Theorem ply1coe 20466

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 2823	. . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅)
2		psr1baslem 20355	. . 3 ⊢ (ℕ0 ↑m 1o) = {𝑑 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑑 “ ℕ) ∈ Fin}
3		eqid 2823	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
4		eqid 2823	. . 3 ⊢ (1r‘𝑅) = (1r‘𝑅)
5		1onn 8267	. . . 4 ⊢ 1o ∈ ω
6	5	a1i 11	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 1o ∈ ω)
7		ply1coe.p	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
8		eqid 2823	. . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅)
9		ply1coe.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
10	7, 8, 9	ply1bas 20365	. . 3 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅))
11		ply1coe.n	. . . 4 ⊢ · = ( ·𝑠 ‘𝑃)
12	7, 1, 11	ply1vsca 20396	. . 3 ⊢ · = ( ·𝑠 ‘(1o mPoly 𝑅))
13		simpl 485	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝑅 ∈ Ring)
14		simpr 487	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐾 ∈ 𝐵)
15	1, 2, 3, 4, 6, 10, 12, 13, 14	mplcoe1 20248	. 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 20377	. . . . . 6 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑎 ∈ (ℕ0 ↑m 1o)) → (𝐾‘𝑎) = (𝐴‘(𝑎‘∅)))
18	17	adantll 712	. . . . 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 2823	. . . . . . 7 ⊢ (mulGrp‘(1o mPoly 𝑅)) = (mulGrp‘(1o mPoly 𝑅))
21		eqid 2823	. . . . . . 7 ⊢ (.g‘(mulGrp‘(1o mPoly 𝑅))) = (.g‘(mulGrp‘(1o mPoly 𝑅)))
22		eqid 2823	. . . . . . 7 ⊢ (1o mVar 𝑅) = (1o mVar 𝑅)
23		simpll 765	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑m 1o)) → 𝑅 ∈ Ring)
24		simpr 487	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑m 1o)) → 𝑎 ∈ (ℕ0 ↑m 1o))
25		eqidd 2824	. . . . . . . . . 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 5213	. . . . . . . . . . 11 ⊢ ∅ ∈ V
27		fveq2 6672	. . . . . . . . . . . . 13 ⊢ (𝑏 = ∅ → ((1o mVar 𝑅)‘𝑏) = ((1o mVar 𝑅)‘∅))
28	27	oveq1d 7173	. . . . . . . . . . . 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 7174	. . . . . . . . . . . 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 2839	. . . . . . . . . . 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 4621	. . . . . . . . . 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 236	. . . . . . . . 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 6672	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → ((1o mVar 𝑅)‘𝑥) = ((1o mVar 𝑅)‘∅))
34	33	oveq2d 7174	. . . . . . . . . . . 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 7173	. . . . . . . . . . . 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 2839	. . . . . . . . . . 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 3199	. . . . . . . . . 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 4621	. . . . . . . . 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 236	. . . . . . . 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 8118	. . . . . . . . 9 ⊢ 1o = {∅}
41	40	raleqi 3415	. . . . . . . . 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 3236	. . . . . . . 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 236	. . . . . . 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 20251	. . . . . 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	40	mpteq1i 5158	. . . . . . . 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)))
46	45	oveq2i 7169	. . . . . . 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))))
47	1	mplring 20234	. . . . . . . . . . 11 ⊢ ((1o ∈ ω ∧ 𝑅 ∈ Ring) → (1o mPoly 𝑅) ∈ Ring)
48	5, 47	mpan 688	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (1o mPoly 𝑅) ∈ Ring)
49	20	ringmgp 19305	. . . . . . . . . 10 ⊢ ((1o mPoly 𝑅) ∈ Ring → (mulGrp‘(1o mPoly 𝑅)) ∈ Mnd)
50	48, 49	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (mulGrp‘(1o mPoly 𝑅)) ∈ Mnd)
51	50	ad2antrr 724	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑m 1o)) → (mulGrp‘(1o mPoly 𝑅)) ∈ Mnd)
52	26	a1i 11	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑m 1o)) → ∅ ∈ V)
53		ply1coe.e	. . . . . . . . . . . 12 ⊢ ↑ = (.g‘𝑀)
54	20, 10	mgpbas 19247	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘(mulGrp‘(1o mPoly 𝑅)))
55	54	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐵 = (Base‘(mulGrp‘(1o mPoly 𝑅))))
56		ply1coe.m	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (mulGrp‘𝑃)
57	56, 9	mgpbas 19247	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑀)
58	57	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐵 = (Base‘𝑀))
59		ssv 3993	. . . . . . . . . . . . 13 ⊢ 𝐵 ⊆ V
60	59	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐵 ⊆ V)
61		ovexd 7193	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(+g‘(mulGrp‘(1o mPoly 𝑅)))𝑏) ∈ V)
62		eqid 2823	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘𝑃) = (.r‘𝑃)
63	7, 1, 62	ply1mulr 20397	. . . . . . . . . . . . . . . 16 ⊢ (.r‘𝑃) = (.r‘(1o mPoly 𝑅))
64	20, 63	mgpplusg 19245	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑃) = (+g‘(mulGrp‘(1o mPoly 𝑅)))
65	56, 62	mgpplusg 19245	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑃) = (+g‘𝑀)
66	64, 65	eqtr3i 2848	. . . . . . . . . . . . . 14 ⊢ (+g‘(mulGrp‘(1o mPoly 𝑅))) = (+g‘𝑀)
67	66	oveqi 7171	. . . . . . . . . . . . 13 ⊢ (𝑎(+g‘(mulGrp‘(1o mPoly 𝑅)))𝑏) = (𝑎(+g‘𝑀)𝑏)
68	67	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(+g‘(mulGrp‘(1o mPoly 𝑅)))𝑏) = (𝑎(+g‘𝑀)𝑏))
69	21, 53, 55, 58, 60, 61, 68	mulgpropd 18271	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (.g‘(mulGrp‘(1o mPoly 𝑅))) = ↑ )
70	69	oveqd 7175	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ((𝑎‘∅)(.g‘(mulGrp‘(1o mPoly 𝑅)))𝑋) = ((𝑎‘∅) ↑ 𝑋))
71	70	adantr 483	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑m 1o)) → ((𝑎‘∅)(.g‘(mulGrp‘(1o mPoly 𝑅)))𝑋) = ((𝑎‘∅) ↑ 𝑋))
72	7	ply1ring 20418	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
73	56	ringmgp 19305	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ Ring → 𝑀 ∈ Mnd)
74	72, 73	syl 17	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
75	74	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑m 1o)) → 𝑀 ∈ Mnd)
76		elmapi 8430	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (ℕ0 ↑m 1o) → 𝑎:1o⟶ℕ0)
77		0lt1o 8131	. . . . . . . . . . . 12 ⊢ ∅ ∈ 1o
78		ffvelrn 6851	. . . . . . . . . . . 12 ⊢ ((𝑎:1o⟶ℕ0 ∧ ∅ ∈ 1o) → (𝑎‘∅) ∈ ℕ0)
79	76, 77, 78	sylancl 588	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (ℕ0 ↑m 1o) → (𝑎‘∅) ∈ ℕ0)
80	79	adantl 484	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑m 1o)) → (𝑎‘∅) ∈ ℕ0)
81		ply1coe.x	. . . . . . . . . . . 12 ⊢ 𝑋 = (var1‘𝑅)
82	81, 7, 9	vr1cl 20387	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵)
83	82	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑m 1o)) → 𝑋 ∈ 𝐵)
84	57, 53	mulgnn0cl 18246	. . . . . . . . . 10 ⊢ ((𝑀 ∈ Mnd ∧ (𝑎‘∅) ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑎‘∅) ↑ 𝑋) ∈ 𝐵)
85	75, 80, 83, 84	syl3anc 1367	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑m 1o)) → ((𝑎‘∅) ↑ 𝑋) ∈ 𝐵)
86	71, 85	eqeltrd 2915	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑m 1o)) → ((𝑎‘∅)(.g‘(mulGrp‘(1o mPoly 𝑅)))𝑋) ∈ 𝐵)
87		fveq2 6672	. . . . . . . . . 10 ⊢ (𝑐 = ∅ → (𝑎‘𝑐) = (𝑎‘∅))
88		fveq2 6672	. . . . . . . . . . 11 ⊢ (𝑐 = ∅ → ((1o mVar 𝑅)‘𝑐) = ((1o mVar 𝑅)‘∅))
89	81	vr1val 20362	. . . . . . . . . . 11 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅)
90	88, 89	syl6eqr 2876	. . . . . . . . . 10 ⊢ (𝑐 = ∅ → ((1o mVar 𝑅)‘𝑐) = 𝑋)
91	87, 90	oveq12d 7176	. . . . . . . . 9 ⊢ (𝑐 = ∅ → ((𝑎‘𝑐)(.g‘(mulGrp‘(1o mPoly 𝑅)))((1o mVar 𝑅)‘𝑐)) = ((𝑎‘∅)(.g‘(mulGrp‘(1o mPoly 𝑅)))𝑋))
92	54, 91	gsumsn 19076	. . . . . . . 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))
93	51, 52, 86, 92	syl3anc 1367	. . . . . . 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))
94	46, 93	syl5eq 2870	. . . . . 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))
95	44, 94, 71	3eqtrd 2862	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑m 1o)) → (𝑏 ∈ (ℕ0 ↑m 1o) ↦ if(𝑏 = 𝑎, (1r‘𝑅), (0g‘𝑅))) = ((𝑎‘∅) ↑ 𝑋))
96	18, 95	oveq12d 7176	. . . 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)))
97	96	mpteq2dva 5163	. . 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))))
98	97	oveq2d 7174	. 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)))))
99		nn0ex 11906	. . . . . 6 ⊢ ℕ0 ∈ V
100	99	mptex 6988	. . . . 5 ⊢ (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) ∈ V
101	100	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) ∈ V)
102	7	fvexi 6686	. . . . 5 ⊢ 𝑃 ∈ V
103	102	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝑃 ∈ V)
104		ovexd 7193	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (1o mPoly 𝑅) ∈ V)
105	9, 10	eqtr3i 2848	. . . . 5 ⊢ (Base‘𝑃) = (Base‘(1o mPoly 𝑅))
106	105	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (Base‘𝑃) = (Base‘(1o mPoly 𝑅)))
107		eqid 2823	. . . . . 6 ⊢ (+g‘𝑃) = (+g‘𝑃)
108	7, 1, 107	ply1plusg 20395	. . . . 5 ⊢ (+g‘𝑃) = (+g‘(1o mPoly 𝑅))
109	108	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (+g‘𝑃) = (+g‘(1o mPoly 𝑅)))
110	101, 103, 104, 106, 109	gsumpropd 17890	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))) = ((1o mPoly 𝑅) Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))))
111		eqid 2823	. . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃)
112	1, 7, 111	ply1mpl0 20425	. . . 4 ⊢ (0g‘𝑃) = (0g‘(1o mPoly 𝑅))
113	1	mpllmod 20233	. . . . . 6 ⊢ ((1o ∈ ω ∧ 𝑅 ∈ Ring) → (1o mPoly 𝑅) ∈ LMod)
114	5, 13, 113	sylancr 589	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (1o mPoly 𝑅) ∈ LMod)
115		lmodcmn 19684	. . . . 5 ⊢ ((1o mPoly 𝑅) ∈ LMod → (1o mPoly 𝑅) ∈ CMnd)
116	114, 115	syl 17	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (1o mPoly 𝑅) ∈ CMnd)
117	99	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ℕ0 ∈ V)
118	7	ply1lmod 20422	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
119	118	ad2antrr 724	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ LMod)
120		eqid 2823	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
121	16, 9, 7, 120	coe1f 20381	. . . . . . . . 9 ⊢ (𝐾 ∈ 𝐵 → 𝐴:ℕ0⟶(Base‘𝑅))
122	121	adantl 484	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐴:ℕ0⟶(Base‘𝑅))
123	122	ffvelrnda 6853	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ (Base‘𝑅))
124	7	ply1sca 20423	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
125	124	eqcomd 2829	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
126	125	ad2antrr 724	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (Scalar‘𝑃) = 𝑅)
127	126	fveq2d 6676	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
128	123, 127	eleqtrrd 2918	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ (Base‘(Scalar‘𝑃)))
129	74	ad2antrr 724	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ Mnd)
130		simpr 487	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
131	82	ad2antrr 724	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑋 ∈ 𝐵)
132	57, 53	mulgnn0cl 18246	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑘 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑘 ↑ 𝑋) ∈ 𝐵)
133	129, 130, 131, 132	syl3anc 1367	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑘 ↑ 𝑋) ∈ 𝐵)
134		eqid 2823	. . . . . . 7 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
135		eqid 2823	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
136	9, 134, 11, 135	lmodvscl 19653	. . . . . 6 ⊢ ((𝑃 ∈ LMod ∧ (𝐴‘𝑘) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑘 ↑ 𝑋) ∈ 𝐵) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)) ∈ 𝐵)
137	119, 128, 133, 136	syl3anc 1367	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)) ∈ 𝐵)
138	137	fmpttd 6881	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))):ℕ0⟶𝐵)
139	7, 81, 9, 11, 56, 53, 16	ply1coefsupp 20465	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) finSupp (0g‘𝑃))
140		eqid 2823	. . . . . 6 ⊢ (𝑎 ∈ (ℕ0 ↑m 1o) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ0 ↑m 1o) ↦ (𝑎‘∅))
141	40, 99, 26, 140	mapsnf1o2 8460	. . . . 5 ⊢ (𝑎 ∈ (ℕ0 ↑m 1o) ↦ (𝑎‘∅)):(ℕ0 ↑m 1o)–1-1-onto→ℕ0
142	141	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑎 ∈ (ℕ0 ↑m 1o) ↦ (𝑎‘∅)):(ℕ0 ↑m 1o)–1-1-onto→ℕ0)
143	10, 112, 116, 117, 138, 139, 142	gsumf1o 19038	. . 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‘∅)))))
144		eqidd 2824	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑎 ∈ (ℕ0 ↑m 1o) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ0 ↑m 1o) ↦ (𝑎‘∅)))
145		eqidd 2824	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))))
146		fveq2 6672	. . . . . 6 ⊢ (𝑘 = (𝑎‘∅) → (𝐴‘𝑘) = (𝐴‘(𝑎‘∅)))
147		oveq1 7165	. . . . . 6 ⊢ (𝑘 = (𝑎‘∅) → (𝑘 ↑ 𝑋) = ((𝑎‘∅) ↑ 𝑋))
148	146, 147	oveq12d 7176	. . . . 5 ⊢ (𝑘 = (𝑎‘∅) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)) = ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋)))
149	80, 144, 145, 148	fmptco 6893	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ((𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) ∘ (𝑎 ∈ (ℕ0 ↑m 1o) ↦ (𝑎‘∅))) = (𝑎 ∈ (ℕ0 ↑m 1o) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋))))
150	149	oveq2d 7174	. . 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)))))
151	110, 143, 150	3eqtrrd 2863	. 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)))))
152	15, 98, 151	3eqtrd 2862	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐾 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  Vcvv 3496   ⊆ wss 3938  ∅c0 4293  ifcif 4469  {csn 4569   ↦ cmpt 5148   ∘ ccom 5561  ⟶wf 6353  –1-1-onto→wf1o 6356  ‘cfv 6357  (class class class)co 7158  ωcom 7582  1_oc1o 8097   ↑_m cmap 8408  ℕ₀cn0 11900  Basecbs 16485  +_gcplusg 16567  ._rcmulr 16568  Scalarcsca 16570   ·_𝑠 cvsca 16571  0_gc0g 16715   Σ_g cgsu 16716  Mndcmnd 17913  ._gcmg 18226  CMndccmn 18908  mulGrpcmgp 19241  1_rcur 19253  Ringcrg 19299  LModclmod 19636   mVar cmvr 20134   mPoly cmpl 20135  PwSer₁cps1 20345  var₁cv1 20346  Poly₁cpl1 20347  coe₁cco1 20348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-of 7411  df-ofr 7412  df-om 7583  df-1st 7691  df-2nd 7692  df-supp 7833  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-fsupp 8836  df-oi 8976  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-fz 12896  df-fzo 13037  df-seq 13373  df-hash 13694  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-mulr 16581  df-sca 16583  df-vsca 16584  df-tset 16586  df-ple 16587  df-0g 16717  df-gsum 16718  df-mre 16859  df-mrc 16860  df-acs 16862  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-mhm 17958  df-submnd 17959  df-grp 18108  df-minusg 18109  df-sbg 18110  df-mulg 18227  df-subg 18278  df-ghm 18358  df-cntz 18449  df-cmn 18910  df-abl 18911  df-mgp 19242  df-ur 19254  df-srg 19258  df-ring 19301  df-subrg 19535  df-lmod 19638  df-lss 19706  df-psr 20138  df-mvr 20139  df-mpl 20140  df-opsr 20142  df-psr1 20350  df-vr1 20351  df-ply1 20352  df-coe1 20353

This theorem is referenced by:  eqcoe1ply1eq  20467  pmatcollpw1lem2  21385  mp2pm2mp  21421  plypf1  24804