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Theorem ply1coe 22213

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 2731	. . 3 ⊢ (1_o mPoly 𝑅) = (1_o mPoly 𝑅)
2		psr1baslem 22097	. . 3 ⊢ (ℕ₀ ↑_m 1_o) = {𝑑 ∈ (ℕ₀ ↑_m 1_o) ∣ (^◡𝑑 “ ℕ) ∈ Fin}
3		eqid 2731	. . 3 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
4		eqid 2731	. . 3 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
5		1onn 8555	. . . 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 22107	. . 3 ⊢ 𝐵 = (Base‘(1_o mPoly 𝑅))
10		ply1coe.n	. . . 4 ⊢ · = ( ·_𝑠 ‘𝑃)
11	7, 1, 10	ply1vsca 22137	. . 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 21972	. 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 22120	. . . . . 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 2731	. . . . . . 7 ⊢ (mulGrp‘(1_o mPoly 𝑅)) = (mulGrp‘(1_o mPoly 𝑅))
20		eqid 2731	. . . . . . 7 ⊢ (._g‘(mulGrp‘(1_o mPoly 𝑅))) = (._g‘(mulGrp‘(1_o mPoly 𝑅)))
21		eqid 2731	. . . . . . 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 2732	. . . . . . . . . 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 5243	. . . . . . . . . . 11 ⊢ ∅ ∈ V
26		fveq2 6822	. . . . . . . . . . . . 13 ⊢ (𝑏 = ∅ → ((1_o mVar 𝑅)‘𝑏) = ((1_o mVar 𝑅)‘∅))
27	26	oveq1d 7361	. . . . . . . . . . . 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 7362	. . . . . . . . . . . 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 2747	. . . . . . . . . . 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 4631	. . . . . . . . . 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 6822	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → ((1_o mVar 𝑅)‘𝑥) = ((1_o mVar 𝑅)‘∅))
33	32	oveq2d 7362	. . . . . . . . . . . 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 7361	. . . . . . . . . . . 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 2747	. . . . . . . . . . 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 3155	. . . . . . . . . 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 4631	. . . . . . . . 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 8392	. . . . . . . . 9 ⊢ 1_o = {∅}
40	39	raleqi 3290	. . . . . . . . 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 3310	. . . . . . . 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 21975	. . . . . 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 5180	. . . . . . . 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 7357	. . . . . . 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 21956	. . . . . . . . . . 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 20157	. . . . . . . . . 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 20063	. . . . . . . . . . . . 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 20063	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑀)
57	56	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐵 = (Base‘𝑀))
58		ssv 3954	. . . . . . . . . . . . 13 ⊢ 𝐵 ⊆ V
59	58	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐵 ⊆ V)
60		ovexd 7381	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(+_g‘(mulGrp‘(1_o mPoly 𝑅)))𝑏) ∈ V)
61		eqid 2731	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑃) = (._r‘𝑃)
62	7, 1, 61	ply1mulr 22138	. . . . . . . . . . . . . . . 16 ⊢ (._r‘𝑃) = (._r‘(1_o mPoly 𝑅))
63	19, 62	mgpplusg 20062	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝑃) = (+_g‘(mulGrp‘(1_o mPoly 𝑅)))
64	55, 61	mgpplusg 20062	. . . . . . . . . . . . . . 15 ⊢ (._r‘𝑃) = (+_g‘𝑀)
65	63, 64	eqtr3i 2756	. . . . . . . . . . . . . 14 ⊢ (+_g‘(mulGrp‘(1_o mPoly 𝑅))) = (+_g‘𝑀)
66	65	oveqi 7359	. . . . . . . . . . . . 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 19029	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (._g‘(mulGrp‘(1_o mPoly 𝑅))) = ↑ )
69	68	oveqd 7363	. . . . . . . . . 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 22160	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
72	55	ringmgp 20157	. . . . . . . . . . . 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 8773	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (ℕ₀ ↑_m 1_o) → 𝑎:1_o⟶ℕ₀)
76		0lt1o 8419	. . . . . . . . . . . 12 ⊢ ∅ ∈ 1_o
77		ffvelcdm 7014	. . . . . . . . . . . 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 22130	. . . . . . . . . . 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 19008	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → ((𝑎‘∅) ↑ 𝑋) ∈ 𝐵)
84	70, 83	eqeltrd 2831	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑎 ∈ (ℕ₀ ↑_m 1_o)) → ((𝑎‘∅)(._g‘(mulGrp‘(1_o mPoly 𝑅)))𝑋) ∈ 𝐵)
85		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑐 = ∅ → (𝑎‘𝑐) = (𝑎‘∅))
86		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑐 = ∅ → ((1_o mVar 𝑅)‘𝑐) = ((1_o mVar 𝑅)‘∅))
87	80	vr1val 22104	. . . . . . . . . . 11 ⊢ 𝑋 = ((1_o mVar 𝑅)‘∅)
88	86, 87	eqtr4di 2784	. . . . . . . . . 10 ⊢ (𝑐 = ∅ → ((1_o mVar 𝑅)‘𝑐) = 𝑋)
89	85, 88	oveq12d 7364	. . . . . . . . 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 19866	. . . . . . . 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 2778	. . . . . 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 2770	. . . . 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 7364	. . . 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 5182	. . 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 7362	. 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 12387	. . . . . 6 ⊢ ℕ₀ ∈ V
98	97	mptex 7157	. . . . 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 6836	. . . . 5 ⊢ 𝑃 ∈ V
101	100	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝑃 ∈ V)
102		ovexd 7381	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (1_o mPoly 𝑅) ∈ V)
103	8, 9	eqtr3i 2756	. . . . 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 2731	. . . . . 6 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
106	7, 1, 105	ply1plusg 22136	. . . . 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 18586	. . 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 2731	. . . . 5 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
110	1, 7, 109	ply1mpl0 22169	. . . 4 ⊢ (0_g‘𝑃) = (0_g‘(1_o mPoly 𝑅))
111	1	mpllmod 21955	. . . . . 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 20843	. . . . 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 22164	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
117	116	ad2antrr 726	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → 𝑃 ∈ LMod)
118		eqid 2731	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
119	15, 8, 7, 118	coe1f 22124	. . . . . . . . 9 ⊢ (𝐾 ∈ 𝐵 → 𝐴:ℕ₀⟶(Base‘𝑅))
120	119	adantl 481	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → 𝐴:ℕ₀⟶(Base‘𝑅))
121	120	ffvelcdmda 7017	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ (Base‘𝑅))
122	7	ply1sca 22165	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
123	122	eqcomd 2737	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
124	123	ad2antrr 726	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → (Scalar‘𝑃) = 𝑅)
125	124	fveq2d 6826	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
126	121, 125	eleqtrrd 2834	. . . . . 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 19008	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → (𝑘 ↑ 𝑋) ∈ 𝐵)
131		eqid 2731	. . . . . . 7 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
132		eqid 2731	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
133	8, 131, 10, 132	lmodvscl 20811	. . . . . 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 7048	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))):ℕ₀⟶𝐵)
136	7, 80, 8, 10, 55, 52, 15	ply1coefsupp 22212	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) finSupp (0_g‘𝑃))
137		eqid 2731	. . . . . 6 ⊢ (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑎‘∅))
138	39, 97, 25, 137	mapsnf1o2 8818	. . . . 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 19828	. . 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 2732	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑎‘∅)))
142		eqidd 2732	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋))))
143		fveq2 6822	. . . . . 6 ⊢ (𝑘 = (𝑎‘∅) → (𝐴‘𝑘) = (𝐴‘(𝑎‘∅)))
144		oveq1 7353	. . . . . 6 ⊢ (𝑘 = (𝑎‘∅) → (𝑘 ↑ 𝑋) = ((𝑎‘∅) ↑ 𝑋))
145	143, 144	oveq12d 7364	. . . . 5 ⊢ (𝑘 = (𝑎‘∅) → ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)) = ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) ↑ 𝑋)))
146	79, 141, 142, 145	fmptco 7062	. . . 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 7362	. . 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 2771	. 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 2770	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 1541 ∈ wcel 2111 ∀wral 3047 Vcvv 3436 ⊆ wss 3897 ∅c0 4280 ifcif 4472 {csn 4573 ↦ cmpt 5170 ∘ ccom 5618 ⟶wf 6477 –1-1-onto→wf1o 6480 ‘cfv 6481 (class class class)co 7346 ωcom 7796 1_oc1o 8378 ↑_m cmap 8750 ℕ₀cn0 12381 Basecbs 17120 +_gcplusg 17161 ._rcmulr 17162 Scalarcsca 17164 ·_𝑠 cvsca 17165 0_gc0g 17343 Σ_g cgsu 17344 Mndcmnd 18642 ._gcmg 18980 CMndccmn 19692 mulGrpcmgp 20058 1_rcur 20099 Ringcrg 20151 LModclmod 20793 mVar cmvr 21842 mPoly cmpl 21843 var₁cv1 22088 Poly₁cpl1 22089 coe₁cco1 22090

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-ofr 7611 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-fz 13408 df-fzo 13555 df-seq 13909 df-hash 14238 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-ghm 19125 df-cntz 19229 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-srg 20105 df-ring 20153 df-subrng 20461 df-subrg 20485 df-lmod 20795 df-lss 20865 df-psr 21846 df-mvr 21847 df-mpl 21848 df-opsr 21850 df-psr1 22092 df-vr1 22093 df-ply1 22094 df-coe1 22095

This theorem is referenced by: eqcoe1ply1eq 22214 evls1fpws 22284 pmatcollpw1lem2 22690 mp2pm2mp 22726 plypf1 26144 ply1degltdimlem 33635

W3C validator