fourierdlem61 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem61 44873

Description: Given a differentiable function 𝐹, with finite limit of the derivative at 𝐴 the derived function 𝐻 has a limit at 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem61.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem61.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem61.altb	⊢ (𝜑 → 𝐴 < 𝐵)
fourierdlem61.f	⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
fourierdlem61.y	⊢ (𝜑 → 𝑌 ∈ (𝐹 lim_ℂ 𝐴))
fourierdlem61.g	⊢ 𝐺 = (ℝ D 𝐹)
fourierdlem61.domg	⊢ (𝜑 → dom 𝐺 = (𝐴(,)𝐵))
fourierdlem61.e	⊢ (𝜑 → 𝐸 ∈ (𝐺 lim_ℂ 𝐴))
fourierdlem61.h	⊢ 𝐻 = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (((𝐹‘(𝐴 + 𝑠)) − 𝑌) / 𝑠))
fourierdlem61.n	⊢ 𝑁 = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))
fourierdlem61.d	⊢ 𝐷 = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)

**Assertion**
Ref	Expression
fourierdlem61	⊢ (𝜑 → 𝐸 ∈ (𝐻 lim_ℂ 0))

Distinct variable groups: 𝐴,𝑠 𝐵,𝑠 𝐷,𝑠 𝐸,𝑠 𝐹,𝑠 𝐺,𝑠 𝑁,𝑠 𝑌,𝑠 𝜑,𝑠 Allowed substitution hint: 𝐻(𝑠)

Proof of Theorem fourierdlem61 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0red 11216	. . 3 ⊢ (𝜑 → 0 ∈ ℝ)
2		fourierdlem61.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		fourierdlem61.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4	2, 3	resubcld 11641	. . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
5	4	rexrd 11263	. . 3 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ^*)
6		fourierdlem61.altb	. . . 4 ⊢ (𝜑 → 𝐴 < 𝐵)
7	3, 2	posdifd 11800	. . . 4 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
8	6, 7	mpbid 231	. . 3 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
9		fourierdlem61.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
10	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
11	3	rexrd 11263	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
12	11	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐴 ∈ ℝ^*)
13	2	rexrd 11263	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
14	13	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐵 ∈ ℝ^*)
15	3	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐴 ∈ ℝ)
16		elioore 13353	. . . . . . . . 9 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) → 𝑠 ∈ ℝ)
17	16	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑠 ∈ ℝ)
18	15, 17	readdcld 11242	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) ∈ ℝ)
19	3	recnd 11241	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ)
20	19	addridd 11413	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 + 0) = 𝐴)
21	20	eqcomd 2738	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = (𝐴 + 0))
22	21	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐴 = (𝐴 + 0))
23		0red 11216	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 0 ∈ ℝ)
24		0xr 11260	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
25	24	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 0 ∈ ℝ^*)
26	5	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐵 − 𝐴) ∈ ℝ^*)
27		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑠 ∈ (0(,)(𝐵 − 𝐴)))
28	25, 26, 27	ioogtlbd 44253	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 0 < 𝑠)
29	23, 17, 15, 28	ltadd2dd 11372	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 0) < (𝐴 + 𝑠))
30	22, 29	eqbrtrd 5170	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐴 < (𝐴 + 𝑠))
31	4	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐵 − 𝐴) ∈ ℝ)
32	25, 26, 27	iooltubd 44247	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑠 < (𝐵 − 𝐴))
33	17, 31, 15, 32	ltadd2dd 11372	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) < (𝐴 + (𝐵 − 𝐴)))
34	2	recnd 11241	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ)
35	19, 34	pncan3d 11573	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 + (𝐵 − 𝐴)) = 𝐵)
36	35	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + (𝐵 − 𝐴)) = 𝐵)
37	33, 36	breqtrd 5174	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) < 𝐵)
38	12, 14, 18, 30, 37	eliood 44201	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) ∈ (𝐴(,)𝐵))
39	10, 38	ffvelcdmd 7087	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐹‘(𝐴 + 𝑠)) ∈ ℝ)
40		ioossre 13384	. . . . . . . . 9 ⊢ (𝐴(,)𝐵) ⊆ ℝ
41	40	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
42		ax-resscn 11166	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
43	41, 42	sstrdi 3994	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
44		eqid 2732	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	44, 13, 3, 6	lptioo1cn 44352	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝐴(,)𝐵)))
46		fourierdlem61.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝐹 lim_ℂ 𝐴))
47	9, 43, 45, 46	limcrecl 44335	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ℝ)
48	47	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑌 ∈ ℝ)
49	39, 48	resubcld 11641	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝐹‘(𝐴 + 𝑠)) − 𝑌) ∈ ℝ)
50		fourierdlem61.n	. . . 4 ⊢ 𝑁 = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))
51	49, 50	fmptd 7113	. . 3 ⊢ (𝜑 → 𝑁:(0(,)(𝐵 − 𝐴))⟶ℝ)
52		fourierdlem61.d	. . . 4 ⊢ 𝐷 = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)
53	17, 52	fmptd 7113	. . 3 ⊢ (𝜑 → 𝐷:(0(,)(𝐵 − 𝐴))⟶ℝ)
54	50	oveq2i 7419	. . . . . 6 ⊢ (ℝ D 𝑁) = (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌)))
55	54	a1i 11	. . . . 5 ⊢ (𝜑 → (ℝ D 𝑁) = (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))))
56	55	dmeqd 5905	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝑁) = dom (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))))
57		reelprrecn 11201	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
58	57	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
59	39	recnd 11241	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐹‘(𝐴 + 𝑠)) ∈ ℂ)
60		dvfre 25467	. . . . . . . . . . 11 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
61	9, 41, 60	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
62		fourierdlem61.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (ℝ D 𝐹)
63	62	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 = (ℝ D 𝐹))
64	63	feq1d 6702	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺:dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ))
65	61, 64	mpbird 256	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:dom (ℝ D 𝐹)⟶ℝ)
66	65	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐺:dom (ℝ D 𝐹)⟶ℝ)
67	63	eqcomd 2738	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)
68	67	dmeqd 5905	. . . . . . . . . . 11 ⊢ (𝜑 → dom (ℝ D 𝐹) = dom 𝐺)
69		fourierdlem61.domg	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐺 = (𝐴(,)𝐵))
70	68, 69	eqtr2d 2773	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,)𝐵) = dom (ℝ D 𝐹))
71	70	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴(,)𝐵) = dom (ℝ D 𝐹))
72	38, 71	eleqtrd 2835	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) ∈ dom (ℝ D 𝐹))
73	66, 72	ffvelcdmd 7087	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐺‘(𝐴 + 𝑠)) ∈ ℝ)
74		1red 11214	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 1 ∈ ℝ)
75	9	ffvelcdmda 7086	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℝ)
76	75	recnd 11241	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ)
77	70	feq2d 6703	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺:(𝐴(,)𝐵)⟶ℝ ↔ 𝐺:dom (ℝ D 𝐹)⟶ℝ))
78	65, 77	mpbird 256	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:(𝐴(,)𝐵)⟶ℝ)
79	78	ffvelcdmda 7086	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑥) ∈ ℝ)
80	19	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐴 ∈ ℂ)
81	19	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝐴 ∈ ℂ)
82		0red 11216	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 ∈ ℝ)
83	58, 19	dvmptc 25474	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝐴)) = (𝑠 ∈ ℝ ↦ 0))
84		ioossre 13384	. . . . . . . . . . . . 13 ⊢ (0(,)(𝐵 − 𝐴)) ⊆ ℝ
85	84	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (0(,)(𝐵 − 𝐴)) ⊆ ℝ)
86		tgioo4 44276	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
87		iooretop 24281	. . . . . . . . . . . . 13 ⊢ (0(,)(𝐵 − 𝐴)) ∈ (topGen‘ran (,))
88	87	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (0(,)(𝐵 − 𝐴)) ∈ (topGen‘ran (,)))
89	58, 81, 82, 83, 85, 86, 44, 88	dvmptres 25479	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝐴)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 0))
90	17	recnd 11241	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑠 ∈ ℂ)
91		recn 11199	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℝ → 𝑠 ∈ ℂ)
92	91	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℂ)
93		1red 11214	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
94	58	dvmptid 25473	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑠)) = (𝑠 ∈ ℝ ↦ 1))
95	58, 92, 93, 94, 85, 86, 44, 88	dvmptres 25479	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
96	58, 80, 23, 89, 90, 74, 95	dvmptadd 25476	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (0 + 1)))
97		0p1e1 12333	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
98	97	mpteq2i 5253	. . . . . . . . . 10 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (0 + 1)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)
99	96, 98	eqtrdi 2788	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
100	9	feqmptd 6960	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)))
101	100	eqcomd 2738	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) = 𝐹)
102	101	oveq2d 7424	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))) = (ℝ D 𝐹))
103	78	feqmptd 6960	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)))
104	102, 67, 103	3eqtrd 2776	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)))
105		fveq2 6891	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 + 𝑠) → (𝐹‘𝑥) = (𝐹‘(𝐴 + 𝑠)))
106		fveq2 6891	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 + 𝑠) → (𝐺‘𝑥) = (𝐺‘(𝐴 + 𝑠)))
107	58, 58, 38, 74, 76, 79, 99, 104, 105, 106	dvmptco 25488	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠)))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐺‘(𝐴 + 𝑠)) · 1)))
108	73	recnd 11241	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐺‘(𝐴 + 𝑠)) ∈ ℂ)
109	108	mulridd 11230	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝐺‘(𝐴 + 𝑠)) · 1) = (𝐺‘(𝐴 + 𝑠)))
110	109	mpteq2dva 5248	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐺‘(𝐴 + 𝑠)) · 1)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
111	107, 110	eqtrd 2772	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠)))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
112		limccl 25391	. . . . . . . . 9 ⊢ (𝐹 lim_ℂ 𝐴) ⊆ ℂ
113	112, 46	sselid 3980	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℂ)
114	113	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑌 ∈ ℂ)
115	113	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑌 ∈ ℂ)
116	58, 113	dvmptc 25474	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑌)) = (𝑠 ∈ ℝ ↦ 0))
117	58, 115, 82, 116, 85, 86, 44, 88	dvmptres 25479	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑌)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 0))
118	58, 59, 73, 111, 114, 23, 117	dvmptsub 25483	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐺‘(𝐴 + 𝑠)) − 0)))
119	108	subid1d 11559	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝐺‘(𝐴 + 𝑠)) − 0) = (𝐺‘(𝐴 + 𝑠)))
120	119	mpteq2dva 5248	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐺‘(𝐴 + 𝑠)) − 0)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
121	118, 120	eqtrd 2772	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
122	121	dmeqd 5905	. . . 4 ⊢ (𝜑 → dom (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))) = dom (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
123	73	ralrimiva 3146	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ (0(,)(𝐵 − 𝐴))(𝐺‘(𝐴 + 𝑠)) ∈ ℝ)
124		dmmptg 6241	. . . . 5 ⊢ (∀𝑠 ∈ (0(,)(𝐵 − 𝐴))(𝐺‘(𝐴 + 𝑠)) ∈ ℝ → dom (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))) = (0(,)(𝐵 − 𝐴)))
125	123, 124	syl 17	. . . 4 ⊢ (𝜑 → dom (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))) = (0(,)(𝐵 − 𝐴)))
126	56, 122, 125	3eqtrd 2776	. . 3 ⊢ (𝜑 → dom (ℝ D 𝑁) = (0(,)(𝐵 − 𝐴)))
127	52	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐷 = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠))
128	127	oveq2d 7424	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐷) = (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)))
129	128, 95	eqtrd 2772	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐷) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
130	129	dmeqd 5905	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝐷) = dom (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
131	74	ralrimiva 3146	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ (0(,)(𝐵 − 𝐴))1 ∈ ℝ)
132		dmmptg 6241	. . . . 5 ⊢ (∀𝑠 ∈ (0(,)(𝐵 − 𝐴))1 ∈ ℝ → dom (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1) = (0(,)(𝐵 − 𝐴)))
133	131, 132	syl 17	. . . 4 ⊢ (𝜑 → dom (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1) = (0(,)(𝐵 − 𝐴)))
134	130, 133	eqtrd 2772	. . 3 ⊢ (𝜑 → dom (ℝ D 𝐷) = (0(,)(𝐵 − 𝐴)))
135		eqid 2732	. . . . 5 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠)))
136		eqid 2732	. . . . 5 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑌) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑌)
137		eqid 2732	. . . . 5 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))
138	38	adantrr 715	. . . . . 6 ⊢ ((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) ≠ 𝐴)) → (𝐴 + 𝑠) ∈ (𝐴(,)𝐵))
139		eqid 2732	. . . . . . . 8 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝐴) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝐴)
140		eqid 2732	. . . . . . . 8 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)
141		eqid 2732	. . . . . . . 8 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠))
142	85, 42	sstrdi 3994	. . . . . . . . 9 ⊢ (𝜑 → (0(,)(𝐵 − 𝐴)) ⊆ ℂ)
143	1	recnd 11241	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℂ)
144	139, 142, 19, 143	constlimc 44330	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝐴) lim_ℂ 0))
145	142, 140, 143	idlimc 44332	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠) lim_ℂ 0))
146	139, 140, 141, 80, 90, 144, 145	addlimc 44354	. . . . . . 7 ⊢ (𝜑 → (𝐴 + 0) ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠)) lim_ℂ 0))
147	21, 146	eqeltrd 2833	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠)) lim_ℂ 0))
148	100	oveq1d 7423	. . . . . . 7 ⊢ (𝜑 → (𝐹 lim_ℂ 𝐴) = ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) lim_ℂ 𝐴))
149	46, 148	eleqtrd 2835	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) lim_ℂ 𝐴))
150		simplrr 776	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) = 𝐴)) ∧ ¬ (𝐹‘(𝐴 + 𝑠)) = 𝑌) → (𝐴 + 𝑠) = 𝐴)
151	15, 30	gtned 11348	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) ≠ 𝐴)
152	151	neneqd 2945	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ¬ (𝐴 + 𝑠) = 𝐴)
153	152	adantrr 715	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) = 𝐴)) → ¬ (𝐴 + 𝑠) = 𝐴)
154	153	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) = 𝐴)) ∧ ¬ (𝐹‘(𝐴 + 𝑠)) = 𝑌) → ¬ (𝐴 + 𝑠) = 𝐴)
155	150, 154	condan 816	. . . . . 6 ⊢ ((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) = 𝐴)) → (𝐹‘(𝐴 + 𝑠)) = 𝑌)
156	138, 76, 147, 149, 105, 155	limcco 25409	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠))) lim_ℂ 0))
157	136, 142, 113, 143	constlimc 44330	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑌) lim_ℂ 0))
158	135, 136, 137, 59, 114, 156, 157	sublimc 44358	. . . 4 ⊢ (𝜑 → (𝑌 − 𝑌) ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌)) lim_ℂ 0))
159	113	subidd 11558	. . . 4 ⊢ (𝜑 → (𝑌 − 𝑌) = 0)
160	50	eqcomi 2741	. . . . . 6 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌)) = 𝑁
161	160	oveq1i 7418	. . . . 5 ⊢ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌)) lim_ℂ 0) = (𝑁 lim_ℂ 0)
162	161	a1i 11	. . . 4 ⊢ (𝜑 → ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌)) lim_ℂ 0) = (𝑁 lim_ℂ 0))
163	158, 159, 162	3eltr3d 2847	. . 3 ⊢ (𝜑 → 0 ∈ (𝑁 lim_ℂ 0))
164	142, 52, 143	idlimc 44332	. . 3 ⊢ (𝜑 → 0 ∈ (𝐷 lim_ℂ 0))
165		lbioo 13354	. . . . 5 ⊢ ¬ 0 ∈ (0(,)(𝐵 − 𝐴))
166	165	a1i 11	. . . 4 ⊢ (𝜑 → ¬ 0 ∈ (0(,)(𝐵 − 𝐴)))
167		mptresid 6050	. . . . . . 7 ⊢ ( I ↾ (0(,)(𝐵 − 𝐴))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)
168	127, 167	eqtr4di 2790	. . . . . 6 ⊢ (𝜑 → 𝐷 = ( I ↾ (0(,)(𝐵 − 𝐴))))
169	168	rneqd 5937	. . . . 5 ⊢ (𝜑 → ran 𝐷 = ran ( I ↾ (0(,)(𝐵 − 𝐴))))
170		rnresi 6074	. . . . 5 ⊢ ran ( I ↾ (0(,)(𝐵 − 𝐴))) = (0(,)(𝐵 − 𝐴))
171	169, 170	eqtr2di 2789	. . . 4 ⊢ (𝜑 → (0(,)(𝐵 − 𝐴)) = ran 𝐷)
172	166, 171	neleqtrd 2855	. . 3 ⊢ (𝜑 → ¬ 0 ∈ ran 𝐷)
173		0ne1 12282	. . . . . 6 ⊢ 0 ≠ 1
174	173	neii 2942	. . . . 5 ⊢ ¬ 0 = 1
175		elsng 4642	. . . . . 6 ⊢ (0 ∈ ℝ → (0 ∈ {1} ↔ 0 = 1))
176	1, 175	syl 17	. . . . 5 ⊢ (𝜑 → (0 ∈ {1} ↔ 0 = 1))
177	174, 176	mtbiri 326	. . . 4 ⊢ (𝜑 → ¬ 0 ∈ {1})
178	129	rneqd 5937	. . . . 5 ⊢ (𝜑 → ran (ℝ D 𝐷) = ran (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
179		eqid 2732	. . . . . 6 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)
180	24	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ^*)
181		ioon0 13349	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ (𝐵 − 𝐴) ∈ ℝ^*) → ((0(,)(𝐵 − 𝐴)) ≠ ∅ ↔ 0 < (𝐵 − 𝐴)))
182	180, 5, 181	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((0(,)(𝐵 − 𝐴)) ≠ ∅ ↔ 0 < (𝐵 − 𝐴)))
183	8, 182	mpbird 256	. . . . . 6 ⊢ (𝜑 → (0(,)(𝐵 − 𝐴)) ≠ ∅)
184	179, 183	rnmptc 7207	. . . . 5 ⊢ (𝜑 → ran (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1) = {1})
185	178, 184	eqtr2d 2773	. . . 4 ⊢ (𝜑 → {1} = ran (ℝ D 𝐷))
186	177, 185	neleqtrd 2855	. . 3 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐷))
187	79	recnd 11241	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑥) ∈ ℂ)
188		fourierdlem61.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝐺 lim_ℂ 𝐴))
189	103	oveq1d 7423	. . . . . 6 ⊢ (𝜑 → (𝐺 lim_ℂ 𝐴) = ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)) lim_ℂ 𝐴))
190	188, 189	eleqtrd 2835	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)) lim_ℂ 𝐴))
191		simplrr 776	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) = 𝐴)) ∧ ¬ (𝐺‘(𝐴 + 𝑠)) = 𝐸) → (𝐴 + 𝑠) = 𝐴)
192	153	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) = 𝐴)) ∧ ¬ (𝐺‘(𝐴 + 𝑠)) = 𝐸) → ¬ (𝐴 + 𝑠) = 𝐴)
193	191, 192	condan 816	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) = 𝐴)) → (𝐺‘(𝐴 + 𝑠)) = 𝐸)
194	138, 187, 147, 190, 106, 193	limcco 25409	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))) lim_ℂ 0))
195	108	div1d 11981	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝐺‘(𝐴 + 𝑠)) / 1) = (𝐺‘(𝐴 + 𝑠)))
196	54, 121	eqtrid 2784	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D 𝑁) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
197	196	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (ℝ D 𝑁) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
198	197	fveq1d 6893	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((ℝ D 𝑁)‘𝑠) = ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠)))‘𝑠))
199		fvmpt4 43931	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐺‘(𝐴 + 𝑠)) ∈ ℝ) → ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠)))‘𝑠) = (𝐺‘(𝐴 + 𝑠)))
200	27, 73, 199	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠)))‘𝑠) = (𝐺‘(𝐴 + 𝑠)))
201	198, 200	eqtr2d 2773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐺‘(𝐴 + 𝑠)) = ((ℝ D 𝑁)‘𝑠))
202	129	fveq1d 6893	. . . . . . . . . 10 ⊢ (𝜑 → ((ℝ D 𝐷)‘𝑠) = ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)‘𝑠))
203	202	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((ℝ D 𝐷)‘𝑠) = ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)‘𝑠))
204		fvmpt4 43931	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ 1 ∈ ℝ) → ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)‘𝑠) = 1)
205	27, 74, 204	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)‘𝑠) = 1)
206	203, 205	eqtr2d 2773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 1 = ((ℝ D 𝐷)‘𝑠))
207	201, 206	oveq12d 7426	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝐺‘(𝐴 + 𝑠)) / 1) = (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠)))
208	195, 207	eqtr3d 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐺‘(𝐴 + 𝑠)) = (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠)))
209	208	mpteq2dva 5248	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠))))
210	209	oveq1d 7423	. . . 4 ⊢ (𝜑 → ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))) lim_ℂ 0) = ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠))) lim_ℂ 0))
211	194, 210	eleqtrd 2835	. . 3 ⊢ (𝜑 → 𝐸 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠))) lim_ℂ 0))
212	1, 5, 8, 51, 53, 126, 134, 163, 164, 172, 186, 211	lhop1 25530	. 2 ⊢ (𝜑 → 𝐸 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) lim_ℂ 0))
213	50	fvmpt2 7009	. . . . . . 7 ⊢ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ ((𝐹‘(𝐴 + 𝑠)) − 𝑌) ∈ ℝ) → (𝑁‘𝑠) = ((𝐹‘(𝐴 + 𝑠)) − 𝑌))
214	27, 49, 213	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝑁‘𝑠) = ((𝐹‘(𝐴 + 𝑠)) − 𝑌))
215	52	fvmpt2 7009	. . . . . . 7 ⊢ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐷‘𝑠) = 𝑠)
216	27, 27, 215	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐷‘𝑠) = 𝑠)
217	214, 216	oveq12d 7426	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝑁‘𝑠) / (𝐷‘𝑠)) = (((𝐹‘(𝐴 + 𝑠)) − 𝑌) / 𝑠))
218	217	mpteq2dva 5248	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (((𝐹‘(𝐴 + 𝑠)) − 𝑌) / 𝑠)))
219		fourierdlem61.h	. . . 4 ⊢ 𝐻 = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (((𝐹‘(𝐴 + 𝑠)) − 𝑌) / 𝑠))
220	218, 219	eqtr4di 2790	. . 3 ⊢ (𝜑 → (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) = 𝐻)
221	220	oveq1d 7423	. 2 ⊢ (𝜑 → ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) lim_ℂ 0) = (𝐻 lim_ℂ 0))
222	212, 221	eleqtrd 2835	1 ⊢ (𝜑 → 𝐸 ∈ (𝐻 lim_ℂ 0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ⊆ wss 3948 ∅c0 4322 {csn 4628 {cpr 4630 class class class wbr 5148 ↦ cmpt 5231 I cid 5573 dom cdm 5676 ran crn 5677 ↾ cres 5678 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ℂcc 11107 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 · cmul 11114 ℝ^*cxr 11246 < clt 11247 − cmin 11443 / cdiv 11870 (,)cioo 13323 TopOpenctopn 17366 topGenctg 17382 ℂ_fldccnfld 20943 lim_ℂ climc 25378 D cdv 25379

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ioo 13327 df-ioc 13328 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-seq 13966 df-exp 14027 df-hash 14290 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-submnd 18671 df-mulg 18950 df-cntz 19180 df-cmn 19649 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-fbas 20940 df-fg 20941 df-cnfld 20944 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 df-nei 22601 df-lp 22639 df-perf 22640 df-cn 22730 df-cnp 22731 df-haus 22818 df-cmp 22890 df-tx 23065 df-hmeo 23258 df-fil 23349 df-fm 23441 df-flim 23442 df-flf 23443 df-xms 23825 df-ms 23826 df-tms 23827 df-cncf 24393 df-limc 25382 df-dv 25383

This theorem is referenced by: fourierdlem75 44887

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