Theorem ply1coe 19666

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 2622	. . 3 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
2		psr1baslem 19555	. . 3 ⊢ (ℕ0 ↑𝑚 1𝑜) = {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ (◡𝑑 “ ℕ) ∈ Fin}
3		eqid 2622	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
4		eqid 2622	. . 3 ⊢ (1r‘𝑅) = (1r‘𝑅)
5		1onn 7719	. . . 4 ⊢ 1𝑜 ∈ ω
6	5	a1i 11	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 1𝑜 ∈ ω)
7		ply1coe.p	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
8		eqid 2622	. . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅)
9		ply1coe.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
10	7, 8, 9	ply1bas 19565	. . 3 ⊢ 𝐵 = (Base‘(1𝑜 mPoly 𝑅))
11		ply1coe.n	. . . 4 ⊢ · = ( ·𝑠 ‘𝑃)
12	7, 1, 11	ply1vsca 19596	. . 3 ⊢ · = ( ·𝑠 ‘(1𝑜 mPoly 𝑅))
13		simpl 473	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝑅 ∈ Ring)
14		simpr 477	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐾 ∈ 𝐵)
15	1, 2, 3, 4, 6, 10, 12, 13, 14	mplcoe1 19465	. 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 19577	. . . . . 6 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝐾‘𝑎) = (𝐴‘(𝑎‘∅)))
18	17	adantll 750	. . . . 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 2622	. . . . . . 7 ⊢ (mulGrp‘(1𝑜 mPoly 𝑅)) = (mulGrp‘(1𝑜 mPoly 𝑅))
21		eqid 2622	. . . . . . 7 ⊢ (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = (.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
22		eqid 2622	. . . . . . 7 ⊢ (1𝑜 mVar 𝑅) = (1𝑜 mVar 𝑅)
23		simpll 790	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → 𝑅 ∈ Ring)
24		simpr 477	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜))
25		eqidd 2623	. . . . . . . . . 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 4790	. . . . . . . . . . 11 ⊢ ∅ ∈ V
27		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑏 = ∅ → ((1𝑜 mVar 𝑅)‘𝑏) = ((1𝑜 mVar 𝑅)‘∅))
28	27	oveq1d 6665	. . . . . . . . . . . 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 6666	. . . . . . . . . . . 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 2637	. . . . . . . . . . 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 4222	. . . . . . . . . 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 224	. . . . . . . . 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 6191	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → ((1𝑜 mVar 𝑅)‘𝑥) = ((1𝑜 mVar 𝑅)‘∅))
34	33	oveq2d 6666	. . . . . . . . . . . 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 6665	. . . . . . . . . . . 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 2637	. . . . . . . . . . 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 2986	. . . . . . . . . 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 4222	. . . . . . . . 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 224	. . . . . . . 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 7572	. . . . . . . . 9 ⊢ 1𝑜 = {∅}
41	40	raleqi 3142	. . . . . . . . 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 2990	. . . . . . . 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 224	. . . . . . 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 19468	. . . . . 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 4737	. . . . . . . . 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 6661	. . . . . . 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 19452	. . . . . . . . . . 11 ⊢ ((1𝑜 ∈ ω ∧ 𝑅 ∈ Ring) → (1𝑜 mPoly 𝑅) ∈ Ring)
49	5, 48	mpan 706	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (1𝑜 mPoly 𝑅) ∈ Ring)
50	20	ringmgp 18553	. . . . . . . . . 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 762	. . . . . . . 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 18495	. . . . . . . . . . . . 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 18495	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑀)
59	58	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐵 = (Base‘𝑀))
60		ssv 3625	. . . . . . . . . . . . 13 ⊢ 𝐵 ⊆ V
61	60	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐵 ⊆ V)
62		ovexd 6680	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑏) ∈ V)
63		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘𝑃) = (.r‘𝑃)
64	7, 1, 63	ply1mulr 19597	. . . . . . . . . . . . . . . 16 ⊢ (.r‘𝑃) = (.r‘(1𝑜 mPoly 𝑅))
65	20, 64	mgpplusg 18493	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑃) = (+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
66	57, 63	mgpplusg 18493	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑃) = (+g‘𝑀)
67	65, 66	eqtr3i 2646	. . . . . . . . . . . . . 14 ⊢ (+g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = (+g‘𝑀)
68	67	oveqi 6663	. . . . . . . . . . . . 13 ⊢ (𝑎(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑏) = (𝑎(+g‘𝑀)𝑏)
69	68	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑏) = (𝑎(+g‘𝑀)𝑏))
70	21, 54, 56, 59, 61, 62, 69	mulgpropd 17584	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = ↑ )
71	70	oveqd 6667	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋) = ((𝑎‘∅) ↑ 𝑋))
72	71	adantr 481	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋) = ((𝑎‘∅) ↑ 𝑋))
73	7	ply1ring 19618	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
74	57	ringmgp 18553	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ Ring → 𝑀 ∈ Mnd)
75	73, 74	syl 17	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
76	75	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → 𝑀 ∈ Mnd)
77		elmapi 7879	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) → 𝑎:1𝑜⟶ℕ0)
78		0lt1o 7584	. . . . . . . . . . . 12 ⊢ ∅ ∈ 1𝑜
79		ffvelrn 6357	. . . . . . . . . . . 12 ⊢ ((𝑎:1𝑜⟶ℕ0 ∧ ∅ ∈ 1𝑜) → (𝑎‘∅) ∈ ℕ0)
80	77, 78, 79	sylancl 694	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) → (𝑎‘∅) ∈ ℕ0)
81	80	adantl 482	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝑎‘∅) ∈ ℕ0)
82		ply1coe.x	. . . . . . . . . . . 12 ⊢ 𝑋 = (var1‘𝑅)
83	82, 7, 9	vr1cl 19587	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵)
84	83	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → 𝑋 ∈ 𝐵)
85	58, 54	mulgnn0cl 17558	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Mnd ∧ (𝑎‘∅) ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑎‘∅) ↑ 𝑋) ∈ 𝐵)
86	76, 81, 84, 85	syl3anc 1326	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → ((𝑎‘∅) ↑ 𝑋) ∈ 𝐵)
87	72, 86	eqeltrd 2701	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋) ∈ 𝐵)
88		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑐 = ∅ → (𝑎‘𝑐) = (𝑎‘∅))
89		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑐 = ∅ → ((1𝑜 mVar 𝑅)‘𝑐) = ((1𝑜 mVar 𝑅)‘∅))
90	82	vr1val 19562	. . . . . . . . . . 11 ⊢ 𝑋 = ((1𝑜 mVar 𝑅)‘∅)
91	89, 90	syl6eqr 2674	. . . . . . . . . 10 ⊢ (𝑐 = ∅ → ((1𝑜 mVar 𝑅)‘𝑐) = 𝑋)
92	88, 91	oveq12d 6668	. . . . . . . . 9 ⊢ (𝑐 = ∅ → ((𝑎‘𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐)) = ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋))
93	55, 92	gsumsn 18354	. . . . . . . 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))
94	52, 53, 87, 93	syl3anc 1326	. . . . . . 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))
95	47, 94	syl5eq 2668	. . . . . 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))
96	44, 95, 72	3eqtrd 2660	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝑏 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r‘𝑅), (0g‘𝑅))) = ((𝑎‘∅) ↑ 𝑋))
97	18, 96	oveq12d 6668	. . . 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)))
98	97	mpteq2dva 4744	. . 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))))
99	98	oveq2d 6666	. 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)))))
100		nn0ex 11298	. . . . . 6 ⊢ ℕ0 ∈ V
101	100	mptex 6486	. . . . 5 ⊢ (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) ∈ V
102	101	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) ∈ V)
103		fvex 6201	. . . . . 6 ⊢ (Poly1‘𝑅) ∈ V
104	7, 103	eqeltri 2697	. . . . 5 ⊢ 𝑃 ∈ V
105	104	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝑃 ∈ V)
106		ovexd 6680	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (1𝑜 mPoly 𝑅) ∈ V)
107	9, 10	eqtr3i 2646	. . . . 5 ⊢ (Base‘𝑃) = (Base‘(1𝑜 mPoly 𝑅))
108	107	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (Base‘𝑃) = (Base‘(1𝑜 mPoly 𝑅)))
109		eqid 2622	. . . . . 6 ⊢ (+g‘𝑃) = (+g‘𝑃)
110	7, 1, 109	ply1plusg 19595	. . . . 5 ⊢ (+g‘𝑃) = (+g‘(1𝑜 mPoly 𝑅))
111	110	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (+g‘𝑃) = (+g‘(1𝑜 mPoly 𝑅)))
112	102, 105, 106, 108, 111	gsumpropd 17272	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))) = ((1𝑜 mPoly 𝑅) Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))))
113		eqid 2622	. . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃)
114	1, 7, 113	ply1mpl0 19625	. . . 4 ⊢ (0g‘𝑃) = (0g‘(1𝑜 mPoly 𝑅))
115	1	mpllmod 19451	. . . . . 6 ⊢ ((1𝑜 ∈ ω ∧ 𝑅 ∈ Ring) → (1𝑜 mPoly 𝑅) ∈ LMod)
116	5, 13, 115	sylancr 695	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (1𝑜 mPoly 𝑅) ∈ LMod)
117		lmodcmn 18911	. . . . 5 ⊢ ((1𝑜 mPoly 𝑅) ∈ LMod → (1𝑜 mPoly 𝑅) ∈ CMnd)
118	116, 117	syl 17	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (1𝑜 mPoly 𝑅) ∈ CMnd)
119	100	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ℕ0 ∈ V)
120	7	ply1lmod 19622	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
121	120	ad2antrr 762	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ LMod)
122		eqid 2622	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
123	16, 9, 7, 122	coe1f 19581	. . . . . . . . 9 ⊢ (𝐾 ∈ 𝐵 → 𝐴:ℕ0⟶(Base‘𝑅))
124	123	adantl 482	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐴:ℕ0⟶(Base‘𝑅))
125	124	ffvelrnda 6359	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ (Base‘𝑅))
126	7	ply1sca 19623	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
127	126	eqcomd 2628	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
128	127	ad2antrr 762	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (Scalar‘𝑃) = 𝑅)
129	128	fveq2d 6195	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
130	125, 129	eleqtrrd 2704	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ (Base‘(Scalar‘𝑃)))
131	75	ad2antrr 762	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ Mnd)
132		simpr 477	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
133	83	ad2antrr 762	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑋 ∈ 𝐵)
134	58, 54	mulgnn0cl 17558	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑘 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑘 ↑ 𝑋) ∈ 𝐵)
135	131, 132, 133, 134	syl3anc 1326	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑘 ↑ 𝑋) ∈ 𝐵)
136		eqid 2622	. . . . . . 7 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
137		eqid 2622	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
138	9, 136, 11, 137	lmodvscl 18880	. . . . . 6 ⊢ ((𝑃 ∈ LMod ∧ (𝐴‘𝑘) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑘 ↑ 𝑋) ∈ 𝐵) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)) ∈ 𝐵)
139	121, 130, 135, 138	syl3anc 1326	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)) ∈ 𝐵)
140		eqid 2622	. . . . 5 ⊢ (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))
141	139, 140	fmptd 6385	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))):ℕ0⟶𝐵)
142	7, 82, 9, 11, 57, 54, 16	ply1coefsupp 19665	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) finSupp (0g‘𝑃))
143		eqid 2622	. . . . . 6 ⊢ (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑎‘∅))
144	40, 100, 26, 143	mapsnf1o2 7905	. . . . 5 ⊢ (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑎‘∅)):(ℕ0 ↑𝑚 1𝑜)–1-1-onto→ℕ0
145	144	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑎‘∅)):(ℕ0 ↑𝑚 1𝑜)–1-1-onto→ℕ0)
146	10, 114, 118, 119, 141, 142, 145	gsumf1o 18317	. . 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‘∅)))))
147		eqidd 2623	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑎‘∅)))
148		eqidd 2623	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))))
149		fveq2 6191	. . . . . 6 ⊢ (𝑘 = (𝑎‘∅) → (𝐴‘𝑘) = (𝐴‘(𝑎‘∅)))
150		oveq1 6657	. . . . . 6 ⊢ (𝑘 = (𝑎‘∅) → (𝑘 ↑ 𝑋) = ((𝑎‘∅) ↑ 𝑋))
151	149, 150	oveq12d 6668	. . . . 5 ⊢ (𝑘 = (𝑎‘∅) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)) = ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋)))
152	81, 147, 148, 151	fmptco 6396	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ((𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) ∘ (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ (𝑎‘∅))) = (𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋))))
153	152	oveq2d 6666	. . 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)))))
154	112, 146, 153	3eqtrrd 2661	. 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)))))
155	15, 99, 154	3eqtrd 2660	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐾 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ifcif 4086  {csn 4177   ↦ cmpt 4729   ∘ ccom 5118  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  ωcom 7065  1_𝑜c1o 7553   ↑_𝑚 cmap 7857  ℕ₀cn0 11292  Basecbs 15857  +_gcplusg 15941  ._rcmulr 15942  Scalarcsca 15944   ·_𝑠 cvsca 15945  0_gc0g 16100   Σ_g cgsu 16101  Mndcmnd 17294  ._gcmg 17540  CMndccmn 18193  mulGrpcmgp 18489  1_rcur 18501  Ringcrg 18547  LModclmod 18863   mVar cmvr 19352   mPoly cmpl 19353  PwSer₁cps1 19545  var₁cv1 19546  Poly₁cpl1 19547  coe₁cco1 19548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-tset 15960  df-ple 15961  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-subrg 18778  df-lmod 18865  df-lss 18933  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553

This theorem is referenced by:  eqcoe1ply1eq  19667  pmatcollpw1lem2  20580  mp2pm2mp  20616  plypf1  23968