dvdivbd - Mathbox for Glauco Siliprandi

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Theorem dvdivbd 45921

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 11204	. . . 4 ⊢ (𝜑 → (𝑈 · 𝑅) ∈ ℝ)
4		dvdivbd.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ)
5		dvdivbd.q	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℝ)
6	4, 5	remulcld 11204	. . . 4 ⊢ (𝜑 → (𝑇 · 𝑄) ∈ ℝ)
7	3, 6	readdcld 11203	. . 3 ⊢ (𝜑 → ((𝑈 · 𝑅) + (𝑇 · 𝑄)) ∈ ℝ)
8		dvdivbd.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
9	8	rpred 12995	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ)
10	9	resqcld 14090	. . 3 ⊢ (𝜑 → (𝐸↑2) ∈ ℝ)
11	8	rpcnd 12997	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ)
12	8	rpgt0d 12998	. . . . 5 ⊢ (𝜑 → 0 < 𝐸)
13	12	gt0ne0d 11742	. . . 4 ⊢ (𝜑 → 𝐸 ≠ 0)
14		2z 12565	. . . . 5 ⊢ 2 ∈ ℤ
15	14	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℤ)
16	11, 13, 15	expne0d 14117	. . 3 ⊢ (𝜑 → (𝐸↑2) ≠ 0)
17	7, 10, 16	redivcld 12010	. 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 15257	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → (abs‘𝐵) = 0)
26		0red 11177	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ)
27	9	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐸 ∈ ℝ)
28	23	abscld 15405	. . . . . . . . . . . . . . 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 3229	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐸 ≤ (abs‘𝐵))
32	26, 27, 28, 29, 31	ltletrd 11334	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 < (abs‘𝐵))
33	32	gt0ne0d 11742	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐵) ≠ 0)
34	33	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → (abs‘𝐵) ≠ 0)
35	34	neneqd 2930	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → ¬ (abs‘𝐵) = 0)
36	25, 35	pm2.65da 816	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ 𝐵 = 0)
37	36	neqned 2932	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ≠ 0)
38		eldifsn 4750	. . . . . . . . 9 ⊢ (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
39	23, 37, 38	sylanbrc 583	. . . . . . . 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 25878	. . . . . . 7 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵))) = (𝑥 ∈ 𝑋 ↦ (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))))
43	18, 42	eqtrid 2776	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))))
44	21, 23	mulcld 11194	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐶 · 𝐵) ∈ ℂ)
45	40, 20	mulcld 11194	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐷 · 𝐴) ∈ ℂ)
46	44, 45	subcld 11533	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐶 · 𝐵) − (𝐷 · 𝐴)) ∈ ℂ)
47	23	sqcld 14109	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ∈ ℂ)
48		sqne0 14088	. . . . . . . . 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 11958	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2)) ∈ ℂ)
52	43, 51	fvmpt2d 6981	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) = (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2)))
53	52	fveq2d 6862	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐹‘𝑥)) = (abs‘(((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))))
54	46, 47, 50	absdivd 15424	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))) = ((abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) / (abs‘(𝐵↑2))))
55	46	abscld 15405	. . . . . 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 14212	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐸↑2) ∈ ℝ⁺)
60	47	abscld 15405	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐵↑2)) ∈ ℝ)
61	46	absge0d 15413	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))))
62	44	abscld 15405	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐶 · 𝐵)) ∈ ℝ)
63	45	abscld 15405	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐷 · 𝐴)) ∈ ℝ)
64	62, 63	readdcld 11203	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘(𝐶 · 𝐵)) + (abs‘(𝐷 · 𝐴))) ∈ ℝ)
65	44, 45	abs2dif2d 15427	. . . . . . 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 15423	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐶 · 𝐵)) = ((abs‘𝐶) · (abs‘𝐵)))
69	21	abscld 15405	. . . . . . . . . 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 15413	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝐶))
73	23	absge0d 15413	. . . . . . . . . 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 12125	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐶) · (abs‘𝐵)) ≤ (𝑈 · 𝑅))
77	68, 76	eqbrtrd 5129	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐶 · 𝐵)) ≤ (𝑈 · 𝑅))
78	40, 20	absmuld 15423	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐷 · 𝐴)) = ((abs‘𝐷) · (abs‘𝐴)))
79	40	abscld 15405	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐷) ∈ ℝ)
80	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑇 ∈ ℝ)
81	20	abscld 15405	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐴) ∈ ℝ)
82	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑄 ∈ ℝ)
83	40	absge0d 15413	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝐷))
84	20	absge0d 15413	. . . . . . . . . 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 12125	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐷) · (abs‘𝐴)) ≤ (𝑇 · 𝑄))
88	78, 87	eqbrtrd 5129	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐷 · 𝐴)) ≤ (𝑇 · 𝑄))
89	62, 63, 66, 67, 77, 88	le2addd 11797	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘(𝐶 · 𝐵)) + (abs‘(𝐷 · 𝐴))) ≤ ((𝑈 · 𝑅) + (𝑇 · 𝑄)))
90	55, 64, 56, 65, 89	letrd 11331	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) ≤ ((𝑈 · 𝑅) + (𝑇 · 𝑄)))
91		2nn0 12459	. . . . . . . . 9 ⊢ 2 ∈ ℕ₀
92	91	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 2 ∈ ℕ₀)
93	26, 27, 29	ltled 11322	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ 𝐸)
94		leexp1a 14140	. . . . . . . 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 1380	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐸↑2) ≤ ((abs‘𝐵)↑2))
96	23, 92	absexpd 15421	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐵↑2)) = ((abs‘𝐵)↑2))
97	95, 96	breqtrrd 5135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐸↑2) ≤ (abs‘(𝐵↑2)))
98	55, 56, 59, 60, 61, 90, 97	lediv12ad 13054	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) / (abs‘(𝐵↑2))) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
99	54, 98	eqbrtrd 5129	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
100	53, 99	eqbrtrd 5129	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐹‘𝑥)) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
101	100	ralrimiva 3125	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (abs‘(𝐹‘𝑥)) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
102		brralrspcev 5167	. 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 584	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 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ∖ cdif 3911 {csn 4589 {cpr 4591 class class class wbr 5107 ↦ cmpt 5188 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 + caddc 11071 · cmul 11073 < clt 11208 ≤ cle 11209 − cmin 11405 / cdiv 11835 2c2 12241 ℕ₀cn0 12442 ℤcz 12529 ℝ⁺crp 12951 ↑cexp 14026 abscabs 15200 D cdv 25764

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-icc 13313 df-fz 13469 df-fzo 13616 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-t1 23201 df-haus 23202 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768

This theorem is referenced by: fourierdlem68 46172

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