dvdivbd - Mathbox for Glauco Siliprandi

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Theorem dvdivbd 47138

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 11366	. . . 4 ⊢ (𝜑 → (𝑈 · 𝑅) ∈ ℝ)
4		dvdivbd.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ)
5		dvdivbd.q	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℝ)
6	4, 5	remulcld 11366	. . . 4 ⊢ (𝜑 → (𝑇 · 𝑄) ∈ ℝ)
7	3, 6	readdcld 11365	. . 3 ⊢ (𝜑 → ((𝑈 · 𝑅) + (𝑇 · 𝑄)) ∈ ℝ)
8		dvdivbd.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
9	8	rpred 13191	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ)
10	9	resqcld 14295	. . 3 ⊢ (𝜑 → (𝐸↑2) ∈ ℝ)
11	8	rpcnd 13193	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ)
12	8	rpgt0d 13194	. . . . 5 ⊢ (𝜑 → 0 < 𝐸)
13	12	gt0ne0d 11907	. . . 4 ⊢ (𝜑 → 𝐸 ≠ 0)
14		2z 12755	. . . . 5 ⊢ 2 ∈ ℤ
15	14	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℤ)
16	11, 13, 15	expne0d 14322	. . 3 ⊢ (𝜑 → (𝐸↑2) ≠ 0)
17	7, 10, 16	redivcld 12172	. 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 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → 𝐵 = 0)
25	24	abs00bd 15487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → (abs‘𝐵) = 0)
26		0red 11338	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ)
27	9	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐸 ∈ ℝ)
28	23	abscld 15635	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐵) ∈ ℝ)
29	12	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 < 𝐸)
30		dvdivbd.ele	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐸 ≤ (abs‘𝐵))
31	30	r19.21bi 3258	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐸 ≤ (abs‘𝐵))
32	26, 27, 28, 29, 31	ltletrd 11497	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 < (abs‘𝐵))
33	32	gt0ne0d 11907	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐵) ≠ 0)
34	33	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → (abs‘𝐵) ≠ 0)
35	34	neneqd 2964	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → ¬ (abs‘𝐵) = 0)
36	25, 35	pm2.65da 829	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ 𝐵 = 0)
37	36	neqned 2966	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ≠ 0)
38		eldifsn 4748	. . . . . . . . 9 ⊢ (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
39	23, 37, 38	sylanbrc 595	. . . . . . . 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 26324	. . . . . . 7 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵))) = (𝑥 ∈ 𝑋 ↦ (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))))
43	18, 42	eqtrid 2810	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))))
44	21, 23	mulcld 11356	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐶 · 𝐵) ∈ ℂ)
45	40, 20	mulcld 11356	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐷 · 𝐴) ∈ ℂ)
46	44, 45	subcld 11696	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐶 · 𝐵) − (𝐷 · 𝐴)) ∈ ℂ)
47	23	sqcld 14314	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ∈ ℂ)
48		sqne0 14293	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → ((𝐵↑2) ≠ 0 ↔ 𝐵 ≠ 0))
49	23, 48	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐵↑2) ≠ 0 ↔ 𝐵 ≠ 0))
50	37, 49	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ≠ 0)
51	46, 47, 50	divcld 12120	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2)) ∈ ℂ)
52	43, 51	fvmpt2d 7030	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) = (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2)))
53	52	fveq2d 6912	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐹‘𝑥)) = (abs‘(((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))))
54	46, 47, 50	absdivd 15654	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))) = ((abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) / (abs‘(𝐵↑2))))
55	46	abscld 15635	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) ∈ ℝ)
56	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑈 · 𝑅) + (𝑇 · 𝑄)) ∈ ℝ)
57	8	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐸 ∈ ℝ⁺)
58	14	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 2 ∈ ℤ)
59	57, 58	rpexpcld 14418	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐸↑2) ∈ ℝ⁺)
60	47	abscld 15635	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐵↑2)) ∈ ℝ)
61	46	absge0d 15643	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))))
62	44	abscld 15635	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐶 · 𝐵)) ∈ ℝ)
63	45	abscld 15635	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐷 · 𝐴)) ∈ ℝ)
64	62, 63	readdcld 11365	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘(𝐶 · 𝐵)) + (abs‘(𝐷 · 𝐴))) ∈ ℝ)
65	44, 45	abs2dif2d 15657	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) ≤ ((abs‘(𝐶 · 𝐵)) + (abs‘(𝐷 · 𝐴))))
66	3	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑈 · 𝑅) ∈ ℝ)
67	6	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑇 · 𝑄) ∈ ℝ)
68	21, 23	absmuld 15653	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐶 · 𝐵)) = ((abs‘𝐶) · (abs‘𝐵)))
69	21	abscld 15635	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐶) ∈ ℝ)
70	1	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑈 ∈ ℝ)
71	2	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ ℝ)
72	21	absge0d 15643	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝐶))
73	23	absge0d 15643	. . . . . . . . . 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 12286	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐶) · (abs‘𝐵)) ≤ (𝑈 · 𝑅))
77	68, 76	eqbrtrd 5130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐶 · 𝐵)) ≤ (𝑈 · 𝑅))
78	40, 20	absmuld 15653	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐷 · 𝐴)) = ((abs‘𝐷) · (abs‘𝐴)))
79	40	abscld 15635	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐷) ∈ ℝ)
80	4	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑇 ∈ ℝ)
81	20	abscld 15635	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐴) ∈ ℝ)
82	5	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑄 ∈ ℝ)
83	40	absge0d 15643	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝐷))
84	20	absge0d 15643	. . . . . . . . . 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 12286	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐷) · (abs‘𝐴)) ≤ (𝑇 · 𝑄))
88	78, 87	eqbrtrd 5130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐷 · 𝐴)) ≤ (𝑇 · 𝑄))
89	62, 63, 66, 67, 77, 88	le2addd 11962	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘(𝐶 · 𝐵)) + (abs‘(𝐷 · 𝐴))) ≤ ((𝑈 · 𝑅) + (𝑇 · 𝑄)))
90	55, 64, 56, 65, 89	letrd 11494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) ≤ ((𝑈 · 𝑅) + (𝑇 · 𝑄)))
91		2nn0 12650	. . . . . . . . 9 ⊢ 2 ∈ ℕ₀
92	91	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 2 ∈ ℕ₀)
93	26, 27, 29	ltled 11485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ 𝐸)
94		leexp1a 14345	. . . . . . . 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 1406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐸↑2) ≤ ((abs‘𝐵)↑2))
96	23, 92	absexpd 15651	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐵↑2)) = ((abs‘𝐵)↑2))
97	95, 96	breqtrrd 5136	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐸↑2) ≤ (abs‘(𝐵↑2)))
98	55, 56, 59, 60, 61, 90, 97	lediv12ad 13250	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) / (abs‘(𝐵↑2))) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
99	54, 98	eqbrtrd 5130	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
100	53, 99	eqbrtrd 5130	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐹‘𝑥)) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
101	100	ralrimiva 3158	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (abs‘(𝐹‘𝑥)) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
102		brralrspcev 5168	. 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 596	1 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘(𝐹‘𝑥)) ≤ 𝑏)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ∃wrex 3090 ∖ cdif 4048 {csn 4588 {cpr 4590 class class class wbr 5106 ↦ cmpt 5189 ‘cfv 6558 (class class class)co 7443 ℂcc 11225 ℝcr 11226 0cc0 11227 + caddc 11230 · cmul 11232 < clt 11370 ≤ cle 11371 − cmin 11568 / cdiv 12000 2c2 12424 ℕ₀cn0 12633 ℤcz 12720 ℝ⁺crp 13147 ↑cexp 14231 abscabs 15430 D cdv 26213

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 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305 ax-addf 11306

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-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-om 7902 df-1st 8025 df-2nd 8026 df-supp 8205 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-er 8745 df-map 8877 df-pm 8878 df-ixp 8954 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-fsupp 9382 df-fi 9430 df-sup 9461 df-inf 9462 df-oi 9531 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-z 12721 df-dec 12842 df-uz 12993 df-q 13103 df-rp 13148 df-xneg 13268 df-xadd 13269 df-xmul 13270 df-icc 13510 df-fz 13667 df-fzo 13816 df-seq 14172 df-exp 14232 df-hash 14502 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-starv 17472 df-sca 17473 df-vsca 17474 df-ip 17475 df-tset 17476 df-ple 17477 df-ds 17479 df-unif 17480 df-hom 17481 df-cco 17482 df-rest 17622 df-topn 17623 df-0g 17641 df-gsum 17642 df-topgen 17643 df-pt 17644 df-prds 17647 df-xrs 17703 df-qtop 17708 df-imas 17709 df-xps 17711 df-mre 17785 df-mrc 17786 df-acs 17788 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-submnd 19008 df-mulg 19307 df-cntz 19561 df-cmn 20026 df-psmet 21700 df-xmet 21701 df-met 21702 df-bl 21703 df-mopn 21704 df-fbas 21705 df-fg 21706 df-cnfld 21709 df-top 23242 df-topon 23259 df-topsp 23281 df-bases 23294 df-cld 23367 df-ntr 23368 df-cls 23369 df-nei 23446 df-lp 23484 df-perf 23485 df-cn 23575 df-cnp 23576 df-t1 23662 df-haus 23663 df-tx 23911 df-hmeo 24104 df-fil 24195 df-fm 24287 df-flim 24288 df-flf 24289 df-xms 24669 df-ms 24670 df-tms 24671 df-cncf 25229 df-limc 26216 df-dv 26217

This theorem is used by: fourierdlem68 47389

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