Definition df-mplcoe 14298

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 14296	. 2 class mPoly
2		vi	. . 3 setvar 𝑖
3		vr	. . 3 setvar 𝑟
4		cvv 2763	. . 3 class V
5		vw	. . . 4 setvar 𝑤
6	2	cv 1363	. . . . 5 class 𝑖
7	3	cv 1363	. . . . 5 class 𝑟
8		cmps 14295	. . . . 5 class mPwSer
9	6, 7, 8	co 5925	. . . 4 class (𝑖 mPwSer 𝑟)
10	5	cv 1363	. . . . 5 class 𝑤
11		vk	. . . . . . . . . . . . 13 setvar 𝑘
12	11	cv 1363	. . . . . . . . . . . 12 class 𝑘
13		va	. . . . . . . . . . . . 13 setvar 𝑎
14	13	cv 1363	. . . . . . . . . . . 12 class 𝑎
15	12, 14	cfv 5259	. . . . . . . . . . 11 class (𝑎‘𝑘)
16		vb	. . . . . . . . . . . . 13 setvar 𝑏
17	16	cv 1363	. . . . . . . . . . . 12 class 𝑏
18	12, 17	cfv 5259	. . . . . . . . . . 11 class (𝑏‘𝑘)
19		clt 8080	. . . . . . . . . . 11 class <
20	15, 18, 19	wbr 4034	. . . . . . . . . 10 wff (𝑎‘𝑘) < (𝑏‘𝑘)
21	20, 11, 6	wral 2475	. . . . . . . . 9 wff ∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘)
22		vf	. . . . . . . . . . . 12 setvar 𝑓
23	22	cv 1363	. . . . . . . . . . 11 class 𝑓
24	17, 23	cfv 5259	. . . . . . . . . 10 class (𝑓‘𝑏)
25		c0g 12960	. . . . . . . . . . 11 class 0g
26	7, 25	cfv 5259	. . . . . . . . . 10 class (0g‘𝑟)
27	24, 26	wceq 1364	. . . . . . . . 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 9268	. . . . . . . . 9 class ℕ0
30		cmap 6716	. . . . . . . . 9 class ↑𝑚
31	29, 6, 30	co 5925	. . . . . . . 8 class (ℕ0 ↑𝑚 𝑖)
32	28, 16, 31	wral 2475	. . . . . . 7 wff ∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))
33	32, 13, 31	wrex 2476	. . . . . 6 wff ∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))
34		cbs 12705	. . . . . . 7 class Base
35	10, 34	cfv 5259	. . . . . 6 class (Base‘𝑤)
36	33, 22, 35	crab 2479	. . . . 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 12706	. . . . 5 class ↾s
38	10, 36, 37	co 5925	. . . 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 3084	. . 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 5927	. 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 1364	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  14323  mplvalcoe  14324  fnmpl  14327