Theorem dvply2g 26264

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 11220. (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 26177	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
2	1	adantl 480	. . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
3	2	feqmptd 6966	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹 = (𝑎 ∈ ℂ ↦ (𝐹‘𝑎)))
4		simplr 767	. . . . . 6 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑎 ∈ ℂ) → 𝐹 ∈ (Poly‘𝑆))
5		dgrcl 26212	. . . . . . . . . 10 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
6	5	adantl 480	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (deg‘𝐹) ∈ ℕ0)
7	6	nn0zd 12617	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (deg‘𝐹) ∈ ℤ)
8	7	adantr 479	. . . . . . 7 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑎 ∈ ℂ) → (deg‘𝐹) ∈ ℤ)
9		uzid 12870	. . . . . . 7 ⊢ ((deg‘𝐹) ∈ ℤ → (deg‘𝐹) ∈ (ℤ≥‘(deg‘𝐹)))
10		peano2uz 12918	. . . . . . 7 ⊢ ((deg‘𝐹) ∈ (ℤ≥‘(deg‘𝐹)) → ((deg‘𝐹) + 1) ∈ (ℤ≥‘(deg‘𝐹)))
11	8, 9, 10	3syl 18	. . . . . 6 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑎 ∈ ℂ) → ((deg‘𝐹) + 1) ∈ (ℤ≥‘(deg‘𝐹)))
12		simpr 483	. . . . . 6 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑎 ∈ ℂ) → 𝑎 ∈ ℂ)
13		eqid 2725	. . . . . . 7 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
14		eqid 2725	. . . . . . 7 ⊢ (deg‘𝐹) = (deg‘𝐹)
15	13, 14	coeid3 26219	. . . . . 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 1368	. . . . 5 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑎 ∈ ℂ) → (𝐹‘𝑎) = Σ𝑏 ∈ (0...((deg‘𝐹) + 1))(((coeff‘𝐹)‘𝑏) · (𝑎↑𝑏)))
17	16	mpteq2dva 5249	. . . 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 2765	. . 3 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹 = (𝑎 ∈ ℂ ↦ Σ𝑏 ∈ (0...((deg‘𝐹) + 1))(((coeff‘𝐹)‘𝑏) · (𝑎↑𝑏))))
19	6	nn0cnd 12567	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (deg‘𝐹) ∈ ℂ)
20		ax-1cn 11198	. . . . . . . 8 ⊢ 1 ∈ ℂ
21		pncan 11498	. . . . . . . 8 ⊢ (((deg‘𝐹) ∈ ℂ ∧ 1 ∈ ℂ) → (((deg‘𝐹) + 1) − 1) = (deg‘𝐹))
22	19, 20, 21	sylancl 584	. . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (((deg‘𝐹) + 1) − 1) = (deg‘𝐹))
23	22	eqcomd 2731	. . . . . 6 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (deg‘𝐹) = (((deg‘𝐹) + 1) − 1))
24	23	oveq2d 7435	. . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (0...(deg‘𝐹)) = (0...(((deg‘𝐹) + 1) − 1)))
25	24	sumeq1d 15683	. . . 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 5251	. . 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 26211	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
28	27	adantl 480	. . 3 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘𝐹):ℕ0⟶ℂ)
29		oveq1 7426	. . . . 5 ⊢ (𝑐 = 𝑏 → (𝑐 + 1) = (𝑏 + 1))
30		fvoveq1 7442	. . . . 5 ⊢ (𝑐 = 𝑏 → ((coeff‘𝐹)‘(𝑐 + 1)) = ((coeff‘𝐹)‘(𝑏 + 1)))
31	29, 30	oveq12d 7437	. . . 4 ⊢ (𝑐 = 𝑏 → ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1))) = ((𝑏 + 1) · ((coeff‘𝐹)‘(𝑏 + 1))))
32	31	cbvmptv 5262	. . 3 ⊢ (𝑐 ∈ ℕ0 ↦ ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1)))) = (𝑏 ∈ ℕ0 ↦ ((𝑏 + 1) · ((coeff‘𝐹)‘(𝑏 + 1))))
33		peano2nn0 12545	. . . 4 ⊢ ((deg‘𝐹) ∈ ℕ0 → ((deg‘𝐹) + 1) ∈ ℕ0)
34	6, 33	syl 17	. . 3 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ((deg‘𝐹) + 1) ∈ ℕ0)
35	18, 26, 28, 32, 34	dvply1 26263	. 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 21300	. . . . 5 ⊢ ℂ = (Base‘ℂfld)
37	36	subrgss 20523	. . . 4 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ⊆ ℂ)
38	37	adantr 479	. . 3 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝑆 ⊆ ℂ)
39		elfznn0 13629	. . . 4 ⊢ (𝑏 ∈ (0...(deg‘𝐹)) → 𝑏 ∈ ℕ0)
40		simpll 765	. . . . . . 7 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → 𝑆 ∈ (SubRing‘ℂfld))
41		zsssubrg 21375	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → ℤ ⊆ 𝑆)
42	41	ad2antrr 724	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → ℤ ⊆ 𝑆)
43		peano2nn0 12545	. . . . . . . . . 10 ⊢ (𝑐 ∈ ℕ0 → (𝑐 + 1) ∈ ℕ0)
44	43	adantl 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → (𝑐 + 1) ∈ ℕ0)
45	44	nn0zd 12617	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → (𝑐 + 1) ∈ ℤ)
46	42, 45	sseldd 3977	. . . . . . 7 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → (𝑐 + 1) ∈ 𝑆)
47		simplr 767	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → 𝐹 ∈ (Poly‘𝑆))
48		subrgsubg 20528	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ∈ (SubGrp‘ℂfld))
49		cnfld0 21337	. . . . . . . . . . . 12 ⊢ 0 = (0g‘ℂfld)
50	49	subg0cl 19097	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (SubGrp‘ℂfld) → 0 ∈ 𝑆)
51	48, 50	syl 17	. . . . . . . . . 10 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 0 ∈ 𝑆)
52	51	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → 0 ∈ 𝑆)
53	13	coef2 26210	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 0 ∈ 𝑆) → (coeff‘𝐹):ℕ0⟶𝑆)
54	47, 52, 53	syl2anc 582	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → (coeff‘𝐹):ℕ0⟶𝑆)
55	54, 44	ffvelcdmd 7094	. . . . . . 7 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆)
56		mpocnfldmul 21303	. . . . . . . . 9 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) = (.r‘ℂfld)
57	56	subrgmcl 20535	. . . . . . . 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 25	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → (((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆 → 𝑆 ⊆ ℂ))
59		ssel 3970	. . . . . . . . . . . . . . 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 47	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → (((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆 → ((𝑐 + 1) ∈ 𝑆 → (𝑐 + 1) ∈ ℂ)))
62	61	com23 86	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → ((𝑐 + 1) ∈ 𝑆 → (((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆 → (𝑐 + 1) ∈ ℂ)))
63	62	3imp 1108	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑐 + 1) ∈ 𝑆 ∧ ((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆) → (𝑐 + 1) ∈ ℂ)
64	37	a1d 25	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → ((𝑐 + 1) ∈ 𝑆 → 𝑆 ⊆ ℂ))
65		ssel 3970	. . . . . . . . . . . . . 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 47	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → ((𝑐 + 1) ∈ 𝑆 → (((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆 → ((coeff‘𝐹)‘(𝑐 + 1)) ∈ ℂ)))
68	67	3imp 1108	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑐 + 1) ∈ 𝑆 ∧ ((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆) → ((coeff‘𝐹)‘(𝑐 + 1)) ∈ ℂ)
69	63, 68	jca 510	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑐 + 1) ∈ 𝑆 ∧ ((coeff‘𝐹)‘(𝑐 + 1)) ∈ 𝑆) → ((𝑐 + 1) ∈ ℂ ∧ ((coeff‘𝐹)‘(𝑐 + 1)) ∈ ℂ))
70		ovmpot 7582	. . . . . . . . . 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 17	. . . . . . . . 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 2810	. . . . . . . 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 231	. . . . . . 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 1368	. . . . . 6 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑐 ∈ ℕ0) → ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1))) ∈ 𝑆)
75	74	fmpttd 7124	. . . . 5 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (𝑐 ∈ ℕ0 ↦ ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1)))):ℕ0⟶𝑆)
76	75	ffvelcdmda 7093	. . . 4 ⊢ (((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑏 ∈ ℕ0) → ((𝑐 ∈ ℕ0 ↦ ((𝑐 + 1) · ((coeff‘𝐹)‘(𝑐 + 1))))‘𝑏) ∈ 𝑆)
77	39, 76	sylan2 591	. . 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 26181	. 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 2825	1 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℂ D 𝐹) ∈ (Poly‘𝑆))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ⊆ wss 3944   ↦ cmpt 5232  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419   ∈ cmpo 7421  ℂcc 11138  0cc0 11140  1c1 11141   + caddc 11143   · cmul 11145   − cmin 11476  ℕ₀cn0 12505  ℤcz 12591  ℤ_≥cuz 12855  ...cfz 13519  ↑cexp 14062  Σcsu 15668  SubGrpcsubg 19083  SubRingcsubrg 20518  ℂ_fldccnfld 21296   D cdv 25836  Polycply 26163  coeffccoe 26165  degcdgr 26166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218  ax-addf 11219

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9388  df-fi 9436  df-sup 9467  df-inf 9468  df-oi 9535  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-z 12592  df-dec 12711  df-uz 12856  df-q 12966  df-rp 13010  df-xneg 13127  df-xadd 13128  df-xmul 13129  df-icc 13366  df-fz 13520  df-fzo 13663  df-fl 13793  df-seq 14003  df-exp 14063  df-hash 14326  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-clim 15468  df-rlim 15469  df-sum 15669  df-struct 17119  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-starv 17251  df-sca 17252  df-vsca 17253  df-ip 17254  df-tset 17255  df-ple 17256  df-ds 17258  df-unif 17259  df-hom 17260  df-cco 17261  df-rest 17407  df-topn 17408  df-0g 17426  df-gsum 17427  df-topgen 17428  df-pt 17429  df-prds 17432  df-xrs 17487  df-qtop 17492  df-imas 17493  df-xps 17495  df-mre 17569  df-mrc 17570  df-acs 17572  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-submnd 18744  df-grp 18901  df-minusg 18902  df-mulg 19032  df-subg 19086  df-cntz 19280  df-cmn 19749  df-abl 19750  df-mgp 20087  df-rng 20105  df-ur 20134  df-ring 20187  df-cring 20188  df-subrng 20495  df-subrg 20520  df-psmet 21288  df-xmet 21289  df-met 21290  df-bl 21291  df-mopn 21292  df-fbas 21293  df-fg 21294  df-cnfld 21297  df-top 22840  df-topon 22857  df-topsp 22879  df-bases 22893  df-cld 22967  df-ntr 22968  df-cls 22969  df-nei 23046  df-lp 23084  df-perf 23085  df-cn 23175  df-cnp 23176  df-haus 23263  df-tx 23510  df-hmeo 23703  df-fil 23794  df-fm 23886  df-flim 23887  df-flf 23888  df-xms 24270  df-ms 24271  df-tms 24272  df-cncf 24842  df-0p 25643  df-limc 25839  df-dv 25840  df-ply 26167  df-coe 26169  df-dgr 26170

This theorem is referenced by:  dvply2  26266  dvnply2  26267