fourier2 - Mathbox for Glauco Siliprandi

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Theorem fourier2 47344

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 47339 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 47329	. . . . 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 4299	. . . 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 4299	. . . . . . . . . 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 47339	. . . . . . . . . . 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 1949	. . . . . . . 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 3087	. . . . . . 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 1949	. . 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 3087	. 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 1569 ∃wex 1811 ∈ wcel 2144 ≠ wne 2955 ∃wrex 3086 ∖ cdif 4044 ∅c0 4277 ↦ cmpt 5185 dom cdm 5650 ↾ cres 5652 ⟶wf 6537 ‘cfv 6541 (class class class)co 7422 Fincfn 8978 ℂcc 11203 ℝcr 11204 0cc0 11205 + caddc 11208 · cmul 11210 +∞cpnf 11345 -∞cmnf 11346 -cneg 11547 / cdiv 11978 ℕcn 12340 2c2 12402 ℕ₀cn0 12611 (,)cioo 13481 (,]cioc 13482 [,)cico 13483 Σcsu 15860 sincsin 16236 cosccos 16237 πcpi 16239 –cn→ccncf 25204 ∫citg 25946 lim_ℂ climc 26189 D cdv 26190

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-inf2 9647 ax-cc 10518 ax-cnex 11261 ax-resscn 11262 ax-1cn 11263 ax-icn 11264 ax-addcl 11265 ax-addrcl 11266 ax-mulcl 11267 ax-mulrcl 11268 ax-mulcom 11269 ax-addass 11270 ax-mulass 11271 ax-distr 11272 ax-i2m1 11273 ax-1ne0 11274 ax-1rid 11275 ax-rnegex 11276 ax-rrecex 11277 ax-cnre 11278 ax-pre-lttri 11279 ax-pre-lttrn 11280 ax-pre-ltadd 11281 ax-pre-mulgt0 11282 ax-pre-sup 11283 ax-addf 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-symdif 4197 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-disj 5070 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7379 df-ov 7425 df-oprab 7426 df-mpo 7427 df-of 7695 df-ofr 7696 df-om 7880 df-1st 8003 df-2nd 8004 df-supp 8182 df-frecs 8304 df-wrecs 8335 df-recs 8384 df-rdg 8423 df-1o 8481 df-2o 8482 df-oadd 8485 df-omul 8486 df-er 8722 df-map 8854 df-pm 8855 df-ixp 8931 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-fsupp 9359 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9509 df-dju 9987 df-card 10025 df-acn 10028 df-pnf 11350 df-mnf 11351 df-xr 11352 df-ltxr 11353 df-le 11354 df-sub 11548 df-neg 11549 df-div 11979 df-nn 12341 df-2 12410 df-3 12411 df-4 12412 df-5 12413 df-6 12414 df-7 12415 df-8 12416 df-9 12417 df-n0 12612 df-xnn0 12685 df-z 12699 df-dec 12820 df-uz 12971 df-q 13081 df-rp 13126 df-xneg 13246 df-xadd 13247 df-xmul 13248 df-ioo 13485 df-ioc 13486 df-ico 13487 df-icc 13488 df-fz 13645 df-fzo 13794 df-fl 13937 df-mod 14015 df-seq 14150 df-exp 14210 df-fac 14423 df-bc 14452 df-hash 14480 df-shft 15227 df-cj 15273 df-re 15274 df-im 15275 df-sqrt 15409 df-abs 15410 df-limsup 15645 df-clim 15662 df-rlim 15663 df-sum 15861 df-ef 16240 df-sin 16242 df-cos 16243 df-pi 16245 df-struct 17332 df-sets 17349 df-slot 17367 df-ndx 17379 df-base 17395 df-ress 17416 df-plusg 17448 df-mulr 17449 df-starv 17450 df-sca 17451 df-vsca 17452 df-ip 17453 df-tset 17454 df-ple 17455 df-ds 17457 df-unif 17458 df-hom 17459 df-cco 17460 df-rest 17600 df-topn 17601 df-0g 17619 df-gsum 17620 df-topgen 17621 df-pt 17622 df-prds 17625 df-xrs 17681 df-qtop 17686 df-imas 17687 df-xps 17689 df-mre 17763 df-mrc 17764 df-acs 17766 df-mgm 18823 df-sgrp 18915 df-mnd 18931 df-submnd 18986 df-mulg 19285 df-cntz 19538 df-cmn 20003 df-psmet 21677 df-xmet 21678 df-met 21679 df-bl 21680 df-mopn 21681 df-fbas 21682 df-fg 21683 df-cnfld 21686 df-top 23219 df-topon 23236 df-topsp 23258 df-bases 23271 df-cld 23344 df-ntr 23345 df-cls 23346 df-nei 23423 df-lp 23461 df-perf 23462 df-cn 23552 df-cnp 23553 df-t1 23639 df-haus 23640 df-cmp 23712 df-tx 23888 df-hmeo 24081 df-fil 24172 df-fm 24264 df-flim 24265 df-flf 24266 df-xms 24646 df-ms 24647 df-tms 24648 df-cncf 25206 df-ovol 25792 df-vol 25793 df-mbf 25947 df-itg1 25948 df-itg2 25949 df-ibl 25950 df-itg 25951 df-0p 25998 df-ditg 26174 df-limc 26193 df-dv 26194

This theorem is used by: (None)

W3C validator