mplvalcoe  14496  fnmpl  14499