fourier2 - Mathbox for Glauco Siliprandi

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Theorem fourier2 46223

Description: Fourier series convergence, for a piecewise smooth function. Here it is also proven the existence of the left and right limits of 𝐹 at any given point 𝑋. See fourierd 46218 for a comparison. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourier2.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourier2.t	⊢ 𝑇 = (2 · π)
fourier2.per	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
fourier2.g	⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))
fourier2.dmdv	⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
fourier2.dvcn	⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ))
fourier2.rlim	⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
fourier2.llim	⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
fourier2.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourier2.a	⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
fourier2.b	⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))

**Assertion**
Ref	Expression
fourier2	⊢ (𝜑 → ∃𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)∃𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)(((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝑙 + 𝑟) / 2))

Distinct variable groups: 𝐴,𝑛 𝐵,𝑛 𝐹,𝑙,𝑛,𝑟,𝑥 𝑥,𝐺 𝑇,𝑛,𝑥 𝑋,𝑙,𝑛,𝑟,𝑥 𝜑,𝑙,𝑛,𝑟,𝑥 Allowed substitution hints: 𝐴(𝑥,𝑟,𝑙) 𝐵(𝑥,𝑟,𝑙) 𝑇(𝑟,𝑙) 𝐺(𝑛,𝑟,𝑙)

**Proof of Theorem fourier2**
Step	Hyp	Ref	Expression
1		fourier2.f	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
2		fourier2.t	. . . . . 6 ⊢ 𝑇 = (2 · π)
3		fourier2.per	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
4		fourier2.g	. . . . . 6 ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))
5		fourier2.dmdv	. . . . . 6 ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
6		fourier2.dvcn	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ))
7		fourier2.rlim	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
8		fourier2.llim	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
9		fourier2.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℝ)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	fourierdlem106 46208	. . . . 5 ⊢ (𝜑 → (((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ≠ ∅ ∧ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ≠ ∅))
11	10	simpld 494	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ≠ ∅)
12		n0 4333	. . . 4 ⊢ (((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ≠ ∅ ↔ ∃𝑙 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
13	11, 12	sylib 218	. . 3 ⊢ (𝜑 → ∃𝑙 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
14		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) → 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
15	10	simprd 495	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ≠ ∅)
16		n0 4333	. . . . . . . . . 10 ⊢ (((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ≠ ∅ ↔ ∃𝑟 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
17	15, 16	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑟 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
18	17	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) → ∃𝑟 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
19		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) → 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
20	1	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) → 𝐹:ℝ⟶ℝ)
21	3	ad4ant14 752	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
22	5	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
23	6	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) → 𝐺 ∈ (dom 𝐺–cn→ℂ))
24	7	ad4ant14 752	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅)
25	8	ad4ant14 752	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅)
26	9	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) → 𝑋 ∈ ℝ)
27	14	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) → 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
28		fourier2.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))
29		fourier2.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))
30	20, 2, 21, 4, 22, 23, 24, 25, 26, 27, 19, 28, 29	fourierd 46218	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) → (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝑙 + 𝑟) / 2))
31	19, 30	jca 511	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) → (𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝑙 + 𝑟) / 2)))
32	31	ex 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) → (𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) → (𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝑙 + 𝑟) / 2))))
33	32	eximdv 1917	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) → (∃𝑟 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) → ∃𝑟(𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝑙 + 𝑟) / 2))))
34	18, 33	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) → ∃𝑟(𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝑙 + 𝑟) / 2)))
35		df-rex 3062	. . . . . . 7 ⊢ (∃𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)(((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝑙 + 𝑟) / 2) ↔ ∃𝑟(𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝑙 + 𝑟) / 2)))
36	34, 35	sylibr 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) → ∃𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)(((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝑙 + 𝑟) / 2))
37	14, 36	jca 511	. . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) → (𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ∧ ∃𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)(((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝑙 + 𝑟) / 2)))
38	37	ex 412	. . . 4 ⊢ (𝜑 → (𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) → (𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ∧ ∃𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)(((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝑙 + 𝑟) / 2))))
39	38	eximdv 1917	. . 3 ⊢ (𝜑 → (∃𝑙 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) → ∃𝑙(𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ∧ ∃𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)(((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝑙 + 𝑟) / 2))))
40	13, 39	mpd 15	. 2 ⊢ (𝜑 → ∃𝑙(𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ∧ ∃𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)(((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝑙 + 𝑟) / 2)))
41		df-rex 3062	. 2 ⊢ (∃𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)∃𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)(((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝑙 + 𝑟) / 2) ↔ ∃𝑙(𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ∧ ∃𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)(((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝑙 + 𝑟) / 2)))
42	40, 41	sylibr 234	1 ⊢ (𝜑 → ∃𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)∃𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)(((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝑙 + 𝑟) / 2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ≠ wne 2933 ∃wrex 3061 ∖ cdif 3928 ∅c0 4313 ↦ cmpt 5206 dom cdm 5659 ↾ cres 5661 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 Fincfn 8964 ℂcc 11132 ℝcr 11133 0cc0 11134 + caddc 11137 · cmul 11139 +∞cpnf 11271 -∞cmnf 11272 -cneg 11472 / cdiv 11899 ℕcn 12245 2c2 12300 ℕ₀cn0 12506 (,)cioo 13367 (,]cioc 13368 [,)cico 13369 Σcsu 15707 sincsin 16084 cosccos 16085 πcpi 16087 –cn→ccncf 24825 ∫citg 25576 lim_ℂ climc 25820 D cdv 25821

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cc 10454 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-symdif 4233 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-disj 5092 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 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 7676 df-ofr 7677 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-omul 8490 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-dju 9920 df-card 9958 df-acn 9961 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 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 12507 df-xnn0 12580 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-shft 15091 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-limsup 15492 df-clim 15509 df-rlim 15510 df-sum 15708 df-ef 16088 df-sin 16090 df-cos 16091 df-pi 16093 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-mulg 19056 df-cntz 19305 df-cmn 19768 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-t1 23257 df-haus 23258 df-cmp 23330 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-ovol 25422 df-vol 25423 df-mbf 25577 df-itg1 25578 df-itg2 25579 df-ibl 25580 df-itg 25581 df-0p 25628 df-ditg 25805 df-limc 25824 df-dv 25825

This theorem is referenced by: (None)

W3C validator