Theorem dvply2g 26446

Description: The derivative of a polynomial with coefficients in a subring is a polynomial with coefficients in the same ring. (Contributed by Mario Carneiro, 1-Jan-2017.) Avoid ax-mulf 11175. (Revised by GG, 30-Apr-2025.)

Assertion

Ref	Expression
dvply2g	⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℂ D 𝐹) ∈ (Poly‘𝑆))

Proof of Theorem dvply2g

Dummy variables 𝑎 𝑏 𝑐 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plyf 26355	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
2	1	adantl 486	. . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
3	2	feqmptd 6949	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹 = (𝑎 ∈ ℂ ↦ (𝐹‘𝑎)))
4		simplr 780	. . . . . 6 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑎 ∈ ℂ) → 𝐹 ∈ (Poly‘𝑆))
5		dgrcl 26390	. . . . . . . . . 10 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
6	5	adantl 486	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (deg‘𝐹) ∈ ℕ0)
7	6	nn0zd 12611	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (deg‘𝐹) ∈ ℤ)
8	7	adantr 485	. . . . . . 7 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑎 ∈ ℂ) → (deg‘𝐹) ∈ ℤ)
9		uzid 12872	. . . . . . 7 ⊢ ((deg‘𝐹) ∈ ℤ → (deg‘𝐹) ∈ (ℤ≥‘(deg‘𝐹)))
10		peano2uz 12920	. . . . . . 7 ⊢ ((deg‘𝐹) ∈ (ℤ≥‘(deg‘𝐹)) → ((deg‘𝐹) + 1) ∈ (ℤ≥‘(deg‘𝐹)))
11	8, 9, 10	3syl 19	. . . . . 6 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑎 ∈ ℂ) → ((deg‘𝐹) + 1) ∈ (ℤ≥‘(deg‘𝐹)))
12		simpr 489	. . . . . 6 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑎 ∈ ℂ) → 𝑎 ∈ ℂ)
13		eqid 2763	. . . . . . 7 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
14		eqid 2763	. . . . . . 7 ⊢ (deg‘𝐹) = (deg‘𝐹)
15	13, 14	coeid3 26397	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ ((deg‘𝐹) + 1) ∈ (ℤ≥‘(deg‘𝐹)) ∧ 𝑎 ∈ ℂ) → (𝐹‘𝑎) = Σ𝑏 ∈ (0...((deg‘𝐹) + 1))(((coeff‘𝐹)‘𝑏) · (𝑎↑𝑏)))
16	4, 11, 12, 15	syl3anc 1398	. . . . 5 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑎 ∈ ℂ) → (𝐹‘𝑎) = Σ𝑏 ∈ (0...((deg‘𝐹) + 1))(((coeff‘𝐹)‘𝑏) · (𝑎↑𝑏)))
17	16	mpteq2dva 5204	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (𝑎 ∈ ℂ ↦ (𝐹‘𝑎)) = (𝑎 ∈ ℂ ↦ Σ𝑏 ∈ (0...((deg‘𝐹) + 1))(((coeff‘𝐹)‘𝑏) · (𝑎↑𝑏))))
18	3, 17	eqtrd 2798	. . 3 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹 = (𝑎 ∈ ℂ ↦ Σ𝑏 ∈ (0...((deg‘𝐹) + 1))(((coeff‘𝐹)‘𝑏) · (𝑎↑𝑏))))
19	6	nn0cnd 12562	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (deg‘𝐹) ∈ ℂ)
20		ax-1cn 11153	. . . . . . . 8 ⊢ 1 ∈ ℂ
21		pncan 11458	. . . . . . . 8 ⊢ (((deg‘𝐹) ∈ ℂ ∧ 1 ∈ ℂ) → (((deg‘𝐹) + 1) − 1) = (deg‘𝐹))
22	19, 20, 21	sylancl 597	. . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (((deg‘𝐹) + 1) − 1) = (deg‘𝐹))
23	22	eqcomd 2769	. . . . . 6 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (deg‘𝐹) = (((deg‘𝐹) + 1) − 1))
24	23	oveq2d 7426	. . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (0...(deg‘𝐹)) = (0...(((deg‘𝐹) + 1) − 1)))
25	24	sumeq1d 15747	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → Σ𝑏 ∈ (0...(deg‘𝐹))(((𝑐 ∈ ℕ0 ↦ ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1))))‘𝑏) · (𝑎↑𝑏)) = Σ𝑏 ∈ (0...(((deg‘𝐹) + 1) − 1))(((𝑐 ∈ ℕ0 ↦ ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1))))‘𝑏) · (𝑎↑𝑏)))
26	25	mpteq2dv 5205	. . 3 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (𝑎 ∈ ℂ ↦ Σ𝑏 ∈ (0...(deg‘𝐹))(((𝑐 ∈ ℕ0 ↦ ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1))))‘𝑏) · (𝑎↑𝑏))) = (𝑎 ∈ ℂ ↦ Σ𝑏 ∈ (0...(((deg‘𝐹) + 1) − 1))(((𝑐 ∈ ℕ0 ↦ ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1))))‘𝑏) · (𝑎↑𝑏))))
27	13	coef3 26389	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
28	27	adantl 486	. . 3 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘𝐹):ℕ0⟶ℂ)
29		oveq1 7417	. . . . 5 ⊢ (𝑐 = 𝑏 → (𝑐 + 1) = (𝑏 + 1))
30		fvoveq1 7433	. . . . 5 ⊢ (𝑐 = 𝑏 → ((coeff‘𝐹)‘(𝑐 + 1)) = ((coeff‘𝐹)‘(𝑏 + 1)))
31	29, 30	oveq12d 7428	. . . 4 ⊢ (𝑐 = 𝑏 → ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1))) = ((𝑏 + 1) · ((coeff‘𝐹)‘(𝑏 + 1))))
32	31	cbvmptv 5215	. . 3 ⊢ (𝑐 ∈ ℕ0 ↦ ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1)))) = (𝑏 ∈ ℕ0 ↦ ((𝑏 + 1) · ((coeff‘𝐹)‘(𝑏 + 1))))
33		peano2nn0 12539	. . . 4 ⊢ ((deg‘𝐹) ∈ ℕ0 → ((deg‘𝐹) + 1) ∈ ℕ0)
34	6, 33	syl 18	. . 3 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((deg‘𝐹) + 1) ∈ ℕ0)
35	18, 26, 28, 32, 34	dvply1 26445	. 2 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℂ D 𝐹) = (𝑎 ∈ ℂ ↦ Σ𝑏 ∈ (0...(deg‘𝐹))(((𝑐 ∈ ℕ0 ↦ ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1))))‘𝑏) · (𝑎↑𝑏))))
36		cnfldbas 21526	. . . . 5 ⊢ ℂ = (Base‘ℂfld)
37	36	subrgss 20671	. . . 4 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ⊆ ℂ)
38	37	adantr 485	. . 3 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝑆 ⊆ ℂ)
39		elfznn0 13644	. . . 4 ⊢ (𝑏 ∈ (0...(deg‘𝐹)) → 𝑏 ∈ ℕ0)
40		simpll 778	. . . . . . 7 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → 𝑆 ∈ (SubRing‘ℂfld))
41		zsssubrg 21575	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → ℤ ⊆ 𝑆)
42	41	ad2antrr 738	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → ℤ ⊆ 𝑆)
43		peano2nn0 12539	. . . . . . . . . 10 ⊢ (𝑐 ∈ ℕ0 → (𝑐 + 1) ∈ ℕ0)
44	43	adantl 486	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → (𝑐 + 1) ∈ ℕ0)
45	44	nn0zd 12611	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → (𝑐 + 1) ∈ ℤ)
46	42, 45	sseldd 3938	. . . . . . 7 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → (𝑐 + 1) ∈ 𝑆)
47		simplr 780	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → 𝐹 ∈ (Poly‘𝑆))
48		subrgsubg 20676	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ∈ (SubGrp‘ℂfld))
49		cnfld0 21546	. . . . . . . . . . . 12 ⊢ 0 = (0g‘ℂfld)
50	49	subg0cl 19195	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubGrp‘ℂfld) → 0 ∈ 𝑆)
51	48, 50	syl 18	. . . . . . . . . 10 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 0 ∈ 𝑆)
52	51	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → 0 ∈ 𝑆)
53	13	coef2 26388	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 0 ∈ 𝑆) → (coeff‘𝐹):ℕ0⟶𝑆)
54	47, 52, 53	syl2anc 595	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → (coeff‘𝐹):ℕ0⟶𝑆)
55	54, 44	ffvelcdmd 7080	. . . . . . 7 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆)
56		mpocnfldmul 21529	. . . . . . . . 9 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) = (.r‘ℂfld)
57	56	subrgmcl 20683	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑐 + 1) ∈ 𝑆 ∧ ((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆) → ((𝑐 + 1)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))((coeff‘𝐹)‘(𝑐 + 1))) ∈ 𝑆)
58	37	a1d 26	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → (((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆 → 𝑆 ⊆ ℂ))
59		ssel 3931	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ⊆ ℂ → ((𝑐 + 1) ∈ 𝑆 → (𝑐 + 1) ∈ ℂ))
60	59	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → (𝑆 ⊆ ℂ → ((𝑐 + 1) ∈ 𝑆 → (𝑐 + 1) ∈ ℂ)))
61	58, 60	syld 48	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → (((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆 → ((𝑐 + 1) ∈ 𝑆 → (𝑐 + 1) ∈ ℂ)))
62	61	com23 87	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → ((𝑐 + 1) ∈ 𝑆 → (((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆 → (𝑐 + 1) ∈ ℂ)))
63	62	3imp 1128	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑐 + 1) ∈ 𝑆 ∧ ((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆) → (𝑐 + 1) ∈ ℂ)
64	37	a1d 26	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → ((𝑐 + 1) ∈ 𝑆 → 𝑆 ⊆ ℂ))
65		ssel 3931	. . . . . . . . . . . . . 14 ⊢ (𝑆 ⊆ ℂ → (((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆 → ((coeff‘𝐹)‘(𝑐 + 1)) ∈ ℂ))
66	65	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → (𝑆 ⊆ ℂ → (((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆 → ((coeff‘𝐹)‘(𝑐 + 1)) ∈ ℂ)))
67	64, 66	syld 48	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → ((𝑐 + 1) ∈ 𝑆 → (((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆 → ((coeff‘𝐹)‘(𝑐 + 1)) ∈ ℂ)))
68	67	3imp 1128	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑐 + 1) ∈ 𝑆 ∧ ((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆) → ((coeff‘𝐹)‘(𝑐 + 1)) ∈ ℂ)
69	63, 68	jca 520	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑐 + 1) ∈ 𝑆 ∧ ((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆) → ((𝑐 + 1) ∈ ℂ ∧ ((coeff‘𝐹)‘(𝑐 + 1)) ∈ ℂ))
70		ovmpot 7571	. . . . . . . . . 10 ⊢ (((𝑐 + 1) ∈ ℂ ∧ ((coeff‘𝐹)‘(𝑐 + 1)) ∈ ℂ) → ((𝑐 + 1)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))((coeff‘𝐹)‘(𝑐 + 1))) = ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1))))
71	69, 70	syl 18	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑐 + 1) ∈ 𝑆 ∧ ((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆) → ((𝑐 + 1)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))((coeff‘𝐹)‘(𝑐 + 1))) = ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1))))
72	71	eleq1d 2848	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑐 + 1) ∈ 𝑆 ∧ ((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆) → (((𝑐 + 1)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))((coeff‘𝐹)‘(𝑐 + 1))) ∈ 𝑆 ↔ ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1))) ∈ 𝑆))
73	57, 72	mpbid 235	. . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑐 + 1) ∈ 𝑆 ∧ ((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆) → ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1))) ∈ 𝑆)
74	40, 46, 55, 73	syl3anc 1398	. . . . . 6 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1))) ∈ 𝑆)
75	74	fmpttd 7110	. . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (𝑐 ∈ ℕ0 ↦ ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1)))):ℕ0⟶𝑆)
76	75	ffvelcdmda 7079	. . . 4 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑏 ∈ ℕ0) → ((𝑐 ∈ ℕ0 ↦ ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1))))‘𝑏) ∈ 𝑆)
77	39, 76	sylan2 604	. . 3 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑏 ∈ (0...(deg‘𝐹))) → ((𝑐 ∈ ℕ0 ↦ ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1))))‘𝑏) ∈ 𝑆)
78	38, 6, 77	elplyd 26359	. 2 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (𝑎 ∈ ℂ ↦ Σ𝑏 ∈ (0...(deg‘𝐹))(((𝑐 ∈ ℕ0 ↦ ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1))))‘𝑏) · (𝑎↑𝑏))) ∈ (Poly‘𝑆))
79	35, 78	eqeltrd 2863	1 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℂ D 𝐹) ∈ (Poly‘𝑆))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ⊆ wss 3905   ↦ cmpt 5192  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  ℂcc 11093  0cc0 11095  1c1 11096   + caddc 11098   · cmul 11100   − cmin 11436  ℕ₀cn0 12499  ℤcz 12586  ℤ_≥cuz 12857  ...cfz 13530  ↑cexp 14093  Σcsu 15733  SubGrpcsubg 19181  SubRingcsubrg 20668  ℂ_fldccnfld 21522   D cdv 26022  Polycply 26341  coeffccoe 26343  degcdgr 26344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173  ax-addf 11174

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-fi 9367  df-sup 9398  df-inf 9399  df-oi 9468  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12858  df-q 12968  df-rp 13012  df-xneg 13132  df-xadd 13133  df-xmul 13134  df-icc 13374  df-fz 13531  df-fzo 13679  df-fl 13821  df-seq 14034  df-exp 14094  df-hash 14363  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-clim 15535  df-rlim 15536  df-sum 15734  df-struct 17202  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-mulr 17319  df-starv 17320  df-sca 17321  df-vsca 17322  df-ip 17323  df-tset 17324  df-ple 17325  df-ds 17327  df-unif 17328  df-hom 17329  df-cco 17330  df-rest 17470  df-topn 17471  df-0g 17489  df-gsum 17490  df-topgen 17491  df-pt 17492  df-prds 17495  df-xrs 17551  df-qtop 17556  df-imas 17557  df-xps 17559  df-mre 17633  df-mrc 17634  df-acs 17636  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-submnd 18837  df-grp 18998  df-minusg 18999  df-mulg 19129  df-subg 19184  df-cntz 19382  df-cmn 19847  df-abl 19848  df-mgp 20212  df-rng 20226  df-ur 20259  df-ring 20312  df-cring 20313  df-subrng 20645  df-subrg 20669  df-psmet 21514  df-xmet 21515  df-met 21516  df-bl 21517  df-mopn 21518  df-fbas 21519  df-fg 21520  df-cnfld 21523  df-top 23051  df-topon 23068  df-topsp 23090  df-bases 23103  df-cld 23176  df-ntr 23177  df-cls 23178  df-nei 23255  df-lp 23293  df-perf 23294  df-cn 23384  df-cnp 23385  df-haus 23472  df-tx 23719  df-hmeo 23912  df-fil 24003  df-fm 24095  df-flim 24096  df-flf 24097  df-xms 24477  df-ms 24478  df-tms 24479  df-cncf 25037  df-0p 25829  df-limc 26025  df-dv 26026  df-ply 26345  df-coe 26347  df-dgr 26348

This theorem is referenced by:  dvply2  26447  dvnply2  26448