Theorem ply1coe 20458

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 2821	. . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅)
2		psr1baslem 20347	. . 3 ⊢ (ℕ0 ↑m 1o) = {𝑑 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑑 “ ℕ) ∈ Fin}
3		eqid 2821	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
4		eqid 2821	. . 3 ⊢ (1r‘𝑅) = (1r‘𝑅)
5		1onn 8259	. . . 4 ⊢ 1o ∈ ω
6	5	a1i 11	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 1o ∈ ω)
7		ply1coe.p	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
8		eqid 2821	. . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅)
9		ply1coe.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
10	7, 8, 9	ply1bas 20357	. . 3 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅))
11		ply1coe.n	. . . 4 ⊢ · = ( ·𝑠 ‘𝑃)
12	7, 1, 11	ply1vsca 20388	. . 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 20240	. 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 20369	. . . . . 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 2821	. . . . . . 7 ⊢ (mulGrp‘(1o mPoly 𝑅)) = (mulGrp‘(1o mPoly 𝑅))
21		eqid 2821	. . . . . . 7 ⊢ (.g‘(mulGrp‘(1o mPoly 𝑅))) = (.g‘(mulGrp‘(1o mPoly 𝑅)))
22		eqid 2821	. . . . . . 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 2822	. . . . . . . . . 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 5203	. . . . . . . . . . 11 ⊢ ∅ ∈ V
27		fveq2 6664	. . . . . . . . . . . . 13 ⊢ (𝑏 = ∅ → ((1o mVar 𝑅)‘𝑏) = ((1o mVar 𝑅)‘∅))
28	27	oveq1d 7165	. . . . . . . . . . . 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 7166	. . . . . . . . . . . 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 2837	. . . . . . . . . . 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 4612	. . . . . . . . . 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 6664	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → ((1o mVar 𝑅)‘𝑥) = ((1o mVar 𝑅)‘∅))
34	33	oveq2d 7166	. . . . . . . . . . . 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 7165	. . . . . . . . . . . 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 2837	. . . . . . . . . . 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 3197	. . . . . . . . . 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 4612	. . . . . . . . 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 8110	. . . . . . . . 9 ⊢ 1o = {∅}
41	40	raleqi 3413	. . . . . . . . 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 3234	. . . . . . . 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 20243	. . . . . 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 5148	. . . . . . . 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 7161	. . . . . . 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 20226	. . . . . . . . . . 11 ⊢ ((1o ∈ ω ∧ 𝑅 ∈ Ring) → (1o mPoly 𝑅) ∈ Ring)
48	5, 47	mpan 688	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (1o mPoly 𝑅) ∈ Ring)
49	20	ringmgp 19297	. . . . . . . . . 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 19239	. . . . . . . . . . . . 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 19239	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑀)
58	57	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐵 = (Base‘𝑀))
59		ssv 3990	. . . . . . . . . . . . 13 ⊢ 𝐵 ⊆ V
60	59	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐵 ⊆ V)
61		ovexd 7185	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(+g‘(mulGrp‘(1o mPoly 𝑅)))𝑏) ∈ V)
62		eqid 2821	. . . . . . . . . . . . . . . . 17 ⊢ (.r‘𝑃) = (.r‘𝑃)
63	7, 1, 62	ply1mulr 20389	. . . . . . . . . . . . . . . 16 ⊢ (.r‘𝑃) = (.r‘(1o mPoly 𝑅))
64	20, 63	mgpplusg 19237	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑃) = (+g‘(mulGrp‘(1o mPoly 𝑅)))
65	56, 62	mgpplusg 19237	. . . . . . . . . . . . . . 15 ⊢ (.r‘𝑃) = (+g‘𝑀)
66	64, 65	eqtr3i 2846	. . . . . . . . . . . . . 14 ⊢ (+g‘(mulGrp‘(1o mPoly 𝑅))) = (+g‘𝑀)
67	66	oveqi 7163	. . . . . . . . . . . . 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 18263	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (.g‘(mulGrp‘(1o mPoly 𝑅))) = ↑ )
70	69	oveqd 7167	. . . . . . . . . 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 20410	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
73	56	ringmgp 19297	. . . . . . . . . . . 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 8422	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (ℕ0 ↑m 1o) → 𝑎:1o⟶ℕ0)
77		0lt1o 8123	. . . . . . . . . . . 12 ⊢ ∅ ∈ 1o
78		ffvelrn 6843	. . . . . . . . . . . 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 20379	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵)
83	82	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑m 1o)) → 𝑋 ∈ 𝐵)
84	57, 53	mulgnn0cl 18238	. . . . . . . . . 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 2913	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ0 ↑m 1o)) → ((𝑎‘∅)(.g‘(mulGrp‘(1o mPoly 𝑅)))𝑋) ∈ 𝐵)
87		fveq2 6664	. . . . . . . . . 10 ⊢ (𝑐 = ∅ → (𝑎‘𝑐) = (𝑎‘∅))
88		fveq2 6664	. . . . . . . . . . 11 ⊢ (𝑐 = ∅ → ((1o mVar 𝑅)‘𝑐) = ((1o mVar 𝑅)‘∅))
89	81	vr1val 20354	. . . . . . . . . . 11 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅)
90	88, 89	syl6eqr 2874	. . . . . . . . . 10 ⊢ (𝑐 = ∅ → ((1o mVar 𝑅)‘𝑐) = 𝑋)
91	87, 90	oveq12d 7168	. . . . . . . . 9 ⊢ (𝑐 = ∅ → ((𝑎‘𝑐)(.g‘(mulGrp‘(1o mPoly 𝑅)))((1o mVar 𝑅)‘𝑐)) = ((𝑎‘∅)(.g‘(mulGrp‘(1o mPoly 𝑅)))𝑋))
92	54, 91	gsumsn 19068	. . . . . . . 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 2868	. . . . . 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 2860	. . . . 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 7168	. . . 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 5153	. . 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 7166	. 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 11897	. . . . . 6 ⊢ ℕ0 ∈ V
100	99	mptex 6980	. . . . 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 6678	. . . . 5 ⊢ 𝑃 ∈ V
103	102	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝑃 ∈ V)
104		ovexd 7185	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (1o mPoly 𝑅) ∈ V)
105	9, 10	eqtr3i 2846	. . . . 5 ⊢ (Base‘𝑃) = (Base‘(1o mPoly 𝑅))
106	105	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (Base‘𝑃) = (Base‘(1o mPoly 𝑅)))
107		eqid 2821	. . . . . 6 ⊢ (+g‘𝑃) = (+g‘𝑃)
108	7, 1, 107	ply1plusg 20387	. . . . 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 17882	. . 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 2821	. . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃)
112	1, 7, 111	ply1mpl0 20417	. . . 4 ⊢ (0g‘𝑃) = (0g‘(1o mPoly 𝑅))
113	1	mpllmod 20225	. . . . . 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 19676	. . . . 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 20414	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
119	118	ad2antrr 724	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ LMod)
120		eqid 2821	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
121	16, 9, 7, 120	coe1f 20373	. . . . . . . . 9 ⊢ (𝐾 ∈ 𝐵 → 𝐴:ℕ0⟶(Base‘𝑅))
122	121	adantl 484	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐴:ℕ0⟶(Base‘𝑅))
123	122	ffvelrnda 6845	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ (Base‘𝑅))
124	7	ply1sca 20415	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
125	124	eqcomd 2827	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
126	125	ad2antrr 724	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (Scalar‘𝑃) = 𝑅)
127	126	fveq2d 6668	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
128	123, 127	eleqtrrd 2916	. . . . . 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 18238	. . . . . . 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 2821	. . . . . . 7 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
135		eqid 2821	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
136	9, 134, 11, 135	lmodvscl 19645	. . . . . 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 6873	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))):ℕ0⟶𝐵)
139	7, 81, 9, 11, 56, 53, 16	ply1coefsupp 20457	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) finSupp (0g‘𝑃))
140		eqid 2821	. . . . . 6 ⊢ (𝑎 ∈ (ℕ0 ↑m 1o) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ0 ↑m 1o) ↦ (𝑎‘∅))
141	40, 99, 26, 140	mapsnf1o2 8452	. . . . 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 19030	. . 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 2822	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑎 ∈ (ℕ0 ↑m 1o) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ0 ↑m 1o) ↦ (𝑎‘∅)))
145		eqidd 2822	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))))
146		fveq2 6664	. . . . . 6 ⊢ (𝑘 = (𝑎‘∅) → (𝐴‘𝑘) = (𝐴‘(𝑎‘∅)))
147		oveq1 7157	. . . . . 6 ⊢ (𝑘 = (𝑎‘∅) → (𝑘 ↑ 𝑋) = ((𝑎‘∅) ↑ 𝑋))
148	146, 147	oveq12d 7168	. . . . 5 ⊢ (𝑘 = (𝑎‘∅) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)) = ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋)))
149	80, 144, 145, 148	fmptco 6885	. . . 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 7166	. . 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 2861	. 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 2860	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐾 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494   ⊆ wss 3935  ∅c0 4290  ifcif 4466  {csn 4560   ↦ cmpt 5138   ∘ ccom 5553  ⟶wf 6345  –1-1-onto→wf1o 6348  ‘cfv 6349  (class class class)co 7150  ωcom 7574  1_oc1o 8089   ↑_m cmap 8400  ℕ₀cn0 11891  Basecbs 16477  +_gcplusg 16559  ._rcmulr 16560  Scalarcsca 16562   ·_𝑠 cvsca 16563  0_gc0g 16707   Σ_g cgsu 16708  Mndcmnd 17905  ._gcmg 18218  CMndccmn 18900  mulGrpcmgp 19233  1_rcur 19245  Ringcrg 19291  LModclmod 19628   mVar cmvr 20126   mPoly cmpl 20127  PwSer₁cps1 20337  var₁cv1 20338  Poly₁cpl1 20339  coe₁cco1 20340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-iin 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-ofr 7404  df-om 7575  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-pm 8403  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fsupp 8828  df-oi 8968  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-fz 12887  df-fzo 13028  df-seq 13364  df-hash 13685  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-mulr 16573  df-sca 16575  df-vsca 16576  df-tset 16578  df-ple 16579  df-0g 16709  df-gsum 16710  df-mre 16851  df-mrc 16852  df-acs 16854  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-mhm 17950  df-submnd 17951  df-grp 18100  df-minusg 18101  df-sbg 18102  df-mulg 18219  df-subg 18270  df-ghm 18350  df-cntz 18441  df-cmn 18902  df-abl 18903  df-mgp 19234  df-ur 19246  df-srg 19250  df-ring 19293  df-subrg 19527  df-lmod 19630  df-lss 19698  df-psr 20130  df-mvr 20131  df-mpl 20132  df-opsr 20134  df-psr1 20342  df-vr1 20343  df-ply1 20344  df-coe1 20345

This theorem is referenced by:  eqcoe1ply1eq  20459  pmatcollpw1lem2  21377  mp2pm2mp  21413  plypf1  24796