Theorem ply1coe 19487

Description: Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 7-Oct-2019.)

Hypotheses

Ref	Expression
ply1coe.p	⊢ 𝑃 = (Poly1‘𝑅)
ply1coe.x	⊢ 𝑋 = (var1‘𝑅)
ply1coe.b	⊢ 𝐵 = (Base‘𝑃)
ply1coe.n	⊢ · = ( ·𝑠 ‘𝑃)
ply1coe.m	⊢ 𝑀 = (mulGrp‘𝑃)
ply1coe.e	⊢ ↑ = (.g‘𝑀)
ply1coe.a	⊢ 𝐴 = (coe1‘𝐾)

Assertion

Ref	Expression
ply1coe	⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐾 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))))

Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝑘,𝐾   𝑘,𝑋   ↑ ,𝑘   𝑅,𝑘   · ,𝑘   𝑃,𝑘

Allowed substitution hint:   𝑀(𝑘)

Proof of Theorem ply1coe

Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2610	. . 3 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
2		psr1baslem 19376	. . 3 ⊢ (ℕ0 ↑𝑚 1𝑜) = {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ (◡𝑑 “ ℕ) ∈ Fin}
3		eqid 2610	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
4		eqid 2610	. . 3 ⊢ (1r‘𝑅) = (1r‘𝑅)
5		1onn 7606	. . . 4 ⊢ 1𝑜 ∈ ω
6	5	a1i 11	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 1𝑜 ∈ ω)
7		ply1coe.p	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
8		eqid 2610	. . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅)
9		ply1coe.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
10	7, 8, 9	ply1bas 19386	. . 3 ⊢ 𝐵 = (Base‘(1𝑜 mPoly 𝑅))
11		ply1coe.n	. . . 4 ⊢ · = ( ·𝑠 ‘𝑃)
12	7, 1, 11	ply1vsca 19417	. . 3 ⊢ · = ( ·𝑠 ‘(1𝑜 mPoly 𝑅))
13		simpl 472	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝑅 ∈ Ring)
14		simpr 476	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐾 ∈ 𝐵)
15	1, 2, 3, 4, 6, 10, 12, 13, 14	mplcoe1 19286	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐾 = ((1𝑜 mPoly 𝑅) Σg (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ ((𝐾‘𝑎) · (𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r‘𝑅), (0g‘𝑅)))))))
16		ply1coe.a	. . . . . . 7 ⊢ 𝐴 = (coe1‘𝐾)
17	16	fvcoe1 19398	. . . . . 6 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝐾‘𝑎) = (𝐴‘(𝑎‘∅)))
18	17	adantll 746	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝐾‘𝑎) = (𝐴‘(𝑎‘∅)))
19	5	a1i 11	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → 1𝑜 ∈ ω)
20		eqid 2610	. . . . . . 7 ⊢ (mulGrp‘(1𝑜 mPoly 𝑅)) = (mulGrp‘(1𝑜 mPoly 𝑅))
21		eqid 2610	. . . . . . 7 ⊢ (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = (.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
22		eqid 2610	. . . . . . 7 ⊢ (1𝑜 mVar 𝑅) = (1𝑜 mVar 𝑅)
23		simpll 786	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → 𝑅 ∈ Ring)
24		simpr 476	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜))
25		eqidd 2611	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)))
26		0ex 4718	. . . . . . . . . . 11 ⊢ ∅ ∈ V
27		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑏 = ∅ → ((1𝑜 mVar 𝑅)‘𝑏) = ((1𝑜 mVar 𝑅)‘∅))
28	27	oveq1d 6564	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)))
29	27	oveq2d 6565	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)))
30	28, 29	eqeq12d 2625	. . . . . . . . . . 11 ⊢ (𝑏 = ∅ → ((((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) ↔ (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅))))
31	26, 30	ralsn 4169	. . . . . . . . . 10 ⊢ (∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) ↔ (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)))
32	25, 31	sylibr 223	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → ∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)))
33		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → ((1𝑜 mVar 𝑅)‘𝑥) = ((1𝑜 mVar 𝑅)‘∅))
34	33	oveq2d 6565	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)))
35	33	oveq1d 6564	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)))
36	34, 35	eqeq12d 2625	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ((((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) ↔ (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏))))
37	36	ralbidv 2969	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) ↔ ∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏))))
38	26, 37	ralsn 4169	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {∅}∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) ↔ ∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)))
39	32, 38	sylibr 223	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → ∀𝑥 ∈ {∅}∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)))
40		df1o2 7459	. . . . . . . . 9 ⊢ 1𝑜 = {∅}
41	40	raleqi 3119	. . . . . . . . 9 ⊢ (∀𝑏 ∈ 1𝑜 (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) ↔ ∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)))
42	40, 41	raleqbii 2973	. . . . . . . 8 ⊢ (∀𝑥 ∈ 1𝑜 ∀𝑏 ∈ 1𝑜 (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) ↔ ∀𝑥 ∈ {∅}∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)))
43	39, 42	sylibr 223	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → ∀𝑥 ∈ 1𝑜 ∀𝑏 ∈ 1𝑜 (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)))
44	1, 2, 3, 4, 19, 20, 21, 22, 23, 24, 43	mplcoe5 19289	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r‘𝑅), (0g‘𝑅))) = ((mulGrp‘(1𝑜 mPoly 𝑅)) Σg (𝑐 ∈ 1𝑜 ↦ ((𝑎‘𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐)))))
45		mpteq1 4665	. . . . . . . . 9 ⊢ (1𝑜 = {∅} → (𝑐 ∈ 1𝑜 ↦ ((𝑎‘𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐))) = (𝑐 ∈ {∅} ↦ ((𝑎‘𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐))))
46	40, 45	ax-mp 5	. . . . . . . 8 ⊢ (𝑐 ∈ 1𝑜 ↦ ((𝑎‘𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐))) = (𝑐 ∈ {∅} ↦ ((𝑎‘𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐)))
47	46	oveq2i 6560	. . . . . . 7 ⊢ ((mulGrp‘(1𝑜 mPoly 𝑅)) Σg (𝑐 ∈ 1𝑜 ↦ ((𝑎‘𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐)))) = ((mulGrp‘(1𝑜 mPoly 𝑅)) Σg (𝑐 ∈ {∅} ↦ ((𝑎‘𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐))))
48	1	mplring 19273	. . . . . . . . . . 11 ⊢ ((1𝑜 ∈ ω ∧ 𝑅 ∈ Ring) → (1𝑜 mPoly 𝑅) ∈ Ring)
49	5, 48	mpan 702	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (1𝑜 mPoly 𝑅) ∈ Ring)
50	20	ringmgp 18376	. . . . . . . . . 10 ⊢ ((1𝑜 mPoly 𝑅) ∈ Ring → (mulGrp‘(1𝑜 mPoly 𝑅)) ∈ Mnd)
51	49, 50	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (mulGrp‘(1𝑜 mPoly 𝑅)) ∈ Mnd)
52	51	ad2antrr 758	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → (mulGrp‘(1𝑜 mPoly 𝑅)) ∈ Mnd)
53	26	a1i 11	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → ∅ ∈ V)
54		ply1coe.e	. . . . . . . . . . . 12 ⊢ ↑ = (.g‘𝑀)
55	20, 10	mgpbas 18318	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘(mulGrp‘(1𝑜 mPoly 𝑅)))
56	55	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐵 = (Base‘(mulGrp‘(1𝑜 mPoly 𝑅))))
57		ply1coe.m	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (mulGrp‘𝑃)
58	57, 9	mgpbas 18318	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑀)
59	58	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐵 = (Base‘𝑀))
60		ssv 3588	. . . . . . . . . . . . 13 ⊢ 𝐵 ⊆ V
61	60	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐵 ⊆ V)
62		ovex 6577	. . . . . . . . . . . . 13 ⊢ (𝑎(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑏) ∈ V
63	62	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑏) ∈ V)
64		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘𝑃) = (.r‘𝑃)
65	7, 1, 64	ply1mulr 19418	. . . . . . . . . . . . . . . 16 ⊢ (.r‘𝑃) = (.r‘(1𝑜 mPoly 𝑅))
66	20, 65	mgpplusg 18316	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑃) = (+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
67	57, 64	mgpplusg 18316	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑃) = (+g‘𝑀)
68	66, 67	eqtr3i 2634	. . . . . . . . . . . . . 14 ⊢ (+g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = (+g‘𝑀)
69	68	oveqi 6562	. . . . . . . . . . . . 13 ⊢ (𝑎(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑏) = (𝑎(+g‘𝑀)𝑏)
70	69	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑏) = (𝑎(+g‘𝑀)𝑏))
71	21, 54, 56, 59, 61, 63, 70	mulgpropd 17407	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = ↑ )
72	71	oveqd 6566	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋) = ((𝑎‘∅) ↑ 𝑋))
73	72	adantr 480	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋) = ((𝑎‘∅) ↑ 𝑋))
74	7	ply1ring 19439	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
75	57	ringmgp 18376	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ Ring → 𝑀 ∈ Mnd)
76	74, 75	syl 17	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
77	76	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → 𝑀 ∈ Mnd)
78		elmapi 7765	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) → 𝑎:1𝑜⟶ℕ0)
79		0lt1o 7471	. . . . . . . . . . . 12 ⊢ ∅ ∈ 1𝑜
80		ffvelrn 6265	. . . . . . . . . . . 12 ⊢ ((𝑎:1𝑜⟶ℕ0 ∧ ∅ ∈ 1𝑜) → (𝑎‘∅) ∈ ℕ0)
81	78, 79, 80	sylancl 693	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) → (𝑎‘∅) ∈ ℕ0)
82	81	adantl 481	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝑎‘∅) ∈ ℕ0)
83		ply1coe.x	. . . . . . . . . . . 12 ⊢ 𝑋 = (var1‘𝑅)
84	83, 7, 9	vr1cl 19408	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵)
85	84	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → 𝑋 ∈ 𝐵)
86	58, 54	mulgnn0cl 17381	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Mnd ∧ (𝑎‘∅) ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑎‘∅) ↑ 𝑋) ∈ 𝐵)
87	77, 82, 85, 86	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → ((𝑎‘∅) ↑ 𝑋) ∈ 𝐵)
88	73, 87	eqeltrd 2688	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋) ∈ 𝐵)
89		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑐 = ∅ → (𝑎‘𝑐) = (𝑎‘∅))
90		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑐 = ∅ → ((1𝑜 mVar 𝑅)‘𝑐) = ((1𝑜 mVar 𝑅)‘∅))
91	83	vr1val 19383	. . . . . . . . . . 11 ⊢ 𝑋 = ((1𝑜 mVar 𝑅)‘∅)
92	90, 91	syl6eqr 2662	. . . . . . . . . 10 ⊢ (𝑐 = ∅ → ((1𝑜 mVar 𝑅)‘𝑐) = 𝑋)
93	89, 92	oveq12d 6567	. . . . . . . . 9 ⊢ (𝑐 = ∅ → ((𝑎‘𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐)) = ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋))
94	55, 93	gsumsn 18177	. . . . . . . 8 ⊢ (((mulGrp‘(1𝑜 mPoly 𝑅)) ∈ Mnd ∧ ∅ ∈ V ∧ ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋) ∈ 𝐵) → ((mulGrp‘(1𝑜 mPoly 𝑅)) Σg (𝑐 ∈ {∅} ↦ ((𝑎‘𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐)))) = ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋))
95	52, 53, 88, 94	syl3anc 1318	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → ((mulGrp‘(1𝑜 mPoly 𝑅)) Σg (𝑐 ∈ {∅} ↦ ((𝑎‘𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐)))) = ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋))
96	47, 95	syl5eq 2656	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → ((mulGrp‘(1𝑜 mPoly 𝑅)) Σg (𝑐 ∈ 1𝑜 ↦ ((𝑎‘𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐)))) = ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋))
97	44, 96, 73	3eqtrd 2648	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r‘𝑅), (0g‘𝑅))) = ((𝑎‘∅) ↑ 𝑋))
98	18, 97	oveq12d 6567	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → ((𝐾‘𝑎) · (𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r‘𝑅), (0g‘𝑅)))) = ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋)))
99	98	mpteq2dva 4672	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ ((𝐾‘𝑎) · (𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r‘𝑅), (0g‘𝑅))))) = (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋))))
100	99	oveq2d 6565	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ((1𝑜 mPoly 𝑅) Σg (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ ((𝐾‘𝑎) · (𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r‘𝑅), (0g‘𝑅)))))) = ((1𝑜 mPoly 𝑅) Σg (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋)))))
101		nn0ex 11175	. . . . . 6 ⊢ ℕ0 ∈ V
102	101	mptex 6390	. . . . 5 ⊢ (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) ∈ V
103	102	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) ∈ V)
104		fvex 6113	. . . . . 6 ⊢ (Poly1‘𝑅) ∈ V
105	7, 104	eqeltri 2684	. . . . 5 ⊢ 𝑃 ∈ V
106	105	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝑃 ∈ V)
107		ovex 6577	. . . . 5 ⊢ (1𝑜 mPoly 𝑅) ∈ V
108	107	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (1𝑜 mPoly 𝑅) ∈ V)
109	9, 10	eqtr3i 2634	. . . . 5 ⊢ (Base‘𝑃) = (Base‘(1𝑜 mPoly 𝑅))
110	109	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (Base‘𝑃) = (Base‘(1𝑜 mPoly 𝑅)))
111		eqid 2610	. . . . . 6 ⊢ (+g‘𝑃) = (+g‘𝑃)
112	7, 1, 111	ply1plusg 19416	. . . . 5 ⊢ (+g‘𝑃) = (+g‘(1𝑜 mPoly 𝑅))
113	112	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (+g‘𝑃) = (+g‘(1𝑜 mPoly 𝑅)))
114	103, 106, 108, 110, 113	gsumpropd 17095	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))) = ((1𝑜 mPoly 𝑅) Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))))
115		eqid 2610	. . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃)
116	1, 7, 115	ply1mpl0 19446	. . . 4 ⊢ (0g‘𝑃) = (0g‘(1𝑜 mPoly 𝑅))
117	1	mpllmod 19272	. . . . . 6 ⊢ ((1𝑜 ∈ ω ∧ 𝑅 ∈ Ring) → (1𝑜 mPoly 𝑅) ∈ LMod)
118	5, 13, 117	sylancr 694	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (1𝑜 mPoly 𝑅) ∈ LMod)
119		lmodcmn 18734	. . . . 5 ⊢ ((1𝑜 mPoly 𝑅) ∈ LMod → (1𝑜 mPoly 𝑅) ∈ CMnd)
120	118, 119	syl 17	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (1𝑜 mPoly 𝑅) ∈ CMnd)
121	101	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ℕ0 ∈ V)
122	7	ply1lmod 19443	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
123	122	ad2antrr 758	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ LMod)
124		eqid 2610	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
125	16, 9, 7, 124	coe1f 19402	. . . . . . . . 9 ⊢ (𝐾 ∈ 𝐵 → 𝐴:ℕ0⟶(Base‘𝑅))
126	125	adantl 481	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐴:ℕ0⟶(Base‘𝑅))
127	126	ffvelrnda 6267	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ (Base‘𝑅))
128	7	ply1sca 19444	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
129	128	eqcomd 2616	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
130	129	ad2antrr 758	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (Scalar‘𝑃) = 𝑅)
131	130	fveq2d 6107	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
132	127, 131	eleqtrrd 2691	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ (Base‘(Scalar‘𝑃)))
133	76	ad2antrr 758	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ Mnd)
134		simpr 476	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
135	84	ad2antrr 758	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑋 ∈ 𝐵)
136	58, 54	mulgnn0cl 17381	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑘 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑘 ↑ 𝑋) ∈ 𝐵)
137	133, 134, 135, 136	syl3anc 1318	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑘 ↑ 𝑋) ∈ 𝐵)
138		eqid 2610	. . . . . . 7 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
139		eqid 2610	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
140	9, 138, 11, 139	lmodvscl 18703	. . . . . 6 ⊢ ((𝑃 ∈ LMod ∧ (𝐴‘𝑘) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑘 ↑ 𝑋) ∈ 𝐵) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)) ∈ 𝐵)
141	123, 132, 137, 140	syl3anc 1318	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)) ∈ 𝐵)
142		eqid 2610	. . . . 5 ⊢ (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))
143	141, 142	fmptd 6292	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))):ℕ0⟶𝐵)
144	7, 83, 9, 11, 57, 54, 16	ply1coefsupp 19486	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) finSupp (0g‘𝑃))
145		eqid 2610	. . . . . 6 ⊢ (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑎‘∅))
146	40, 101, 26, 145	mapsnf1o2 7791	. . . . 5 ⊢ (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑎‘∅)):(ℕ0 ↑𝑚 1𝑜)–1-1-onto→ℕ0
147	146	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑎‘∅)):(ℕ0 ↑𝑚 1𝑜)–1-1-onto→ℕ0)
148	10, 116, 120, 121, 143, 144, 147	gsumf1o 18140	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ((1𝑜 mPoly 𝑅) Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))) = ((1𝑜 mPoly 𝑅) Σg ((𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) ∘ (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑎‘∅)))))
149		eqidd 2611	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑎‘∅)))
150		eqidd 2611	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))))
151		fveq2 6103	. . . . . 6 ⊢ (𝑘 = (𝑎‘∅) → (𝐴‘𝑘) = (𝐴‘(𝑎‘∅)))
152		oveq1 6556	. . . . . 6 ⊢ (𝑘 = (𝑎‘∅) → (𝑘 ↑ 𝑋) = ((𝑎‘∅) ↑ 𝑋))
153	151, 152	oveq12d 6567	. . . . 5 ⊢ (𝑘 = (𝑎‘∅) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)) = ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋)))
154	82, 149, 150, 153	fmptco 6303	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ((𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) ∘ (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑎‘∅))) = (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋))))
155	154	oveq2d 6565	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ((1𝑜 mPoly 𝑅) Σg ((𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) ∘ (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑎‘∅)))) = ((1𝑜 mPoly 𝑅) Σg (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋)))))
156	114, 148, 155	3eqtrrd 2649	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ((1𝑜 mPoly 𝑅) Σg (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋)))) = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))))
157	15, 100, 156	3eqtrd 2648	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐾 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {csn 4125   ↦ cmpt 4643   ∘ ccom 5042  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ωcom 6957  1_𝑜c1o 7440   ↑_𝑚 cmap 7744  ℕ₀cn0 11169  Basecbs 15695  +_gcplusg 15768  ._rcmulr 15769  Scalarcsca 15771   ·_𝑠 cvsca 15772  0_gc0g 15923   Σ_g cgsu 15924  Mndcmnd 17117  ._gcmg 17363  CMndccmn 18016  mulGrpcmgp 18312  1_rcur 18324  Ringcrg 18370  LModclmod 18686   mVar cmvr 19173   mPoly cmpl 19174  PwSer₁cps1 19366  var₁cv1 19367  Poly₁cpl1 19368  coe₁cco1 19369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-tset 15787  df-ple 15788  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-srg 18329  df-ring 18372  df-subrg 18601  df-lmod 18688  df-lss 18754  df-psr 19177  df-mvr 19178  df-mpl 19179  df-opsr 19181  df-psr1 19371  df-vr1 19372  df-ply1 19373  df-coe1 19374

This theorem is referenced by:  eqcoe1ply1eq  19488  pmatcollpw1lem2  20399  mp2pm2mp  20435  plypf1  23772