dvdivbd - Mathbox for Glauco Siliprandi

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Theorem dvdivbd 42200

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 10665	. . . 4 ⊢ (𝜑 → (𝑈 · 𝑅) ∈ ℝ)
4		dvdivbd.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ)
5		dvdivbd.q	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℝ)
6	4, 5	remulcld 10665	. . . 4 ⊢ (𝜑 → (𝑇 · 𝑄) ∈ ℝ)
7	3, 6	readdcld 10664	. . 3 ⊢ (𝜑 → ((𝑈 · 𝑅) + (𝑇 · 𝑄)) ∈ ℝ)
8		dvdivbd.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
9	8	rpred 12425	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ)
10	9	resqcld 13605	. . 3 ⊢ (𝜑 → (𝐸↑2) ∈ ℝ)
11	8	rpcnd 12427	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ)
12	8	rpgt0d 12428	. . . . 5 ⊢ (𝜑 → 0 < 𝐸)
13	12	gt0ne0d 11198	. . . 4 ⊢ (𝜑 → 𝐸 ≠ 0)
14		2z 12008	. . . . 5 ⊢ 2 ∈ ℤ
15	14	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℤ)
16	11, 13, 15	expne0d 13510	. . 3 ⊢ (𝜑 → (𝐸↑2) ≠ 0)
17	7, 10, 16	redivcld 11462	. 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 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → 𝐵 = 0)
25	24	abs00bd 14645	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → (abs‘𝐵) = 0)
26		0red 10638	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ)
27	9	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐸 ∈ ℝ)
28	23	abscld 14790	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐵) ∈ ℝ)
29	12	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 < 𝐸)
30		dvdivbd.ele	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐸 ≤ (abs‘𝐵))
31	30	r19.21bi 3208	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐸 ≤ (abs‘𝐵))
32	26, 27, 28, 29, 31	ltletrd 10794	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 < (abs‘𝐵))
33	32	gt0ne0d 11198	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐵) ≠ 0)
34	33	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → (abs‘𝐵) ≠ 0)
35	34	neneqd 3021	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → ¬ (abs‘𝐵) = 0)
36	25, 35	pm2.65da 815	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ 𝐵 = 0)
37	36	neqned 3023	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ≠ 0)
38		eldifsn 4713	. . . . . . . . 9 ⊢ (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
39	23, 37, 38	sylanbrc 585	. . . . . . . 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 24565	. . . . . . 7 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵))) = (𝑥 ∈ 𝑋 ↦ (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))))
43	18, 42	syl5eq 2868	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))))
44	21, 23	mulcld 10655	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐶 · 𝐵) ∈ ℂ)
45	40, 20	mulcld 10655	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐷 · 𝐴) ∈ ℂ)
46	44, 45	subcld 10991	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐶 · 𝐵) − (𝐷 · 𝐴)) ∈ ℂ)
47	23	sqcld 13502	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ∈ ℂ)
48		sqne0 13483	. . . . . . . . 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 259	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ≠ 0)
51	46, 47, 50	divcld 11410	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2)) ∈ ℂ)
52	43, 51	fvmpt2d 6776	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) = (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2)))
53	52	fveq2d 6669	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐹‘𝑥)) = (abs‘(((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))))
54	46, 47, 50	absdivd 14809	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))) = ((abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) / (abs‘(𝐵↑2))))
55	46	abscld 14790	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) ∈ ℝ)
56	7	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑈 · 𝑅) + (𝑇 · 𝑄)) ∈ ℝ)
57	8	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐸 ∈ ℝ⁺)
58	14	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 2 ∈ ℤ)
59	57, 58	rpexpcld 13602	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐸↑2) ∈ ℝ⁺)
60	47	abscld 14790	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐵↑2)) ∈ ℝ)
61	46	absge0d 14798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))))
62	44	abscld 14790	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐶 · 𝐵)) ∈ ℝ)
63	45	abscld 14790	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐷 · 𝐴)) ∈ ℝ)
64	62, 63	readdcld 10664	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘(𝐶 · 𝐵)) + (abs‘(𝐷 · 𝐴))) ∈ ℝ)
65	44, 45	abs2dif2d 14812	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) ≤ ((abs‘(𝐶 · 𝐵)) + (abs‘(𝐷 · 𝐴))))
66	3	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑈 · 𝑅) ∈ ℝ)
67	6	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑇 · 𝑄) ∈ ℝ)
68	21, 23	absmuld 14808	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐶 · 𝐵)) = ((abs‘𝐶) · (abs‘𝐵)))
69	21	abscld 14790	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐶) ∈ ℝ)
70	1	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑈 ∈ ℝ)
71	2	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ ℝ)
72	21	absge0d 14798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝐶))
73	23	absge0d 14798	. . . . . . . . . 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 11576	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐶) · (abs‘𝐵)) ≤ (𝑈 · 𝑅))
77	68, 76	eqbrtrd 5081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐶 · 𝐵)) ≤ (𝑈 · 𝑅))
78	40, 20	absmuld 14808	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐷 · 𝐴)) = ((abs‘𝐷) · (abs‘𝐴)))
79	40	abscld 14790	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐷) ∈ ℝ)
80	4	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑇 ∈ ℝ)
81	20	abscld 14790	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐴) ∈ ℝ)
82	5	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑄 ∈ ℝ)
83	40	absge0d 14798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝐷))
84	20	absge0d 14798	. . . . . . . . . 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 11576	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐷) · (abs‘𝐴)) ≤ (𝑇 · 𝑄))
88	78, 87	eqbrtrd 5081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐷 · 𝐴)) ≤ (𝑇 · 𝑄))
89	62, 63, 66, 67, 77, 88	le2addd 11253	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘(𝐶 · 𝐵)) + (abs‘(𝐷 · 𝐴))) ≤ ((𝑈 · 𝑅) + (𝑇 · 𝑄)))
90	55, 64, 56, 65, 89	letrd 10791	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) ≤ ((𝑈 · 𝑅) + (𝑇 · 𝑄)))
91		2nn0 11908	. . . . . . . . 9 ⊢ 2 ∈ ℕ₀
92	91	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 2 ∈ ℕ₀)
93	26, 27, 29	ltled 10782	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ 𝐸)
94		leexp1a 13533	. . . . . . . 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 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐸↑2) ≤ ((abs‘𝐵)↑2))
96	23, 92	absexpd 14806	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐵↑2)) = ((abs‘𝐵)↑2))
97	95, 96	breqtrrd 5087	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐸↑2) ≤ (abs‘(𝐵↑2)))
98	55, 56, 59, 60, 61, 90, 97	lediv12ad 12484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) / (abs‘(𝐵↑2))) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
99	54, 98	eqbrtrd 5081	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
100	53, 99	eqbrtrd 5081	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐹‘𝑥)) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
101	100	ralrimiva 3182	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (abs‘(𝐹‘𝑥)) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
102		brralrspcev 5119	. 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 586	1 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘(𝐹‘𝑥)) ≤ 𝑏)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ∖ cdif 3933 {csn 4561 {cpr 4563 class class class wbr 5059 ↦ cmpt 5139 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 + caddc 10534 · cmul 10536 < clt 10669 ≤ cle 10670 − cmin 10864 / cdiv 11291 2c2 11686 ℕ₀cn0 11891 ℤcz 11975 ℝ⁺crp 12383 ↑cexp 13423 abscabs 14587 D cdv 24455

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-icc 12739 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-t1 21916 df-haus 21917 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-limc 24458 df-dv 24459

This theorem is referenced by: fourierdlem68 42452

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