fourier2 - Mathbox for Glauco Siliprandi

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Theorem fourier2 47510

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 47505 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 47495	. . . . 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 4303	. . . 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 4303	. . . . . . . . . 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 47505	. . . . . . . . . . 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 1952	. . . . . . . 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 3091	. . . . . . 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 1952	. . 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 3091	. 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 2147 ≠ wne 2959 ∃wrex 3090 ∖ cdif 4048 ∅c0 4281 ↦ cmpt 5192 dom cdm 5658 ↾ cres 5660 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 Fincfn 9011 ℂcc 11237 ℝcr 11238 0cc0 11239 + caddc 11242 · cmul 11244 +∞cpnf 11379 -∞cmnf 11380 -cneg 11581 / cdiv 12013 ℕcn 12375 2c2 12437 ℕ₀cn0 12646 (,)cioo 13516 (,]cioc 13517 [,)cico 13518 Σcsu 15897 sincsin 16273 cosccos 16274 πcpi 16276 –cn→ccncf 25252 ∫citg 25994 lim_ℂ climc 26237 D cdv 26238

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cc 10552 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-symdif 4201 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-disj 5076 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-ofr 7726 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-oadd 8518 df-omul 8519 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-dju 10019 df-card 10057 df-acn 10060 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-xnn0 12720 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ioc 13521 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-fac 14459 df-bc 14488 df-hash 14516 df-shft 15263 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-limsup 15682 df-clim 15699 df-rlim 15700 df-sum 15898 df-ef 16277 df-sin 16279 df-cos 16280 df-pi 16282 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-mulg 19328 df-cntz 19582 df-cmn 20047 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-t1 23687 df-haus 23688 df-cmp 23760 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-ovol 25840 df-vol 25841 df-mbf 25995 df-itg1 25996 df-itg2 25997 df-ibl 25998 df-itg 25999 df-0p 26046 df-ditg 26222 df-limc 26241 df-dv 26242

This theorem is used by: (None)

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