fourier2 - Mathbox for Glauco Siliprandi

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Theorem fourier2 47206

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 47201 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 47191	. . . . 5 ⊢ (𝜑 → (((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ≠ ∅ ∧ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ≠ ∅))
11	10	simpld 500	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ≠ ∅)
12		n0 4300	. . . 4 ⊢ (((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ≠ ∅ ↔ ∃𝑙 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
13	11, 12	sylib 221	. . 3 ⊢ (𝜑 → ∃𝑙 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
14		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) → 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋))
15	10	simprd 501	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ≠ ∅)
16		n0 4300	. . . . . . . . . 10 ⊢ (((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ≠ ∅ ↔ ∃𝑟 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
17	15, 16	sylib 221	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑟 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
18	17	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) → ∃𝑟 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
19		simpr 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) → 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋))
20	1	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) → 𝐹:ℝ⟶ℝ)
21	3	ad4ant14 765	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))
22	5	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)
23	6	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) → 𝐺 ∈ (dom 𝐺–cn→ℂ))
24	7	ad4ant14 765	. . . . . . . . . . . 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 765	. . . . . . . . . . . 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 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) ∧ 𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) → 𝑋 ∈ ℝ)
27	14	adantr 486	. . . . . . . . . . . 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 47201	. . . . . . . . . . 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 521	. . . . . . . . . 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 418	. . . . . . . . 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 1950	. . . . . . . 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 16	. . . . . . 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 3088	. . . . . . 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 237	. . . . . 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 521	. . . . 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 418	. . . 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 1950	. . 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 16	. 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 3088	. 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 237	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2145 ≠ wne 2956 ∃wrex 3087 ∖ cdif 3896 ∅c0 4279 ↦ cmpt 5186 dom cdm 5651 ↾ cres 5653 ⟶wf 6533 ‘cfv 6537 (class class class)co 7418 Fincfn 8966 ℂcc 11191 ℝcr 11192 0cc0 11193 + caddc 11196 · cmul 11198 +∞cpnf 11333 -∞cmnf 11334 -cneg 11535 / cdiv 11966 ℕcn 12328 2c2 12390 ℕ₀cn0 12599 (,)cioo 13469 (,]cioc 13470 [,)cico 13471 Σcsu 15846 sincsin 16222 cosccos 16223 πcpi 16225 –cn→ccncf 25190 ∫citg 25932 lim_ℂ climc 26175 D cdv 26176

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-inf2 9635 ax-cc 10506 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271 ax-addf 11272

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-symdif 4199 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-ofr 7692 df-om 7876 df-1st 7999 df-2nd 8000 df-supp 8171 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-oadd 8473 df-omul 8474 df-er 8710 df-map 8842 df-pm 8843 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9347 df-fi 9396 df-sup 9427 df-inf 9428 df-oi 9497 df-dju 9975 df-card 10013 df-acn 10016 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-xnn0 12673 df-z 12687 df-dec 12808 df-uz 12959 df-q 13069 df-rp 13114 df-xneg 13234 df-xadd 13235 df-xmul 13236 df-ioo 13473 df-ioc 13474 df-ico 13475 df-icc 13476 df-fz 13633 df-fzo 13782 df-fl 13925 df-mod 14003 df-seq 14138 df-exp 14198 df-fac 14411 df-bc 14440 df-hash 14468 df-shft 15213 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-limsup 15631 df-clim 15648 df-rlim 15649 df-sum 15847 df-ef 16226 df-sin 16228 df-cos 16229 df-pi 16231 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-starv 17436 df-sca 17437 df-vsca 17438 df-ip 17439 df-tset 17440 df-ple 17441 df-ds 17443 df-unif 17444 df-hom 17445 df-cco 17446 df-rest 17586 df-topn 17587 df-0g 17605 df-gsum 17606 df-topgen 17607 df-pt 17608 df-prds 17611 df-xrs 17667 df-qtop 17672 df-imas 17673 df-xps 17675 df-mre 17749 df-mrc 17750 df-acs 17752 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-submnd 18972 df-mulg 19271 df-cntz 19524 df-cmn 19989 df-psmet 21663 df-xmet 21664 df-met 21665 df-bl 21666 df-mopn 21667 df-fbas 21668 df-fg 21669 df-cnfld 21672 df-top 23205 df-topon 23222 df-topsp 23244 df-bases 23257 df-cld 23330 df-ntr 23331 df-cls 23332 df-nei 23409 df-lp 23447 df-perf 23448 df-cn 23538 df-cnp 23539 df-t1 23625 df-haus 23626 df-cmp 23698 df-tx 23874 df-hmeo 24067 df-fil 24158 df-fm 24250 df-flim 24251 df-flf 24252 df-xms 24632 df-ms 24633 df-tms 24634 df-cncf 25192 df-ovol 25778 df-vol 25779 df-mbf 25933 df-itg1 25934 df-itg2 25935 df-ibl 25936 df-itg 25937 df-0p 25984 df-ditg 26160 df-limc 26179 df-dv 26180

This theorem is used by: (None)

W3C validator