dvdivbd - Mathbox for Glauco Siliprandi

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Theorem dvdivbd 45844

Description: A sufficient condition for the derivative to be bounded, for the quotient of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
dvdivbd.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvdivbd.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
dvdivbd.adv	⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
dvdivbd.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)
dvdivbd.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
dvdivbd.u	⊢ (𝜑 → 𝑈 ∈ ℝ)
dvdivbd.r	⊢ (𝜑 → 𝑅 ∈ ℝ)
dvdivbd.t	⊢ (𝜑 → 𝑇 ∈ ℝ)
dvdivbd.q	⊢ (𝜑 → 𝑄 ∈ ℝ)
dvdivbd.cbd	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐶) ≤ 𝑈)
dvdivbd.bbd	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐵) ≤ 𝑅)
dvdivbd.dbd	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐷) ≤ 𝑇)
dvdivbd.abd	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐴) ≤ 𝑄)
dvdivbd.bdv	⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐷))
dvdivbd.d	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ ℂ)
dvdivbd.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
dvdivbd.ele	⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐸 ≤ (abs‘𝐵))
dvdivbd.f	⊢ 𝐹 = (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵)))

**Assertion**
Ref	Expression
dvdivbd	⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘(𝐹‘𝑥)) ≤ 𝑏)

Distinct variable groups: 𝐸,𝑏,𝑥 𝐹,𝑏 𝑄,𝑏,𝑥 𝑅,𝑏,𝑥 𝑥,𝑆 𝑇,𝑏,𝑥 𝑈,𝑏,𝑥 𝑋,𝑏,𝑥 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑏) 𝐴(𝑥,𝑏) 𝐵(𝑥,𝑏) 𝐶(𝑥,𝑏) 𝐷(𝑥,𝑏) 𝑆(𝑏) 𝐹(𝑥)

**Proof of Theorem dvdivbd**
Step	Hyp	Ref	Expression
1		dvdivbd.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ ℝ)
2		dvdivbd.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ)
3	1, 2	remulcld 11320	. . . 4 ⊢ (𝜑 → (𝑈 · 𝑅) ∈ ℝ)
4		dvdivbd.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ)
5		dvdivbd.q	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℝ)
6	4, 5	remulcld 11320	. . . 4 ⊢ (𝜑 → (𝑇 · 𝑄) ∈ ℝ)
7	3, 6	readdcld 11319	. . 3 ⊢ (𝜑 → ((𝑈 · 𝑅) + (𝑇 · 𝑄)) ∈ ℝ)
8		dvdivbd.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
9	8	rpred 13099	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ)
10	9	resqcld 14175	. . 3 ⊢ (𝜑 → (𝐸↑2) ∈ ℝ)
11	8	rpcnd 13101	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ)
12	8	rpgt0d 13102	. . . . 5 ⊢ (𝜑 → 0 < 𝐸)
13	12	gt0ne0d 11854	. . . 4 ⊢ (𝜑 → 𝐸 ≠ 0)
14		2z 12675	. . . . 5 ⊢ 2 ∈ ℤ
15	14	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℤ)
16	11, 13, 15	expne0d 14202	. . 3 ⊢ (𝜑 → (𝐸↑2) ≠ 0)
17	7, 10, 16	redivcld 12122	. 2 ⊢ (𝜑 → (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)) ∈ ℝ)
18		dvdivbd.f	. . . . . . 7 ⊢ 𝐹 = (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵)))
19		dvdivbd.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
20		dvdivbd.a	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
21		dvdivbd.c	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)
22		dvdivbd.adv	. . . . . . . 8 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
23		dvdivbd.b	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
24		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → 𝐵 = 0)
25	24	abs00bd 15340	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → (abs‘𝐵) = 0)
26		0red 11293	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ)
27	9	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐸 ∈ ℝ)
28	23	abscld 15485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐵) ∈ ℝ)
29	12	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 < 𝐸)
30		dvdivbd.ele	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐸 ≤ (abs‘𝐵))
31	30	r19.21bi 3257	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐸 ≤ (abs‘𝐵))
32	26, 27, 28, 29, 31	ltletrd 11450	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 < (abs‘𝐵))
33	32	gt0ne0d 11854	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐵) ≠ 0)
34	33	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → (abs‘𝐵) ≠ 0)
35	34	neneqd 2951	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → ¬ (abs‘𝐵) = 0)
36	25, 35	pm2.65da 816	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ 𝐵 = 0)
37	36	neqned 2953	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ≠ 0)
38		eldifsn 4811	. . . . . . . . 9 ⊢ (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
39	23, 37, 38	sylanbrc 582	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (ℂ ∖ {0}))
40		dvdivbd.d	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ ℂ)
41		dvdivbd.bdv	. . . . . . . 8 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐷))
42	19, 20, 21, 22, 39, 40, 41	dvmptdiv 26032	. . . . . . 7 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵))) = (𝑥 ∈ 𝑋 ↦ (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))))
43	18, 42	eqtrid 2792	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))))
44	21, 23	mulcld 11310	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐶 · 𝐵) ∈ ℂ)
45	40, 20	mulcld 11310	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐷 · 𝐴) ∈ ℂ)
46	44, 45	subcld 11647	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐶 · 𝐵) − (𝐷 · 𝐴)) ∈ ℂ)
47	23	sqcld 14194	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ∈ ℂ)
48		sqne0 14173	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → ((𝐵↑2) ≠ 0 ↔ 𝐵 ≠ 0))
49	23, 48	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐵↑2) ≠ 0 ↔ 𝐵 ≠ 0))
50	37, 49	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ≠ 0)
51	46, 47, 50	divcld 12070	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2)) ∈ ℂ)
52	43, 51	fvmpt2d 7042	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) = (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2)))
53	52	fveq2d 6924	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐹‘𝑥)) = (abs‘(((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))))
54	46, 47, 50	absdivd 15504	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))) = ((abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) / (abs‘(𝐵↑2))))
55	46	abscld 15485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) ∈ ℝ)
56	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑈 · 𝑅) + (𝑇 · 𝑄)) ∈ ℝ)
57	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐸 ∈ ℝ⁺)
58	14	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 2 ∈ ℤ)
59	57, 58	rpexpcld 14296	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐸↑2) ∈ ℝ⁺)
60	47	abscld 15485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐵↑2)) ∈ ℝ)
61	46	absge0d 15493	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))))
62	44	abscld 15485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐶 · 𝐵)) ∈ ℝ)
63	45	abscld 15485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐷 · 𝐴)) ∈ ℝ)
64	62, 63	readdcld 11319	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘(𝐶 · 𝐵)) + (abs‘(𝐷 · 𝐴))) ∈ ℝ)
65	44, 45	abs2dif2d 15507	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) ≤ ((abs‘(𝐶 · 𝐵)) + (abs‘(𝐷 · 𝐴))))
66	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑈 · 𝑅) ∈ ℝ)
67	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑇 · 𝑄) ∈ ℝ)
68	21, 23	absmuld 15503	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐶 · 𝐵)) = ((abs‘𝐶) · (abs‘𝐵)))
69	21	abscld 15485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐶) ∈ ℝ)
70	1	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑈 ∈ ℝ)
71	2	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ ℝ)
72	21	absge0d 15493	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝐶))
73	23	absge0d 15493	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝐵))
74		dvdivbd.cbd	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐶) ≤ 𝑈)
75		dvdivbd.bbd	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐵) ≤ 𝑅)
76	69, 70, 28, 71, 72, 73, 74, 75	lemul12ad 12237	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐶) · (abs‘𝐵)) ≤ (𝑈 · 𝑅))
77	68, 76	eqbrtrd 5188	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐶 · 𝐵)) ≤ (𝑈 · 𝑅))
78	40, 20	absmuld 15503	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐷 · 𝐴)) = ((abs‘𝐷) · (abs‘𝐴)))
79	40	abscld 15485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐷) ∈ ℝ)
80	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑇 ∈ ℝ)
81	20	abscld 15485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐴) ∈ ℝ)
82	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑄 ∈ ℝ)
83	40	absge0d 15493	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝐷))
84	20	absge0d 15493	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝐴))
85		dvdivbd.dbd	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐷) ≤ 𝑇)
86		dvdivbd.abd	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐴) ≤ 𝑄)
87	79, 80, 81, 82, 83, 84, 85, 86	lemul12ad 12237	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐷) · (abs‘𝐴)) ≤ (𝑇 · 𝑄))
88	78, 87	eqbrtrd 5188	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐷 · 𝐴)) ≤ (𝑇 · 𝑄))
89	62, 63, 66, 67, 77, 88	le2addd 11909	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘(𝐶 · 𝐵)) + (abs‘(𝐷 · 𝐴))) ≤ ((𝑈 · 𝑅) + (𝑇 · 𝑄)))
90	55, 64, 56, 65, 89	letrd 11447	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) ≤ ((𝑈 · 𝑅) + (𝑇 · 𝑄)))
91		2nn0 12570	. . . . . . . . 9 ⊢ 2 ∈ ℕ₀
92	91	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 2 ∈ ℕ₀)
93	26, 27, 29	ltled 11438	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ 𝐸)
94		leexp1a 14225	. . . . . . . 8 ⊢ (((𝐸 ∈ ℝ ∧ (abs‘𝐵) ∈ ℝ ∧ 2 ∈ ℕ₀) ∧ (0 ≤ 𝐸 ∧ 𝐸 ≤ (abs‘𝐵))) → (𝐸↑2) ≤ ((abs‘𝐵)↑2))
95	27, 28, 92, 93, 31, 94	syl32anc 1378	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐸↑2) ≤ ((abs‘𝐵)↑2))
96	23, 92	absexpd 15501	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐵↑2)) = ((abs‘𝐵)↑2))
97	95, 96	breqtrrd 5194	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐸↑2) ≤ (abs‘(𝐵↑2)))
98	55, 56, 59, 60, 61, 90, 97	lediv12ad 13158	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) / (abs‘(𝐵↑2))) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
99	54, 98	eqbrtrd 5188	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
100	53, 99	eqbrtrd 5188	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐹‘𝑥)) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
101	100	ralrimiva 3152	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (abs‘(𝐹‘𝑥)) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
102		brralrspcev 5226	. 2 ⊢ (((((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)) ∈ ℝ ∧ ∀𝑥 ∈ 𝑋 (abs‘(𝐹‘𝑥)) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2))) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘(𝐹‘𝑥)) ≤ 𝑏)
103	17, 101, 102	syl2anc 583	1 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘(𝐹‘𝑥)) ≤ 𝑏)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 ∖ cdif 3973 {csn 4648 {cpr 4650 class class class wbr 5166 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 + caddc 11187 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ℝ⁺crp 13057 ↑cexp 14112 abscabs 15283 D cdv 25918

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-icc 13414 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-t1 23343 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922

This theorem is referenced by: fourierdlem68 46095

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