Definition df-mplcoe 14671

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 14669	. 2 class mPoly
2		vi	. . 3 setvar 𝑖
3		vr	. . 3 setvar 𝑟
4		cvv 2800	. . 3 class V
5		vw	. . . 4 setvar 𝑤
6	2	cv 1394	. . . . 5 class 𝑖
7	3	cv 1394	. . . . 5 class 𝑟
8		cmps 14668	. . . . 5 class mPwSer
9	6, 7, 8	co 6013	. . . 4 class (𝑖 mPwSer 𝑟)
10	5	cv 1394	. . . . 5 class 𝑤
11		vk	. . . . . . . . . . . . 13 setvar 𝑘
12	11	cv 1394	. . . . . . . . . . . 12 class 𝑘
13		va	. . . . . . . . . . . . 13 setvar 𝑎
14	13	cv 1394	. . . . . . . . . . . 12 class 𝑎
15	12, 14	cfv 5324	. . . . . . . . . . 11 class (𝑎‘𝑘)
16		vb	. . . . . . . . . . . . 13 setvar 𝑏
17	16	cv 1394	. . . . . . . . . . . 12 class 𝑏
18	12, 17	cfv 5324	. . . . . . . . . . 11 class (𝑏‘𝑘)
19		clt 8207	. . . . . . . . . . 11 class <
20	15, 18, 19	wbr 4086	. . . . . . . . . 10 wff (𝑎‘𝑘) < (𝑏‘𝑘)
21	20, 11, 6	wral 2508	. . . . . . . . 9 wff ∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘)
22		vf	. . . . . . . . . . . 12 setvar 𝑓
23	22	cv 1394	. . . . . . . . . . 11 class 𝑓
24	17, 23	cfv 5324	. . . . . . . . . 10 class (𝑓‘𝑏)
25		c0g 13332	. . . . . . . . . . 11 class 0g
26	7, 25	cfv 5324	. . . . . . . . . 10 class (0g‘𝑟)
27	24, 26	wceq 1395	. . . . . . . . 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 9395	. . . . . . . . 9 class ℕ0
30		cmap 6812	. . . . . . . . 9 class ↑𝑚
31	29, 6, 30	co 6013	. . . . . . . 8 class (ℕ0 ↑𝑚 𝑖)
32	28, 16, 31	wral 2508	. . . . . . 7 wff ∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))
33	32, 13, 31	wrex 2509	. . . . . 6 wff ∃𝑎 ∈ (ℕ0 ↑𝑚 𝑖)∀𝑏 ∈ (ℕ0 ↑𝑚 𝑖)(∀𝑘 ∈ 𝑖 (𝑎‘𝑘) < (𝑏‘𝑘) → (𝑓‘𝑏) = (0g‘𝑟))
34		cbs 13075	. . . . . . 7 class Base
35	10, 34	cfv 5324	. . . . . 6 class (Base‘𝑤)
36	33, 22, 35	crab 2512	. . . . 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 13076	. . . . 5 class ↾s
38	10, 36, 37	co 6013	. . . 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 3125	. . 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 6015	. 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 1395	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  14696  mplvalcoe  14697  fnmpl  14700