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Theorem ply1coe 22192

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 Σ_g (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 2730	. . 3 ⊢ (1_o mPoly 𝑅) = (1_o mPoly 𝑅)
2		psr1baslem 22076	. . 3 ⊢ (ℕ₀ ↑_m 1_o) = {𝑑 ∈ (ℕ₀ ↑_m 1_o) ∣ (^◡𝑑 “ ℕ) ∈ Fin}
3		eqid 2730	. . 3 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
4		eqid 2730	. . 3 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
5		1onn 8607	. . . 4 ⊢ 1_o ∈ ω
6	5	a1i 11	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 1_o ∈ ω)
7		ply1coe.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
8		ply1coe.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
9	7, 8	ply1bas 22086	. . 3 ⊢ 𝐵 = (Base‘(1_o mPoly 𝑅))
10		ply1coe.n	. . . 4 ⊢ · = ( ·_𝑠 ‘𝑃)
11	7, 1, 10	ply1vsca 22116	. . 3 ⊢ · = ( ·_𝑠 ‘(1_o mPoly 𝑅))
12		simpl 482	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝑅 ∈ Ring)
13		simpr 484	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐾 ∈ 𝐵)
14	1, 2, 3, 4, 6, 9, 11, 12, 13	mplcoe1 21951	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐾 = ((1_o mPoly 𝑅) Σ_g (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ ((𝐾‘𝑎) · (𝑏 ∈ (ℕ₀ ↑_m 1_o) ↦ if(𝑏 = 𝑎, (1_r‘𝑅), (0_g‘𝑅)))))))
15		ply1coe.a	. . . . . . 7 ⊢ 𝐴 = (coe₁‘𝐾)
16	15	fvcoe1 22099	. . . . . 6 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → (𝐾‘𝑎) = (𝐴‘(𝑎‘∅)))
17	16	adantll 714	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → (𝐾‘𝑎) = (𝐴‘(𝑎‘∅)))
18	5	a1i 11	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → 1_o ∈ ω)
19		eqid 2730	. . . . . . 7 ⊢ (mulGrp‘(1_o mPoly 𝑅)) = (mulGrp‘(1_o mPoly 𝑅))
20		eqid 2730	. . . . . . 7 ⊢ (._g‘(mulGrp‘(1_o mPoly 𝑅))) = (._g‘(mulGrp‘(1_o mPoly 𝑅)))
21		eqid 2730	. . . . . . 7 ⊢ (1_o mVar 𝑅) = (1_o mVar 𝑅)
22		simpll 766	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → 𝑅 ∈ Ring)
23		simpr 484	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → 𝑎 ∈ (ℕ₀ ↑_m 1_o))
24		eqidd 2731	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → (((1_o mVar 𝑅)‘∅)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘∅)) = (((1_o mVar 𝑅)‘∅)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘∅)))
25		0ex 5265	. . . . . . . . . . 11 ⊢ ∅ ∈ V
26		fveq2 6861	. . . . . . . . . . . . 13 ⊢ (𝑏 = ∅ → ((1_o mVar 𝑅)‘𝑏) = ((1_o mVar 𝑅)‘∅))
27	26	oveq1d 7405	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → (((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘∅)) = (((1_o mVar 𝑅)‘∅)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘∅)))
28	26	oveq2d 7406	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → (((1_o mVar 𝑅)‘∅)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏)) = (((1_o mVar 𝑅)‘∅)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘∅)))
29	27, 28	eqeq12d 2746	. . . . . . . . . . 11 ⊢ (𝑏 = ∅ → ((((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘∅)) = (((1_o mVar 𝑅)‘∅)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏)) ↔ (((1_o mVar 𝑅)‘∅)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘∅)) = (((1_o mVar 𝑅)‘∅)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘∅))))
30	25, 29	ralsn 4648	. . . . . . . . . 10 ⊢ (∀𝑏 ∈ {∅} (((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘∅)) = (((1_o mVar 𝑅)‘∅)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏)) ↔ (((1_o mVar 𝑅)‘∅)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘∅)) = (((1_o mVar 𝑅)‘∅)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘∅)))
31	24, 30	sylibr 234	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → ∀𝑏 ∈ {∅} (((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘∅)) = (((1_o mVar 𝑅)‘∅)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏)))
32		fveq2 6861	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → ((1_o mVar 𝑅)‘𝑥) = ((1_o mVar 𝑅)‘∅))
33	32	oveq2d 7406	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑥)) = (((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘∅)))
34	32	oveq1d 7405	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (((1_o mVar 𝑅)‘𝑥)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏)) = (((1_o mVar 𝑅)‘∅)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏)))
35	33, 34	eqeq12d 2746	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → ((((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑥)) = (((1_o mVar 𝑅)‘𝑥)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏)) ↔ (((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘∅)) = (((1_o mVar 𝑅)‘∅)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏))))
36	35	ralbidv 3157	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (∀𝑏 ∈ {∅} (((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑥)) = (((1_o mVar 𝑅)‘𝑥)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏)) ↔ ∀𝑏 ∈ {∅} (((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘∅)) = (((1_o mVar 𝑅)‘∅)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏))))
37	25, 36	ralsn 4648	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {∅}∀𝑏 ∈ {∅} (((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑥)) = (((1_o mVar 𝑅)‘𝑥)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏)) ↔ ∀𝑏 ∈ {∅} (((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘∅)) = (((1_o mVar 𝑅)‘∅)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏)))
38	31, 37	sylibr 234	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → ∀𝑥 ∈ {∅}∀𝑏 ∈ {∅} (((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑥)) = (((1_o mVar 𝑅)‘𝑥)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏)))
39		df1o2 8444	. . . . . . . . 9 ⊢ 1_o = {∅}
40	39	raleqi 3299	. . . . . . . . 9 ⊢ (∀𝑏 ∈ 1_o (((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑥)) = (((1_o mVar 𝑅)‘𝑥)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏)) ↔ ∀𝑏 ∈ {∅} (((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑥)) = (((1_o mVar 𝑅)‘𝑥)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏)))
41	39, 40	raleqbii 3319	. . . . . . . 8 ⊢ (∀𝑥 ∈ 1_o ∀𝑏 ∈ 1_o (((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑥)) = (((1_o mVar 𝑅)‘𝑥)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏)) ↔ ∀𝑥 ∈ {∅}∀𝑏 ∈ {∅} (((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑥)) = (((1_o mVar 𝑅)‘𝑥)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏)))
42	38, 41	sylibr 234	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → ∀𝑥 ∈ 1_o ∀𝑏 ∈ 1_o (((1_o mVar 𝑅)‘𝑏)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑥)) = (((1_o mVar 𝑅)‘𝑥)(+_g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑏)))
43	1, 2, 3, 4, 18, 19, 20, 21, 22, 23, 42	mplcoe5 21954	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → (𝑏 ∈ (ℕ₀ ↑_m 1_o) ↦ if(𝑏 = 𝑎, (1_r‘𝑅), (0_g‘𝑅))) = ((mulGrp‘(1_o mPoly 𝑅)) Σ_g (𝑐 ∈ 1_o ↦ ((𝑎‘𝑐)(._g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑐)))))
44	39	mpteq1i 5201	. . . . . . . 8 ⊢ (𝑐 ∈ 1_o ↦ ((𝑎‘𝑐)(._g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑐))) = (𝑐 ∈ {∅} ↦ ((𝑎‘𝑐)(._g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑐)))
45	44	oveq2i 7401	. . . . . . 7 ⊢ ((mulGrp‘(1_o mPoly 𝑅)) Σ_g (𝑐 ∈ 1_o ↦ ((𝑎‘𝑐)(._g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑐)))) = ((mulGrp‘(1_o mPoly 𝑅)) Σ_g (𝑐 ∈ {∅} ↦ ((𝑎‘𝑐)(._g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑐))))
46	1	mplring 21935	. . . . . . . . . . 11 ⊢ ((1_o ∈ ω ∧ 𝑅 ∈ Ring) → (1_o mPoly 𝑅) ∈ Ring)
47	5, 46	mpan 690	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (1_o mPoly 𝑅) ∈ Ring)
48	19	ringmgp 20155	. . . . . . . . . 10 ⊢ ((1_o mPoly 𝑅) ∈ Ring → (mulGrp‘(1_o mPoly 𝑅)) ∈ Mnd)
49	47, 48	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (mulGrp‘(1_o mPoly 𝑅)) ∈ Mnd)
50	49	ad2antrr 726	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → (mulGrp‘(1_o mPoly 𝑅)) ∈ Mnd)
51	25	a1i 11	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → ∅ ∈ V)
52		ply1coe.e	. . . . . . . . . . . 12 ⊢ ↑ = (._g‘𝑀)
53	19, 9	mgpbas 20061	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘(mulGrp‘(1_o mPoly 𝑅)))
54	53	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐵 = (Base‘(mulGrp‘(1_o mPoly 𝑅))))
55		ply1coe.m	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (mulGrp‘𝑃)
56	55, 8	mgpbas 20061	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑀)
57	56	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐵 = (Base‘𝑀))
58		ssv 3974	. . . . . . . . . . . . 13 ⊢ 𝐵 ⊆ V
59	58	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐵 ⊆ V)
60		ovexd 7425	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(+_g‘(mulGrp‘(1_o mPoly 𝑅)))𝑏) ∈ V)
61		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑃) = (._r‘𝑃)
62	7, 1, 61	ply1mulr 22117	. . . . . . . . . . . . . . . 16 ⊢ (._r‘𝑃) = (._r‘(1_o mPoly 𝑅))
63	19, 62	mgpplusg 20060	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝑃) = (+_g‘(mulGrp‘(1_o mPoly 𝑅)))
64	55, 61	mgpplusg 20060	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝑃) = (+_g‘𝑀)
65	63, 64	eqtr3i 2755	. . . . . . . . . . . . . 14 ⊢ (+_g‘(mulGrp‘(1_o mPoly 𝑅))) = (+_g‘𝑀)
66	65	oveqi 7403	. . . . . . . . . . . . 13 ⊢ (𝑎(+_g‘(mulGrp‘(1_o mPoly 𝑅)))𝑏) = (𝑎(+_g‘𝑀)𝑏)
67	66	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(+_g‘(mulGrp‘(1_o mPoly 𝑅)))𝑏) = (𝑎(+_g‘𝑀)𝑏))
68	20, 52, 54, 57, 59, 60, 67	mulgpropd 19055	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (._g‘(mulGrp‘(1_o mPoly 𝑅))) = ↑ )
69	68	oveqd 7407	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ((𝑎‘∅)(._g‘(mulGrp‘(1_o mPoly 𝑅)))𝑋) = ((𝑎‘∅) ↑ 𝑋))
70	69	adantr 480	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → ((𝑎‘∅)(._g‘(mulGrp‘(1_o mPoly 𝑅)))𝑋) = ((𝑎‘∅) ↑ 𝑋))
71	7	ply1ring 22139	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
72	55	ringmgp 20155	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ Ring → 𝑀 ∈ Mnd)
73	71, 72	syl 17	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
74	73	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → 𝑀 ∈ Mnd)
75		elmapi 8825	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (ℕ₀ ↑_m 1_o) → 𝑎:1_o⟶ℕ₀)
76		0lt1o 8471	. . . . . . . . . . . 12 ⊢ ∅ ∈ 1_o
77		ffvelcdm 7056	. . . . . . . . . . . 12 ⊢ ((𝑎:1_o⟶ℕ₀ ∧ ∅ ∈ 1_o) → (𝑎‘∅) ∈ ℕ₀)
78	75, 76, 77	sylancl 586	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (ℕ₀ ↑_m 1_o) → (𝑎‘∅) ∈ ℕ₀)
79	78	adantl 481	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → (𝑎‘∅) ∈ ℕ₀)
80		ply1coe.x	. . . . . . . . . . . 12 ⊢ 𝑋 = (var₁‘𝑅)
81	80, 7, 8	vr1cl 22109	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵)
82	81	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → 𝑋 ∈ 𝐵)
83	56, 52, 74, 79, 82	mulgnn0cld 19034	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → ((𝑎‘∅) ↑ 𝑋) ∈ 𝐵)
84	70, 83	eqeltrd 2829	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → ((𝑎‘∅)(._g‘(mulGrp‘(1_o mPoly 𝑅)))𝑋) ∈ 𝐵)
85		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑐 = ∅ → (𝑎‘𝑐) = (𝑎‘∅))
86		fveq2 6861	. . . . . . . . . . 11 ⊢ (𝑐 = ∅ → ((1_o mVar 𝑅)‘𝑐) = ((1_o mVar 𝑅)‘∅))
87	80	vr1val 22083	. . . . . . . . . . 11 ⊢ 𝑋 = ((1_o mVar 𝑅)‘∅)
88	86, 87	eqtr4di 2783	. . . . . . . . . 10 ⊢ (𝑐 = ∅ → ((1_o mVar 𝑅)‘𝑐) = 𝑋)
89	85, 88	oveq12d 7408	. . . . . . . . 9 ⊢ (𝑐 = ∅ → ((𝑎‘𝑐)(._g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑐)) = ((𝑎‘∅)(._g‘(mulGrp‘(1_o mPoly 𝑅)))𝑋))
90	53, 89	gsumsn 19891	. . . . . . . 8 ⊢ (((mulGrp‘(1_o mPoly 𝑅)) ∈ Mnd ∧ ∅ ∈ V ∧ ((𝑎‘∅)(._g‘(mulGrp‘(1_o mPoly 𝑅)))𝑋) ∈ 𝐵) → ((mulGrp‘(1_o mPoly 𝑅)) Σ_g (𝑐 ∈ {∅} ↦ ((𝑎‘𝑐)(._g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑐)))) = ((𝑎‘∅)(._g‘(mulGrp‘(1_o mPoly 𝑅)))𝑋))
91	50, 51, 84, 90	syl3anc 1373	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → ((mulGrp‘(1_o mPoly 𝑅)) Σ_g (𝑐 ∈ {∅} ↦ ((𝑎‘𝑐)(._g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑐)))) = ((𝑎‘∅)(._g‘(mulGrp‘(1_o mPoly 𝑅)))𝑋))
92	45, 91	eqtrid 2777	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → ((mulGrp‘(1_o mPoly 𝑅)) Σ_g (𝑐 ∈ 1_o ↦ ((𝑎‘𝑐)(._g‘(mulGrp‘(1_o mPoly 𝑅)))((1_o mVar 𝑅)‘𝑐)))) = ((𝑎‘∅)(._g‘(mulGrp‘(1_o mPoly 𝑅)))𝑋))
93	43, 92, 70	3eqtrd 2769	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → (𝑏 ∈ (ℕ₀ ↑_m 1_o) ↦ if(𝑏 = 𝑎, (1_r‘𝑅), (0_g‘𝑅))) = ((𝑎‘∅) ↑ 𝑋))
94	17, 93	oveq12d 7408	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → ((𝐾‘𝑎) · (𝑏 ∈ (ℕ₀ ↑_m 1_o) ↦ if(𝑏 = 𝑎, (1_r‘𝑅), (0_g‘𝑅)))) = ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋)))
95	94	mpteq2dva 5203	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ ((𝐾‘𝑎) · (𝑏 ∈ (ℕ₀ ↑_m 1_o) ↦ if(𝑏 = 𝑎, (1_r‘𝑅), (0_g‘𝑅))))) = (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋))))
96	95	oveq2d 7406	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ((1_o mPoly 𝑅) Σ_g (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ ((𝐾‘𝑎) · (𝑏 ∈ (ℕ₀ ↑_m 1_o) ↦ if(𝑏 = 𝑎, (1_r‘𝑅), (0_g‘𝑅)))))) = ((1_o mPoly 𝑅) Σ_g (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋)))))
97		nn0ex 12455	. . . . . 6 ⊢ ℕ₀ ∈ V
98	97	mptex 7200	. . . . 5 ⊢ (𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) ∈ V
99	98	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) ∈ V)
100	7	fvexi 6875	. . . . 5 ⊢ 𝑃 ∈ V
101	100	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝑃 ∈ V)
102		ovexd 7425	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (1_o mPoly 𝑅) ∈ V)
103	8, 9	eqtr3i 2755	. . . . 5 ⊢ (Base‘𝑃) = (Base‘(1_o mPoly 𝑅))
104	103	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (Base‘𝑃) = (Base‘(1_o mPoly 𝑅)))
105		eqid 2730	. . . . . 6 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
106	7, 1, 105	ply1plusg 22115	. . . . 5 ⊢ (+_g‘𝑃) = (+_g‘(1_o mPoly 𝑅))
107	106	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (+_g‘𝑃) = (+_g‘(1_o mPoly 𝑅)))
108	99, 101, 102, 104, 107	gsumpropd 18612	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))) = ((1_o mPoly 𝑅) Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))))
109		eqid 2730	. . . . 5 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
110	1, 7, 109	ply1mpl0 22148	. . . 4 ⊢ (0_g‘𝑃) = (0_g‘(1_o mPoly 𝑅))
111	1	mpllmod 21934	. . . . . 6 ⊢ ((1_o ∈ ω ∧ 𝑅 ∈ Ring) → (1_o mPoly 𝑅) ∈ LMod)
112	5, 12, 111	sylancr 587	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (1_o mPoly 𝑅) ∈ LMod)
113		lmodcmn 20823	. . . . 5 ⊢ ((1_o mPoly 𝑅) ∈ LMod → (1_o mPoly 𝑅) ∈ CMnd)
114	112, 113	syl 17	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (1_o mPoly 𝑅) ∈ CMnd)
115	97	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ℕ₀ ∈ V)
116	7	ply1lmod 22143	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
117	116	ad2antrr 726	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑃 ∈ LMod)
118		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
119	15, 8, 7, 118	coe1f 22103	. . . . . . . . 9 ⊢ (𝐾 ∈ 𝐵 → 𝐴:ℕ₀⟶(Base‘𝑅))
120	119	adantl 481	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐴:ℕ₀⟶(Base‘𝑅))
121	120	ffvelcdmda 7059	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ (Base‘𝑅))
122	7	ply1sca 22144	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
123	122	eqcomd 2736	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
124	123	ad2antrr 726	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → (Scalar‘𝑃) = 𝑅)
125	124	fveq2d 6865	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
126	121, 125	eleqtrrd 2832	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ (Base‘(Scalar‘𝑃)))
127	73	ad2antrr 726	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑀 ∈ Mnd)
128		simpr 484	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
129	81	ad2antrr 726	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑋 ∈ 𝐵)
130	56, 52, 127, 128, 129	mulgnn0cld 19034	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ 𝐵)
131		eqid 2730	. . . . . . 7 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
132		eqid 2730	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
133	8, 131, 10, 132	lmodvscl 20791	. . . . . 6 ⊢ ((𝑃 ∈ LMod ∧ (𝐴‘𝑘) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑘 ↑ 𝑋) ∈ 𝐵) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)) ∈ 𝐵)
134	117, 126, 130, 133	syl3anc 1373	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)) ∈ 𝐵)
135	134	fmpttd 7090	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))):ℕ₀⟶𝐵)
136	7, 80, 8, 10, 55, 52, 15	ply1coefsupp 22191	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) finSupp (0_g‘𝑃))
137		eqid 2730	. . . . . 6 ⊢ (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑎‘∅))
138	39, 97, 25, 137	mapsnf1o2 8870	. . . . 5 ⊢ (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑎‘∅)):(ℕ₀ ↑_m 1_o)–1-1-onto→ℕ₀
139	138	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑎‘∅)):(ℕ₀ ↑_m 1_o)–1-1-onto→ℕ₀)
140	9, 110, 114, 115, 135, 136, 139	gsumf1o 19853	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ((1_o mPoly 𝑅) Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))) = ((1_o mPoly 𝑅) Σ_g ((𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) ∘ (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑎‘∅)))))
141		eqidd 2731	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑎‘∅)))
142		eqidd 2731	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))))
143		fveq2 6861	. . . . . 6 ⊢ (𝑘 = (𝑎‘∅) → (𝐴‘𝑘) = (𝐴‘(𝑎‘∅)))
144		oveq1 7397	. . . . . 6 ⊢ (𝑘 = (𝑎‘∅) → (𝑘 ↑ 𝑋) = ((𝑎‘∅) ↑ 𝑋))
145	143, 144	oveq12d 7408	. . . . 5 ⊢ (𝑘 = (𝑎‘∅) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)) = ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋)))
146	79, 141, 142, 145	fmptco 7104	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ((𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) ∘ (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑎‘∅))) = (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋))))
147	146	oveq2d 7406	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ((1_o mPoly 𝑅) Σ_g ((𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) ∘ (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑎‘∅)))) = ((1_o mPoly 𝑅) Σ_g (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋)))))
148	108, 140, 147	3eqtrrd 2770	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → ((1_o mPoly 𝑅) Σ_g (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋)))) = (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))))
149	14, 96, 148	3eqtrd 2769	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐾 = (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3045 Vcvv 3450 ⊆ wss 3917 ∅c0 4299 ifcif 4491 {csn 4592 ↦ cmpt 5191 ∘ ccom 5645 ⟶wf 6510 –1-1-onto→wf1o 6513 ‘cfv 6514 (class class class)co 7390 ωcom 7845 1_oc1o 8430 ↑_m cmap 8802 ℕ₀cn0 12449 Basecbs 17186 +_gcplusg 17227 ._rcmulr 17228 Scalarcsca 17230 ·_𝑠 cvsca 17231 0_gc0g 17409 Σ_g cgsu 17410 Mndcmnd 18668 ._gcmg 19006 CMndccmn 19717 mulGrpcmgp 20056 1_rcur 20097 Ringcrg 20149 LModclmod 20773 mVar cmvr 21821 mPoly cmpl 21822 var₁cv1 22067 Poly₁cpl1 22068 coe₁cco1 22069

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-ofr 7657 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-sup 9400 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-fz 13476 df-fzo 13623 df-seq 13974 df-hash 14303 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-hom 17251 df-cco 17252 df-0g 17411 df-gsum 17412 df-prds 17417 df-pws 17419 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-mhm 18717 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-mulg 19007 df-subg 19062 df-ghm 19152 df-cntz 19256 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-srg 20103 df-ring 20151 df-subrng 20462 df-subrg 20486 df-lmod 20775 df-lss 20845 df-psr 21825 df-mvr 21826 df-mpl 21827 df-opsr 21829 df-psr1 22071 df-vr1 22072 df-ply1 22073 df-coe1 22074

This theorem is referenced by: eqcoe1ply1eq 22193 evls1fpws 22263 pmatcollpw1lem2 22669 mp2pm2mp 22705 plypf1 26124 ply1degltdimlem 33625

W3C validator