signsply0 - Mathbox for Thierry Arnoux

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Theorem signsply0 34017

Description: Lemma for the rule of signs, based on Bolzano's intermediate value theorem for polynomials : If the lowest and highest coefficient 𝐴 and 𝐵 are of opposite signs, the polynomial admits a positive root. (Contributed by Thierry Arnoux, 19-Sep-2018.)

**Hypotheses**
Ref	Expression
signsply0.d	⊢ 𝐷 = (deg‘𝐹)
signsply0.c	⊢ 𝐶 = (coeff‘𝐹)
signsply0.b	⊢ 𝐵 = (𝐶‘𝐷)
signsply0.a	⊢ 𝐴 = (𝐶‘0)
signsply0.1	⊢ (𝜑 → 𝐹 ∈ (Poly‘ℝ))
signsply0.2	⊢ (𝜑 → 𝐹 ≠ 0_𝑝)
signsply0.3	⊢ (𝜑 → (𝐴 · 𝐵) < 0)

**Assertion**
Ref	Expression
signsply0	⊢ (𝜑 → ∃𝑧 ∈ ℝ⁺ (𝐹‘𝑧) = 0)

Distinct variable groups: 𝑧,𝐵 𝑧,𝐹 𝜑,𝑧 Allowed substitution hints: 𝐴(𝑧) 𝐶(𝑧) 𝐷(𝑧)

Proof of Theorem signsply0 Dummy variables 𝑒 𝑑 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 766	. . . . . 6 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < -𝐵)) → 𝑑 ∈ ℝ⁺)
2		simpr 484	. . . . . 6 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < -𝐵)) → ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < -𝐵))
3		rpxr 12979	. . . . . . . 8 ⊢ (𝑑 ∈ ℝ⁺ → 𝑑 ∈ ℝ^*)
4	3	xrleidd 13127	. . . . . . 7 ⊢ (𝑑 ∈ ℝ⁺ → 𝑑 ≤ 𝑑)
5	4	ad2antlr 724	. . . . . 6 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < -𝐵)) → 𝑑 ≤ 𝑑)
6		id 22	. . . . . . 7 ⊢ (𝑑 ∈ ℝ⁺ → 𝑑 ∈ ℝ⁺)
7		simpr 484	. . . . . . . . 9 ⊢ ((𝑑 ∈ ℝ⁺ ∧ 𝑓 = 𝑑) → 𝑓 = 𝑑)
8	7	breq2d 5150	. . . . . . . 8 ⊢ ((𝑑 ∈ ℝ⁺ ∧ 𝑓 = 𝑑) → (𝑑 ≤ 𝑓 ↔ 𝑑 ≤ 𝑑))
9	7	fveq2d 6885	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ ℝ⁺ ∧ 𝑓 = 𝑑) → (𝐹‘𝑓) = (𝐹‘𝑑))
10	7	oveq1d 7416	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ ℝ⁺ ∧ 𝑓 = 𝑑) → (𝑓↑𝐷) = (𝑑↑𝐷))
11	9, 10	oveq12d 7419	. . . . . . . . . 10 ⊢ ((𝑑 ∈ ℝ⁺ ∧ 𝑓 = 𝑑) → ((𝐹‘𝑓) / (𝑓↑𝐷)) = ((𝐹‘𝑑) / (𝑑↑𝐷)))
12	11	fvoveq1d 7423	. . . . . . . . 9 ⊢ ((𝑑 ∈ ℝ⁺ ∧ 𝑓 = 𝑑) → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) = (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)))
13	12	breq1d 5148	. . . . . . . 8 ⊢ ((𝑑 ∈ ℝ⁺ ∧ 𝑓 = 𝑑) → ((abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < -𝐵 ↔ (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < -𝐵))
14	8, 13	imbi12d 344	. . . . . . 7 ⊢ ((𝑑 ∈ ℝ⁺ ∧ 𝑓 = 𝑑) → ((𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < -𝐵) ↔ (𝑑 ≤ 𝑑 → (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < -𝐵)))
15	6, 14	rspcdv 3596	. . . . . 6 ⊢ (𝑑 ∈ ℝ⁺ → (∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < -𝐵) → (𝑑 ≤ 𝑑 → (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < -𝐵)))
16	1, 2, 5, 15	syl3c 66	. . . . 5 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < -𝐵)) → (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < -𝐵)
17		signsply0.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℝ))
18	17	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝐹 ∈ (Poly‘ℝ))
19		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℝ⁺)
20	19	rpred 13012	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℝ)
21	18, 20	plyrecld 34015	. . . . . . . . . 10 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝐹‘𝑑) ∈ ℝ)
22		signsply0.d	. . . . . . . . . . . . 13 ⊢ 𝐷 = (deg‘𝐹)
23		dgrcl 26075	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (Poly‘ℝ) → (deg‘𝐹) ∈ ℕ₀)
24	17, 23	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ₀)
25	22, 24	eqeltrid 2829	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ∈ ℕ₀)
26	25	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝐷 ∈ ℕ₀)
27	20, 26	reexpcld 14124	. . . . . . . . . 10 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝑑↑𝐷) ∈ ℝ)
28	19	rpcnd 13014	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℂ)
29	19	rpne0d 13017	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ≠ 0)
30	25	nn0zd 12580	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ∈ ℤ)
31	30	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝐷 ∈ ℤ)
32	28, 29, 31	expne0d 14113	. . . . . . . . . 10 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝑑↑𝐷) ≠ 0)
33	21, 27, 32	redivcld 12038	. . . . . . . . 9 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → ((𝐹‘𝑑) / (𝑑↑𝐷)) ∈ ℝ)
34		signsply0.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (𝐶‘𝐷)
35		0re 11212	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
36		signsply0.c	. . . . . . . . . . . . . . 15 ⊢ 𝐶 = (coeff‘𝐹)
37	36	coef2 26073	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → 𝐶:ℕ₀⟶ℝ)
38	35, 37	mpan2 688	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐶:ℕ₀⟶ℝ)
39	38	ffvelcdmda 7076	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐷 ∈ ℕ₀) → (𝐶‘𝐷) ∈ ℝ)
40	34, 39	eqeltrid 2829	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐷 ∈ ℕ₀) → 𝐵 ∈ ℝ)
41	17, 25, 40	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
42	41	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
43	42	renegcld 11637	. . . . . . . . 9 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → -𝐵 ∈ ℝ)
44	33, 42, 43	absdifltd 15376	. . . . . . . 8 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → ((abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < -𝐵 ↔ ((𝐵 − -𝐵) < ((𝐹‘𝑑) / (𝑑↑𝐷)) ∧ ((𝐹‘𝑑) / (𝑑↑𝐷)) < (𝐵 + -𝐵))))
45	44	simplbda 499	. . . . . . 7 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < -𝐵) → ((𝐹‘𝑑) / (𝑑↑𝐷)) < (𝐵 + -𝐵))
46	41	recnd 11238	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ)
47	46	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝐵 ∈ ℂ)
48	47	negidd 11557	. . . . . . . 8 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝐵 + -𝐵) = 0)
49	48	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < -𝐵) → (𝐵 + -𝐵) = 0)
50	45, 49	breqtrd 5164	. . . . . 6 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < -𝐵) → ((𝐹‘𝑑) / (𝑑↑𝐷)) < 0)
51	19, 31	rpexpcld 14206	. . . . . . . . . 10 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝑑↑𝐷) ∈ ℝ⁺)
52	21, 51	ge0divd 13050	. . . . . . . . 9 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (0 ≤ (𝐹‘𝑑) ↔ 0 ≤ ((𝐹‘𝑑) / (𝑑↑𝐷))))
53	52	notbid 318	. . . . . . . 8 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (¬ 0 ≤ (𝐹‘𝑑) ↔ ¬ 0 ≤ ((𝐹‘𝑑) / (𝑑↑𝐷))))
54		0red 11213	. . . . . . . . 9 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 0 ∈ ℝ)
55	21, 54	ltnled 11357	. . . . . . . 8 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → ((𝐹‘𝑑) < 0 ↔ ¬ 0 ≤ (𝐹‘𝑑)))
56	33, 54	ltnled 11357	. . . . . . . 8 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (((𝐹‘𝑑) / (𝑑↑𝐷)) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝑑) / (𝑑↑𝐷))))
57	53, 55, 56	3bitr4d 311	. . . . . . 7 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → ((𝐹‘𝑑) < 0 ↔ ((𝐹‘𝑑) / (𝑑↑𝐷)) < 0))
58	57	adantr 480	. . . . . 6 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < -𝐵) → ((𝐹‘𝑑) < 0 ↔ ((𝐹‘𝑑) / (𝑑↑𝐷)) < 0))
59	50, 58	mpbird 257	. . . . 5 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < -𝐵) → (𝐹‘𝑑) < 0)
60	16, 59	syldan 590	. . . 4 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < -𝐵)) → (𝐹‘𝑑) < 0)
61		0red 11213	. . . . . 6 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → 0 ∈ ℝ)
62		simplr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → 𝑑 ∈ ℝ⁺)
63	62	rpred 13012	. . . . . 6 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → 𝑑 ∈ ℝ)
64	62	rpgt0d 13015	. . . . . 6 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → 0 < 𝑑)
65		iccssre 13402	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (0[,]𝑑) ⊆ ℝ)
66	35, 63, 65	sylancr 586	. . . . . . 7 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → (0[,]𝑑) ⊆ ℝ)
67		ax-resscn 11162	. . . . . . 7 ⊢ ℝ ⊆ ℂ
68	66, 67	sstrdi 3986	. . . . . 6 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → (0[,]𝑑) ⊆ ℂ)
69		plycn 26103	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐹 ∈ (ℂ–cn→ℂ))
70	17, 69	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ))
71	70	ad3antrrr 727	. . . . . 6 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → 𝐹 ∈ (ℂ–cn→ℂ))
72	17	ad4antr 729	. . . . . . 7 ⊢ (((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) ∧ 𝑥 ∈ (0[,]𝑑)) → 𝐹 ∈ (Poly‘ℝ))
73	66	sselda 3974	. . . . . . 7 ⊢ (((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) ∧ 𝑥 ∈ (0[,]𝑑)) → 𝑥 ∈ ℝ)
74	72, 73	plyrecld 34015	. . . . . 6 ⊢ (((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) ∧ 𝑥 ∈ (0[,]𝑑)) → (𝐹‘𝑥) ∈ ℝ)
75		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → (𝐹‘𝑑) < 0)
76		simplll 772	. . . . . . . . 9 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → 𝜑)
77	76, 41	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → 𝐵 ∈ ℝ)
78		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ -𝐵 ∈ ℝ⁺) → -𝐵 ∈ ℝ⁺)
79	78	ad2antrr 723	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → -𝐵 ∈ ℝ⁺)
80		negelrp 13003	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → (-𝐵 ∈ ℝ⁺ ↔ 𝐵 < 0))
81	80	biimpa 476	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ -𝐵 ∈ ℝ⁺) → 𝐵 < 0)
82	77, 79, 81	syl2anc 583	. . . . . . . . 9 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → 𝐵 < 0)
83		signsply0.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (𝐶‘0)
84	17, 35, 37	sylancl 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶:ℕ₀⟶ℝ)
85		0nn0 12483	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ₀
86	85	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℕ₀)
87	84, 86	ffvelcdmd 7077	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶‘0) ∈ ℝ)
88	83, 87	eqeltrid 2829	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
89		signsply0.3	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 · 𝐵) < 0)
90	88, 41, 89	mul2lt0rlt0 13072	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 < 0) → 0 < 𝐴)
91	90, 83	breqtrdi 5179	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 < 0) → 0 < (𝐶‘0))
92	76, 82, 91	syl2anc 583	. . . . . . . 8 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → 0 < (𝐶‘0))
93	36	coefv0 26090	. . . . . . . . . 10 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹‘0) = (𝐶‘0))
94	17, 93	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘0) = (𝐶‘0))
95	94	ad3antrrr 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → (𝐹‘0) = (𝐶‘0))
96	92, 95	breqtrrd 5166	. . . . . . 7 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → 0 < (𝐹‘0))
97	75, 96	jca 511	. . . . . 6 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → ((𝐹‘𝑑) < 0 ∧ 0 < (𝐹‘0)))
98	61, 63, 61, 64, 68, 71, 74, 97	ivth2 25294	. . . . 5 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → ∃𝑧 ∈ (0(,)𝑑)(𝐹‘𝑧) = 0)
99		0le0 12309	. . . . . . . 8 ⊢ 0 ≤ 0
100		pnfge 13106	. . . . . . . . 9 ⊢ (𝑑 ∈ ℝ^* → 𝑑 ≤ +∞)
101	3, 100	syl 17	. . . . . . . 8 ⊢ (𝑑 ∈ ℝ⁺ → 𝑑 ≤ +∞)
102		0xr 11257	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
103		pnfxr 11264	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
104		ioossioo 13414	. . . . . . . . 9 ⊢ (((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (0 ≤ 0 ∧ 𝑑 ≤ +∞)) → (0(,)𝑑) ⊆ (0(,)+∞))
105	102, 103, 104	mpanl12 699	. . . . . . . 8 ⊢ ((0 ≤ 0 ∧ 𝑑 ≤ +∞) → (0(,)𝑑) ⊆ (0(,)+∞))
106	99, 101, 105	sylancr 586	. . . . . . 7 ⊢ (𝑑 ∈ ℝ⁺ → (0(,)𝑑) ⊆ (0(,)+∞))
107		ioorp 13398	. . . . . . 7 ⊢ (0(,)+∞) = ℝ⁺
108	106, 107	sseqtrdi 4024	. . . . . 6 ⊢ (𝑑 ∈ ℝ⁺ → (0(,)𝑑) ⊆ ℝ⁺)
109		ssrexv 4043	. . . . . 6 ⊢ ((0(,)𝑑) ⊆ ℝ⁺ → (∃𝑧 ∈ (0(,)𝑑)(𝐹‘𝑧) = 0 → ∃𝑧 ∈ ℝ⁺ (𝐹‘𝑧) = 0))
110	62, 108, 109	3syl 18	. . . . 5 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → (∃𝑧 ∈ (0(,)𝑑)(𝐹‘𝑧) = 0 → ∃𝑧 ∈ ℝ⁺ (𝐹‘𝑧) = 0))
111	98, 110	mpd 15	. . . 4 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝐹‘𝑑) < 0) → ∃𝑧 ∈ ℝ⁺ (𝐹‘𝑧) = 0)
112	60, 111	syldan 590	. . 3 ⊢ ((((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < -𝐵)) → ∃𝑧 ∈ ℝ⁺ (𝐹‘𝑧) = 0)
113		plyf 26040	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐹:ℂ⟶ℂ)
114	17, 113	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
115	114	ffnd 6708	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn ℂ)
116		ovex 7434	. . . . . . . . . . 11 ⊢ (𝑥↑𝐷) ∈ V
117	116	rgenw 3057	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ ℝ⁺ (𝑥↑𝐷) ∈ V
118		eqid 2724	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ ↦ (𝑥↑𝐷)) = (𝑥 ∈ ℝ⁺ ↦ (𝑥↑𝐷))
119	118	fnmpt 6680	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ ℝ⁺ (𝑥↑𝐷) ∈ V → (𝑥 ∈ ℝ⁺ ↦ (𝑥↑𝐷)) Fn ℝ⁺)
120	117, 119	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝑥↑𝐷)) Fn ℝ⁺)
121		cnex 11186	. . . . . . . . . 10 ⊢ ℂ ∈ V
122	121	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℂ ∈ V)
123		rpssre 12977	. . . . . . . . . . . 12 ⊢ ℝ⁺ ⊆ ℝ
124	123, 67	sstri 3983	. . . . . . . . . . 11 ⊢ ℝ⁺ ⊆ ℂ
125	121, 124	ssexi 5312	. . . . . . . . . 10 ⊢ ℝ⁺ ∈ V
126	125	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ⁺ ∈ V)
127		sseqin2 4207	. . . . . . . . . 10 ⊢ (ℝ⁺ ⊆ ℂ ↔ (ℂ ∩ ℝ⁺) = ℝ⁺)
128	124, 127	mpbi 229	. . . . . . . . 9 ⊢ (ℂ ∩ ℝ⁺) = ℝ⁺
129		eqidd 2725	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ ℂ) → (𝐹‘𝑓) = (𝐹‘𝑓))
130		eqidd 2725	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ ℝ⁺) → (𝑥 ∈ ℝ⁺ ↦ (𝑥↑𝐷)) = (𝑥 ∈ ℝ⁺ ↦ (𝑥↑𝐷)))
131		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ ℝ⁺) ∧ 𝑥 = 𝑓) → 𝑥 = 𝑓)
132	131	oveq1d 7416	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ ℝ⁺) ∧ 𝑥 = 𝑓) → (𝑥↑𝐷) = (𝑓↑𝐷))
133		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ ℝ⁺) → 𝑓 ∈ ℝ⁺)
134		ovexd 7436	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ ℝ⁺) → (𝑓↑𝐷) ∈ V)
135	130, 132, 133, 134	fvmptd 6995	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ ℝ⁺) → ((𝑥 ∈ ℝ⁺ ↦ (𝑥↑𝐷))‘𝑓) = (𝑓↑𝐷))
136	115, 120, 122, 126, 128, 129, 135	offval 7672	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘_f / (𝑥 ∈ ℝ⁺ ↦ (𝑥↑𝐷))) = (𝑓 ∈ ℝ⁺ ↦ ((𝐹‘𝑓) / (𝑓↑𝐷))))
137		oveq1 7408	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑓 → (𝑥↑𝐷) = (𝑓↑𝐷))
138	137	cbvmptv 5251	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ ↦ (𝑥↑𝐷)) = (𝑓 ∈ ℝ⁺ ↦ (𝑓↑𝐷))
139	22, 36, 34, 138	signsplypnf 34016	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∘_f / (𝑥 ∈ ℝ⁺ ↦ (𝑥↑𝐷))) ⇝_𝑟 𝐵)
140	17, 139	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘_f / (𝑥 ∈ ℝ⁺ ↦ (𝑥↑𝐷))) ⇝_𝑟 𝐵)
141	136, 140	eqbrtrrd 5162	. . . . . . 7 ⊢ (𝜑 → (𝑓 ∈ ℝ⁺ ↦ ((𝐹‘𝑓) / (𝑓↑𝐷))) ⇝_𝑟 𝐵)
142	114	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ ℝ⁺) → 𝐹:ℂ⟶ℂ)
143	133	rpcnd 13014	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ ℝ⁺) → 𝑓 ∈ ℂ)
144	142, 143	ffvelcdmd 7077	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ ℝ⁺) → (𝐹‘𝑓) ∈ ℂ)
145	25	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ ℝ⁺) → 𝐷 ∈ ℕ₀)
146	143, 145	expcld 14107	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ ℝ⁺) → (𝑓↑𝐷) ∈ ℂ)
147	133	rpne0d 13017	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ ℝ⁺) → 𝑓 ≠ 0)
148	30	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ ℝ⁺) → 𝐷 ∈ ℤ)
149	143, 147, 148	expne0d 14113	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ ℝ⁺) → (𝑓↑𝐷) ≠ 0)
150	144, 146, 149	divcld 11986	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ ℝ⁺) → ((𝐹‘𝑓) / (𝑓↑𝐷)) ∈ ℂ)
151	150	ralrimiva 3138	. . . . . . . 8 ⊢ (𝜑 → ∀𝑓 ∈ ℝ⁺ ((𝐹‘𝑓) / (𝑓↑𝐷)) ∈ ℂ)
152	123	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
153		1red 11211	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
154	151, 152, 46, 153	rlim3 15438	. . . . . . 7 ⊢ (𝜑 → ((𝑓 ∈ ℝ⁺ ↦ ((𝐹‘𝑓) / (𝑓↑𝐷))) ⇝_𝑟 𝐵 ↔ ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ (1[,)+∞)∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒)))
155	141, 154	mpbid 231	. . . . . 6 ⊢ (𝜑 → ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ (1[,)+∞)∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒))
156		0lt1 11732	. . . . . . . . . 10 ⊢ 0 < 1
157		pnfge 13106	. . . . . . . . . . 11 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
158	103, 157	ax-mp 5	. . . . . . . . . 10 ⊢ +∞ ≤ +∞
159		icossioo 13413	. . . . . . . . . 10 ⊢ (((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (0 < 1 ∧ +∞ ≤ +∞)) → (1[,)+∞) ⊆ (0(,)+∞))
160	102, 103, 156, 158, 159	mp4an 690	. . . . . . . . 9 ⊢ (1[,)+∞) ⊆ (0(,)+∞)
161	160, 107	sseqtri 4010	. . . . . . . 8 ⊢ (1[,)+∞) ⊆ ℝ⁺
162		ssrexv 4043	. . . . . . . 8 ⊢ ((1[,)+∞) ⊆ ℝ⁺ → (∃𝑑 ∈ (1[,)+∞)∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒) → ∃𝑑 ∈ ℝ⁺ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒)))
163	161, 162	ax-mp 5	. . . . . . 7 ⊢ (∃𝑑 ∈ (1[,)+∞)∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒) → ∃𝑑 ∈ ℝ⁺ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒))
164	163	ralimi 3075	. . . . . 6 ⊢ (∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ (1[,)+∞)∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒) → ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒))
165	155, 164	syl 17	. . . . 5 ⊢ (𝜑 → ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒))
166	165	adantr 480	. . . 4 ⊢ ((𝜑 ∧ -𝐵 ∈ ℝ⁺) → ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒))
167		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑒 = -𝐵) → 𝑒 = -𝐵)
168	167	breq2d 5150	. . . . . . 7 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑒 = -𝐵) → ((abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒 ↔ (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < -𝐵))
169	168	imbi2d 340	. . . . . 6 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑒 = -𝐵) → ((𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒) ↔ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < -𝐵)))
170	169	rexralbidv 3212	. . . . 5 ⊢ (((𝜑 ∧ -𝐵 ∈ ℝ⁺) ∧ 𝑒 = -𝐵) → (∃𝑑 ∈ ℝ⁺ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒) ↔ ∃𝑑 ∈ ℝ⁺ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < -𝐵)))
171	78, 170	rspcdv 3596	. . . 4 ⊢ ((𝜑 ∧ -𝐵 ∈ ℝ⁺) → (∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒) → ∃𝑑 ∈ ℝ⁺ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < -𝐵)))
172	166, 171	mpd 15	. . 3 ⊢ ((𝜑 ∧ -𝐵 ∈ ℝ⁺) → ∃𝑑 ∈ ℝ⁺ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < -𝐵))
173	112, 172	r19.29a 3154	. 2 ⊢ ((𝜑 ∧ -𝐵 ∈ ℝ⁺) → ∃𝑧 ∈ ℝ⁺ (𝐹‘𝑧) = 0)
174		simplr 766	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝐵)) → 𝑑 ∈ ℝ⁺)
175		simpr 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝐵)) → ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝐵))
176	4	ad2antlr 724	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝐵)) → 𝑑 ≤ 𝑑)
177	12	breq1d 5148	. . . . . . . 8 ⊢ ((𝑑 ∈ ℝ⁺ ∧ 𝑓 = 𝑑) → ((abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝐵 ↔ (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < 𝐵))
178	8, 177	imbi12d 344	. . . . . . 7 ⊢ ((𝑑 ∈ ℝ⁺ ∧ 𝑓 = 𝑑) → ((𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝐵) ↔ (𝑑 ≤ 𝑑 → (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < 𝐵)))
179	6, 178	rspcdv 3596	. . . . . 6 ⊢ (𝑑 ∈ ℝ⁺ → (∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝐵) → (𝑑 ≤ 𝑑 → (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < 𝐵)))
180	174, 175, 176, 179	syl3c 66	. . . . 5 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝐵)) → (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < 𝐵)
181	46	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝐵 ∈ ℂ)
182	181	subidd 11555	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝐵 − 𝐵) = 0)
183	182	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < 𝐵) → (𝐵 − 𝐵) = 0)
184	17	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝐹 ∈ (Poly‘ℝ))
185	123	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ⁺) → ℝ⁺ ⊆ ℝ)
186	185	sselda 3974	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℝ)
187	184, 186	plyrecld 34015	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝐹‘𝑑) ∈ ℝ)
188	25	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝐷 ∈ ℕ₀)
189	186, 188	reexpcld 14124	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝑑↑𝐷) ∈ ℝ)
190	186	recnd 11238	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℂ)
191		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℝ⁺)
192	191	rpne0d 13017	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ≠ 0)
193	30	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝐷 ∈ ℤ)
194	190, 192, 193	expne0d 14113	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝑑↑𝐷) ≠ 0)
195	187, 189, 194	redivcld 12038	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → ((𝐹‘𝑑) / (𝑑↑𝐷)) ∈ ℝ)
196	41	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
197	195, 196, 196	absdifltd 15376	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → ((abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < 𝐵 ↔ ((𝐵 − 𝐵) < ((𝐹‘𝑑) / (𝑑↑𝐷)) ∧ ((𝐹‘𝑑) / (𝑑↑𝐷)) < (𝐵 + 𝐵))))
198	197	simprbda 498	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < 𝐵) → (𝐵 − 𝐵) < ((𝐹‘𝑑) / (𝑑↑𝐷)))
199	183, 198	eqbrtrrd 5162	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < 𝐵) → 0 < ((𝐹‘𝑑) / (𝑑↑𝐷)))
200	191, 193	rpexpcld 14206	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝑑↑𝐷) ∈ ℝ⁺)
201	187, 200	gt0divd 13049	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (0 < (𝐹‘𝑑) ↔ 0 < ((𝐹‘𝑑) / (𝑑↑𝐷))))
202	201	adantr 480	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < 𝐵) → (0 < (𝐹‘𝑑) ↔ 0 < ((𝐹‘𝑑) / (𝑑↑𝐷))))
203	199, 202	mpbird 257	. . . . 5 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ (abs‘(((𝐹‘𝑑) / (𝑑↑𝐷)) − 𝐵)) < 𝐵) → 0 < (𝐹‘𝑑))
204	180, 203	syldan 590	. . . 4 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝐵)) → 0 < (𝐹‘𝑑))
205		0red 11213	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → 0 ∈ ℝ)
206		simplr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → 𝑑 ∈ ℝ⁺)
207	206	rpred 13012	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → 𝑑 ∈ ℝ)
208	206	rpgt0d 13015	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → 0 < 𝑑)
209	35, 207, 65	sylancr 586	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → (0[,]𝑑) ⊆ ℝ)
210	209, 67	sstrdi 3986	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → (0[,]𝑑) ⊆ ℂ)
211	70	ad3antrrr 727	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → 𝐹 ∈ (ℂ–cn→ℂ))
212	17	ad4antr 729	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) ∧ 𝑥 ∈ (0[,]𝑑)) → 𝐹 ∈ (Poly‘ℝ))
213	209	sselda 3974	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) ∧ 𝑥 ∈ (0[,]𝑑)) → 𝑥 ∈ ℝ)
214	212, 213	plyrecld 34015	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) ∧ 𝑥 ∈ (0[,]𝑑)) → (𝐹‘𝑥) ∈ ℝ)
215	94	ad3antrrr 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → (𝐹‘0) = (𝐶‘0))
216		simplll 772	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → 𝜑)
217		simpr1 1191	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐵 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺ ∧ 0 < (𝐹‘𝑑))) → 𝐵 ∈ ℝ⁺)
218	217	rpgt0d 13015	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐵 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺ ∧ 0 < (𝐹‘𝑑))) → 0 < 𝐵)
219	218	3anassrs 1357	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → 0 < 𝐵)
220	88, 41, 89	mul2lt0rgt0 13073	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < 𝐵) → 𝐴 < 0)
221	216, 219, 220	syl2anc 583	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → 𝐴 < 0)
222	83, 221	eqbrtrrid 5174	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → (𝐶‘0) < 0)
223	215, 222	eqbrtrd 5160	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → (𝐹‘0) < 0)
224		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → 0 < (𝐹‘𝑑))
225	223, 224	jca 511	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → ((𝐹‘0) < 0 ∧ 0 < (𝐹‘𝑑)))
226	205, 207, 205, 208, 210, 211, 214, 225	ivth 25293	. . . . 5 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → ∃𝑧 ∈ (0(,)𝑑)(𝐹‘𝑧) = 0)
227	206, 108, 109	3syl 18	. . . . 5 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → (∃𝑧 ∈ (0(,)𝑑)(𝐹‘𝑧) = 0 → ∃𝑧 ∈ ℝ⁺ (𝐹‘𝑧) = 0))
228	226, 227	mpd 15	. . . 4 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 0 < (𝐹‘𝑑)) → ∃𝑧 ∈ ℝ⁺ (𝐹‘𝑧) = 0)
229	204, 228	syldan 590	. . 3 ⊢ ((((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝐵)) → ∃𝑧 ∈ ℝ⁺ (𝐹‘𝑧) = 0)
230	165	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ⁺) → ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒))
231		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ⁺) → 𝐵 ∈ ℝ⁺)
232		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑒 = 𝐵) → 𝑒 = 𝐵)
233	232	breq2d 5150	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑒 = 𝐵) → ((abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒 ↔ (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝐵))
234	233	imbi2d 340	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑒 = 𝐵) → ((𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒) ↔ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝐵)))
235	234	rexralbidv 3212	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑒 = 𝐵) → (∃𝑑 ∈ ℝ⁺ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒) ↔ ∃𝑑 ∈ ℝ⁺ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝐵)))
236	231, 235	rspcdv 3596	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ⁺) → (∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝑒) → ∃𝑑 ∈ ℝ⁺ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝐵)))
237	230, 236	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ⁺) → ∃𝑑 ∈ ℝ⁺ ∀𝑓 ∈ ℝ⁺ (𝑑 ≤ 𝑓 → (abs‘(((𝐹‘𝑓) / (𝑓↑𝐷)) − 𝐵)) < 𝐵))
238	229, 237	r19.29a 3154	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ⁺) → ∃𝑧 ∈ ℝ⁺ (𝐹‘𝑧) = 0)
239		signsply0.2	. . . . 5 ⊢ (𝜑 → 𝐹 ≠ 0_𝑝)
240	22, 36	dgreq0 26108	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 = 0_𝑝 ↔ (𝐶‘𝐷) = 0))
241	17, 240	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹 = 0_𝑝 ↔ (𝐶‘𝐷) = 0))
242	241	necon3bid 2977	. . . . 5 ⊢ (𝜑 → (𝐹 ≠ 0_𝑝 ↔ (𝐶‘𝐷) ≠ 0))
243	239, 242	mpbid 231	. . . 4 ⊢ (𝜑 → (𝐶‘𝐷) ≠ 0)
244	34	neeq1i 2997	. . . 4 ⊢ (𝐵 ≠ 0 ↔ (𝐶‘𝐷) ≠ 0)
245	243, 244	sylibr 233	. . 3 ⊢ (𝜑 → 𝐵 ≠ 0)
246		rpneg 13002	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐵 ∈ ℝ⁺ ↔ ¬ -𝐵 ∈ ℝ⁺))
247	246	biimprd 247	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (¬ -𝐵 ∈ ℝ⁺ → 𝐵 ∈ ℝ⁺))
248	247	orrd 860	. . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (-𝐵 ∈ ℝ⁺ ∨ 𝐵 ∈ ℝ⁺))
249	41, 245, 248	syl2anc 583	. 2 ⊢ (𝜑 → (-𝐵 ∈ ℝ⁺ ∨ 𝐵 ∈ ℝ⁺))
250	173, 238, 249	mpjaodan 955	1 ⊢ (𝜑 → ∃𝑧 ∈ ℝ⁺ (𝐹‘𝑧) = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ∃wrex 3062 Vcvv 3466 ∩ cin 3939 ⊆ wss 3940 class class class wbr 5138 ↦ cmpt 5221 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 ∘_f cof 7661 ℂcc 11103 ℝcr 11104 0cc0 11105 1c1 11106 + caddc 11108 · cmul 11110 +∞cpnf 11241 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 − cmin 11440 -cneg 11441 / cdiv 11867 ℕ₀cn0 12468 ℤcz 12554 ℝ⁺crp 12970 (,)cioo 13320 [,)cico 13322 [,]cicc 13323 ↑cexp 14023 abscabs 15177 ⇝_𝑟 crli 15425 –cn→ccncf 24706 0_𝑝c0p 25508 Polycply 26026 coeffccoe 26028 degcdgr 26029

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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-submnd 18701 df-mulg 18983 df-cntz 19218 df-cmn 19687 df-psmet 21215 df-xmet 21216 df-met 21217 df-bl 21218 df-mopn 21219 df-fbas 21220 df-fg 21221 df-cnfld 21224 df-top 22706 df-topon 22723 df-topsp 22745 df-bases 22759 df-cld 22833 df-ntr 22834 df-cls 22835 df-nei 22912 df-lp 22950 df-perf 22951 df-cn 23041 df-cnp 23042 df-haus 23129 df-tx 23376 df-hmeo 23569 df-fil 23660 df-fm 23752 df-flim 23753 df-flf 23754 df-xms 24136 df-ms 24137 df-tms 24138 df-cncf 24708 df-0p 25509 df-limc 25705 df-dv 25706 df-ply 26030 df-coe 26032 df-dgr 26033 df-log 26395 df-cxp 26396

This theorem is referenced by: (None)

W3C validator