dvdivbd - Mathbox for Glauco Siliprandi

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Theorem dvdivbd 42565

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 10660	. . . 4 ⊢ (𝜑 → (𝑈 · 𝑅) ∈ ℝ)
4		dvdivbd.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ)
5		dvdivbd.q	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℝ)
6	4, 5	remulcld 10660	. . . 4 ⊢ (𝜑 → (𝑇 · 𝑄) ∈ ℝ)
7	3, 6	readdcld 10659	. . 3 ⊢ (𝜑 → ((𝑈 · 𝑅) + (𝑇 · 𝑄)) ∈ ℝ)
8		dvdivbd.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
9	8	rpred 12419	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ)
10	9	resqcld 13607	. . 3 ⊢ (𝜑 → (𝐸↑2) ∈ ℝ)
11	8	rpcnd 12421	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ)
12	8	rpgt0d 12422	. . . . 5 ⊢ (𝜑 → 0 < 𝐸)
13	12	gt0ne0d 11193	. . . 4 ⊢ (𝜑 → 𝐸 ≠ 0)
14		2z 12002	. . . . 5 ⊢ 2 ∈ ℤ
15	14	a1i 11	. . . 4 ⊢ (𝜑 → 2 ∈ ℤ)
16	11, 13, 15	expne0d 13512	. . 3 ⊢ (𝜑 → (𝐸↑2) ≠ 0)
17	7, 10, 16	redivcld 11457	. 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 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → 𝐵 = 0)
25	24	abs00bd 14643	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → (abs‘𝐵) = 0)
26		0red 10633	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ)
27	9	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐸 ∈ ℝ)
28	23	abscld 14788	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐵) ∈ ℝ)
29	12	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 < 𝐸)
30		dvdivbd.ele	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐸 ≤ (abs‘𝐵))
31	30	r19.21bi 3173	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐸 ≤ (abs‘𝐵))
32	26, 27, 28, 29, 31	ltletrd 10789	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 < (abs‘𝐵))
33	32	gt0ne0d 11193	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐵) ≠ 0)
34	33	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → (abs‘𝐵) ≠ 0)
35	34	neneqd 2992	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝐵 = 0) → ¬ (abs‘𝐵) = 0)
36	25, 35	pm2.65da 816	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ 𝐵 = 0)
37	36	neqned 2994	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ≠ 0)
38		eldifsn 4680	. . . . . . . . 9 ⊢ (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
39	23, 37, 38	sylanbrc 586	. . . . . . . 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 24577	. . . . . . 7 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵))) = (𝑥 ∈ 𝑋 ↦ (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))))
43	18, 42	syl5eq 2845	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))))
44	21, 23	mulcld 10650	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐶 · 𝐵) ∈ ℂ)
45	40, 20	mulcld 10650	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐷 · 𝐴) ∈ ℂ)
46	44, 45	subcld 10986	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐶 · 𝐵) − (𝐷 · 𝐴)) ∈ ℂ)
47	23	sqcld 13504	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ∈ ℂ)
48		sqne0 13485	. . . . . . . . 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 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵↑2) ≠ 0)
51	46, 47, 50	divcld 11405	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2)) ∈ ℂ)
52	43, 51	fvmpt2d 6758	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) = (((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2)))
53	52	fveq2d 6649	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐹‘𝑥)) = (abs‘(((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))))
54	46, 47, 50	absdivd 14807	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))) = ((abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) / (abs‘(𝐵↑2))))
55	46	abscld 14788	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) ∈ ℝ)
56	7	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑈 · 𝑅) + (𝑇 · 𝑄)) ∈ ℝ)
57	8	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐸 ∈ ℝ⁺)
58	14	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 2 ∈ ℤ)
59	57, 58	rpexpcld 13604	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐸↑2) ∈ ℝ⁺)
60	47	abscld 14788	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐵↑2)) ∈ ℝ)
61	46	absge0d 14796	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))))
62	44	abscld 14788	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐶 · 𝐵)) ∈ ℝ)
63	45	abscld 14788	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐷 · 𝐴)) ∈ ℝ)
64	62, 63	readdcld 10659	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘(𝐶 · 𝐵)) + (abs‘(𝐷 · 𝐴))) ∈ ℝ)
65	44, 45	abs2dif2d 14810	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) ≤ ((abs‘(𝐶 · 𝐵)) + (abs‘(𝐷 · 𝐴))))
66	3	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑈 · 𝑅) ∈ ℝ)
67	6	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑇 · 𝑄) ∈ ℝ)
68	21, 23	absmuld 14806	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐶 · 𝐵)) = ((abs‘𝐶) · (abs‘𝐵)))
69	21	abscld 14788	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐶) ∈ ℝ)
70	1	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑈 ∈ ℝ)
71	2	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ ℝ)
72	21	absge0d 14796	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝐶))
73	23	absge0d 14796	. . . . . . . . . 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 11571	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐶) · (abs‘𝐵)) ≤ (𝑈 · 𝑅))
77	68, 76	eqbrtrd 5052	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐶 · 𝐵)) ≤ (𝑈 · 𝑅))
78	40, 20	absmuld 14806	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐷 · 𝐴)) = ((abs‘𝐷) · (abs‘𝐴)))
79	40	abscld 14788	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐷) ∈ ℝ)
80	4	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑇 ∈ ℝ)
81	20	abscld 14788	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐴) ∈ ℝ)
82	5	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑄 ∈ ℝ)
83	40	absge0d 14796	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝐷))
84	20	absge0d 14796	. . . . . . . . . 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 11571	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘𝐷) · (abs‘𝐴)) ≤ (𝑇 · 𝑄))
88	78, 87	eqbrtrd 5052	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐷 · 𝐴)) ≤ (𝑇 · 𝑄))
89	62, 63, 66, 67, 77, 88	le2addd 11248	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘(𝐶 · 𝐵)) + (abs‘(𝐷 · 𝐴))) ≤ ((𝑈 · 𝑅) + (𝑇 · 𝑄)))
90	55, 64, 56, 65, 89	letrd 10786	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) ≤ ((𝑈 · 𝑅) + (𝑇 · 𝑄)))
91		2nn0 11902	. . . . . . . . 9 ⊢ 2 ∈ ℕ₀
92	91	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 2 ∈ ℕ₀)
93	26, 27, 29	ltled 10777	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ≤ 𝐸)
94		leexp1a 13535	. . . . . . . 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 1375	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐸↑2) ≤ ((abs‘𝐵)↑2))
96	23, 92	absexpd 14804	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐵↑2)) = ((abs‘𝐵)↑2))
97	95, 96	breqtrrd 5058	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐸↑2) ≤ (abs‘(𝐵↑2)))
98	55, 56, 59, 60, 61, 90, 97	lediv12ad 12478	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((abs‘((𝐶 · 𝐵) − (𝐷 · 𝐴))) / (abs‘(𝐵↑2))) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
99	54, 98	eqbrtrd 5052	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(((𝐶 · 𝐵) − (𝐷 · 𝐴)) / (𝐵↑2))) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
100	53, 99	eqbrtrd 5052	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘(𝐹‘𝑥)) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
101	100	ralrimiva 3149	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (abs‘(𝐹‘𝑥)) ≤ (((𝑈 · 𝑅) + (𝑇 · 𝑄)) / (𝐸↑2)))
102		brralrspcev 5090	. 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 587	1 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘(𝐹‘𝑥)) ≤ 𝑏)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 ∖ cdif 3878 {csn 4525 {cpr 4527 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 2c2 11680 ℕ₀cn0 11885 ℤcz 11969 ℝ⁺crp 12377 ↑cexp 13425 abscabs 14585 D cdv 24466

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-t1 21919 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470

This theorem is referenced by: fourierdlem68 42816

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