fourierdlem61 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem61 42501

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 10644	. . 3 ⊢ (𝜑 → 0 ∈ ℝ)
2		fourierdlem61.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		fourierdlem61.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4	2, 3	resubcld 11068	. . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
5	4	rexrd 10691	. . 3 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ^*)
6		fourierdlem61.altb	. . . 4 ⊢ (𝜑 → 𝐴 < 𝐵)
7	3, 2	posdifd 11227	. . . 4 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
8	6, 7	mpbid 234	. . 3 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
9		fourierdlem61.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
10	9	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
11	3	rexrd 10691	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
12	11	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐴 ∈ ℝ^*)
13	2	rexrd 10691	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
14	13	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐵 ∈ ℝ^*)
15	3	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐴 ∈ ℝ)
16		elioore 12769	. . . . . . . . 9 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) → 𝑠 ∈ ℝ)
17	16	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑠 ∈ ℝ)
18	15, 17	readdcld 10670	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) ∈ ℝ)
19	3	recnd 10669	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ)
20	19	addid1d 10840	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 + 0) = 𝐴)
21	20	eqcomd 2827	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = (𝐴 + 0))
22	21	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐴 = (𝐴 + 0))
23		0red 10644	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 0 ∈ ℝ)
24		0xr 10688	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
25	24	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 0 ∈ ℝ^*)
26	5	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐵 − 𝐴) ∈ ℝ^*)
27		simpr 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑠 ∈ (0(,)(𝐵 − 𝐴)))
28	25, 26, 27	ioogtlbd 41875	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 0 < 𝑠)
29	23, 17, 15, 28	ltadd2dd 10799	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 0) < (𝐴 + 𝑠))
30	22, 29	eqbrtrd 5088	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐴 < (𝐴 + 𝑠))
31	4	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐵 − 𝐴) ∈ ℝ)
32	25, 26, 27	iooltubd 41869	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑠 < (𝐵 − 𝐴))
33	17, 31, 15, 32	ltadd2dd 10799	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) < (𝐴 + (𝐵 − 𝐴)))
34	2	recnd 10669	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ)
35	19, 34	pncan3d 11000	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 + (𝐵 − 𝐴)) = 𝐵)
36	35	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + (𝐵 − 𝐴)) = 𝐵)
37	33, 36	breqtrd 5092	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) < 𝐵)
38	12, 14, 18, 30, 37	eliood 41822	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) ∈ (𝐴(,)𝐵))
39	10, 38	ffvelrnd 6852	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐹‘(𝐴 + 𝑠)) ∈ ℝ)
40		ioossre 12799	. . . . . . . . 9 ⊢ (𝐴(,)𝐵) ⊆ ℝ
41	40	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
42		ax-resscn 10594	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
43	41, 42	sstrdi 3979	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
44		eqid 2821	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	44, 13, 3, 6	lptioo1cn 41976	. . . . . . 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 41959	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ℝ)
48	47	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑌 ∈ ℝ)
49	39, 48	resubcld 11068	. . . 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 6878	. . 3 ⊢ (𝜑 → 𝑁:(0(,)(𝐵 − 𝐴))⟶ℝ)
52		fourierdlem61.d	. . . 4 ⊢ 𝐷 = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)
53	17, 52	fmptd 6878	. . 3 ⊢ (𝜑 → 𝐷:(0(,)(𝐵 − 𝐴))⟶ℝ)
54	50	oveq2i 7167	. . . . . 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 5774	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝑁) = dom (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))))
57		reelprrecn 10629	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
58	57	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
59	39	recnd 10669	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐹‘(𝐴 + 𝑠)) ∈ ℂ)
60		dvfre 24548	. . . . . . . . . . 11 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
61	9, 41, 60	syl2anc 586	. . . . . . . . . 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 6499	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺:dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ))
65	61, 64	mpbird 259	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:dom (ℝ D 𝐹)⟶ℝ)
66	65	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐺:dom (ℝ D 𝐹)⟶ℝ)
67	63	eqcomd 2827	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)
68	67	dmeqd 5774	. . . . . . . . . . 11 ⊢ (𝜑 → dom (ℝ D 𝐹) = dom 𝐺)
69		fourierdlem61.domg	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐺 = (𝐴(,)𝐵))
70	68, 69	eqtr2d 2857	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,)𝐵) = dom (ℝ D 𝐹))
71	70	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴(,)𝐵) = dom (ℝ D 𝐹))
72	38, 71	eleqtrd 2915	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) ∈ dom (ℝ D 𝐹))
73	66, 72	ffvelrnd 6852	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐺‘(𝐴 + 𝑠)) ∈ ℝ)
74		1red 10642	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 1 ∈ ℝ)
75	9	ffvelrnda 6851	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℝ)
76	75	recnd 10669	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ)
77	70	feq2d 6500	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺:(𝐴(,)𝐵)⟶ℝ ↔ 𝐺:dom (ℝ D 𝐹)⟶ℝ))
78	65, 77	mpbird 259	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:(𝐴(,)𝐵)⟶ℝ)
79	78	ffvelrnda 6851	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑥) ∈ ℝ)
80	19	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐴 ∈ ℂ)
81	19	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝐴 ∈ ℂ)
82		0red 10644	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 ∈ ℝ)
83	58, 19	dvmptc 24555	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝐴)) = (𝑠 ∈ ℝ ↦ 0))
84		ioossre 12799	. . . . . . . . . . . . 13 ⊢ (0(,)(𝐵 − 𝐴)) ⊆ ℝ
85	84	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (0(,)(𝐵 − 𝐴)) ⊆ ℝ)
86		tgioo4 41898	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
87		iooretop 23374	. . . . . . . . . . . . 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 24560	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝐴)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 0))
90	17	recnd 10669	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑠 ∈ ℂ)
91		recn 10627	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℝ → 𝑠 ∈ ℂ)
92	91	adantl 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℂ)
93		1red 10642	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
94	58	dvmptid 24554	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑠)) = (𝑠 ∈ ℝ ↦ 1))
95	58, 92, 93, 94, 85, 86, 44, 88	dvmptres 24560	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
96	58, 80, 23, 89, 90, 74, 95	dvmptadd 24557	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (0 + 1)))
97		0p1e1 11760	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
98	97	mpteq2i 5158	. . . . . . . . . 10 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (0 + 1)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)
99	96, 98	syl6eq 2872	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
100	9	feqmptd 6733	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)))
101	100	eqcomd 2827	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) = 𝐹)
102	101	oveq2d 7172	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))) = (ℝ D 𝐹))
103	78	feqmptd 6733	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)))
104	102, 67, 103	3eqtrd 2860	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)))
105		fveq2 6670	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 + 𝑠) → (𝐹‘𝑥) = (𝐹‘(𝐴 + 𝑠)))
106		fveq2 6670	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 + 𝑠) → (𝐺‘𝑥) = (𝐺‘(𝐴 + 𝑠)))
107	58, 58, 38, 74, 76, 79, 99, 104, 105, 106	dvmptco 24569	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠)))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐺‘(𝐴 + 𝑠)) · 1)))
108	73	recnd 10669	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐺‘(𝐴 + 𝑠)) ∈ ℂ)
109	108	mulid1d 10658	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝐺‘(𝐴 + 𝑠)) · 1) = (𝐺‘(𝐴 + 𝑠)))
110	109	mpteq2dva 5161	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐺‘(𝐴 + 𝑠)) · 1)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
111	107, 110	eqtrd 2856	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠)))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
112		limccl 24473	. . . . . . . . 9 ⊢ (𝐹 lim_ℂ 𝐴) ⊆ ℂ
113	112, 46	sseldi 3965	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℂ)
114	113	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑌 ∈ ℂ)
115	113	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑌 ∈ ℂ)
116	58, 113	dvmptc 24555	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑌)) = (𝑠 ∈ ℝ ↦ 0))
117	58, 115, 82, 116, 85, 86, 44, 88	dvmptres 24560	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑌)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 0))
118	58, 59, 73, 111, 114, 23, 117	dvmptsub 24564	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐺‘(𝐴 + 𝑠)) − 0)))
119	108	subid1d 10986	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝐺‘(𝐴 + 𝑠)) − 0) = (𝐺‘(𝐴 + 𝑠)))
120	119	mpteq2dva 5161	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐺‘(𝐴 + 𝑠)) − 0)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
121	118, 120	eqtrd 2856	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
122	121	dmeqd 5774	. . . 4 ⊢ (𝜑 → dom (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))) = dom (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
123	73	ralrimiva 3182	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ (0(,)(𝐵 − 𝐴))(𝐺‘(𝐴 + 𝑠)) ∈ ℝ)
124		dmmptg 6096	. . . . 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 2860	. . 3 ⊢ (𝜑 → dom (ℝ D 𝑁) = (0(,)(𝐵 − 𝐴)))
127	52	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐷 = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠))
128	127	oveq2d 7172	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐷) = (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)))
129	128, 95	eqtrd 2856	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐷) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
130	129	dmeqd 5774	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝐷) = dom (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
131	74	ralrimiva 3182	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ (0(,)(𝐵 − 𝐴))1 ∈ ℝ)
132		dmmptg 6096	. . . . 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 2856	. . 3 ⊢ (𝜑 → dom (ℝ D 𝐷) = (0(,)(𝐵 − 𝐴)))
135		eqid 2821	. . . . 5 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠)))
136		eqid 2821	. . . . 5 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑌) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑌)
137		eqid 2821	. . . . 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 2821	. . . . . . . 8 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝐴) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝐴)
140		eqid 2821	. . . . . . . 8 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)
141		eqid 2821	. . . . . . . 8 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠))
142	85, 42	sstrdi 3979	. . . . . . . . 9 ⊢ (𝜑 → (0(,)(𝐵 − 𝐴)) ⊆ ℂ)
143	1	recnd 10669	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℂ)
144	139, 142, 19, 143	constlimc 41954	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝐴) lim_ℂ 0))
145	142, 140, 143	idlimc 41956	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠) lim_ℂ 0))
146	139, 140, 141, 80, 90, 144, 145	addlimc 41978	. . . . . . 7 ⊢ (𝜑 → (𝐴 + 0) ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠)) lim_ℂ 0))
147	21, 146	eqeltrd 2913	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠)) lim_ℂ 0))
148	100	oveq1d 7171	. . . . . . 7 ⊢ (𝜑 → (𝐹 lim_ℂ 𝐴) = ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) lim_ℂ 𝐴))
149	46, 148	eleqtrd 2915	. . . . . 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 10775	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) ≠ 𝐴)
152	151	neneqd 3021	. . . . . . . . 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 483	. . . . . . 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 24491	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠))) lim_ℂ 0))
157	136, 142, 113, 143	constlimc 41954	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑌) lim_ℂ 0))
158	135, 136, 137, 59, 114, 156, 157	sublimc 41982	. . . 4 ⊢ (𝜑 → (𝑌 − 𝑌) ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌)) lim_ℂ 0))
159	113	subidd 10985	. . . 4 ⊢ (𝜑 → (𝑌 − 𝑌) = 0)
160	50	eqcomi 2830	. . . . . 6 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌)) = 𝑁
161	160	oveq1i 7166	. . . . 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 2927	. . 3 ⊢ (𝜑 → 0 ∈ (𝑁 lim_ℂ 0))
164	142, 52, 143	idlimc 41956	. . 3 ⊢ (𝜑 → 0 ∈ (𝐷 lim_ℂ 0))
165		lbioo 12770	. . . . 5 ⊢ ¬ 0 ∈ (0(,)(𝐵 − 𝐴))
166	165	a1i 11	. . . 4 ⊢ (𝜑 → ¬ 0 ∈ (0(,)(𝐵 − 𝐴)))
167		mptresid 5918	. . . . . . 7 ⊢ ( I ↾ (0(,)(𝐵 − 𝐴))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)
168	127, 167	syl6eqr 2874	. . . . . 6 ⊢ (𝜑 → 𝐷 = ( I ↾ (0(,)(𝐵 − 𝐴))))
169	168	rneqd 5808	. . . . 5 ⊢ (𝜑 → ran 𝐷 = ran ( I ↾ (0(,)(𝐵 − 𝐴))))
170		rnresi 5943	. . . . 5 ⊢ ran ( I ↾ (0(,)(𝐵 − 𝐴))) = (0(,)(𝐵 − 𝐴))
171	169, 170	syl6req 2873	. . . 4 ⊢ (𝜑 → (0(,)(𝐵 − 𝐴)) = ran 𝐷)
172	166, 171	neleqtrd 2934	. . 3 ⊢ (𝜑 → ¬ 0 ∈ ran 𝐷)
173		0ne1 11709	. . . . . 6 ⊢ 0 ≠ 1
174	173	neii 3018	. . . . 5 ⊢ ¬ 0 = 1
175		elsng 4581	. . . . . 6 ⊢ (0 ∈ ℝ → (0 ∈ {1} ↔ 0 = 1))
176	1, 175	syl 17	. . . . 5 ⊢ (𝜑 → (0 ∈ {1} ↔ 0 = 1))
177	174, 176	mtbiri 329	. . . 4 ⊢ (𝜑 → ¬ 0 ∈ {1})
178	129	rneqd 5808	. . . . 5 ⊢ (𝜑 → ran (ℝ D 𝐷) = ran (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
179		eqid 2821	. . . . . 6 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)
180	24	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ^*)
181		ioon0 12765	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ (𝐵 − 𝐴) ∈ ℝ^*) → ((0(,)(𝐵 − 𝐴)) ≠ ∅ ↔ 0 < (𝐵 − 𝐴)))
182	180, 5, 181	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → ((0(,)(𝐵 − 𝐴)) ≠ ∅ ↔ 0 < (𝐵 − 𝐴)))
183	8, 182	mpbird 259	. . . . . 6 ⊢ (𝜑 → (0(,)(𝐵 − 𝐴)) ≠ ∅)
184	179, 183	rnmptc 6969	. . . . 5 ⊢ (𝜑 → ran (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1) = {1})
185	178, 184	eqtr2d 2857	. . . 4 ⊢ (𝜑 → {1} = ran (ℝ D 𝐷))
186	177, 185	neleqtrd 2934	. . 3 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐷))
187	79	recnd 10669	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑥) ∈ ℂ)
188		fourierdlem61.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝐺 lim_ℂ 𝐴))
189	103	oveq1d 7171	. . . . . 6 ⊢ (𝜑 → (𝐺 lim_ℂ 𝐴) = ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)) lim_ℂ 𝐴))
190	188, 189	eleqtrd 2915	. . . . 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 483	. . . . . 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 24491	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))) lim_ℂ 0))
195	108	div1d 11408	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝐺‘(𝐴 + 𝑠)) / 1) = (𝐺‘(𝐴 + 𝑠)))
196	54, 121	syl5eq 2868	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D 𝑁) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
197	196	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (ℝ D 𝑁) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
198	197	fveq1d 6672	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((ℝ D 𝑁)‘𝑠) = ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠)))‘𝑠))
199		fvmpt4 41557	. . . . . . . . . 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 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠)))‘𝑠) = (𝐺‘(𝐴 + 𝑠)))
201	198, 200	eqtr2d 2857	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐺‘(𝐴 + 𝑠)) = ((ℝ D 𝑁)‘𝑠))
202	129	fveq1d 6672	. . . . . . . . . 10 ⊢ (𝜑 → ((ℝ D 𝐷)‘𝑠) = ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)‘𝑠))
203	202	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((ℝ D 𝐷)‘𝑠) = ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)‘𝑠))
204		fvmpt4 41557	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ 1 ∈ ℝ) → ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)‘𝑠) = 1)
205	27, 74, 204	syl2anc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)‘𝑠) = 1)
206	203, 205	eqtr2d 2857	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 1 = ((ℝ D 𝐷)‘𝑠))
207	201, 206	oveq12d 7174	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝐺‘(𝐴 + 𝑠)) / 1) = (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠)))
208	195, 207	eqtr3d 2858	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐺‘(𝐴 + 𝑠)) = (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠)))
209	208	mpteq2dva 5161	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠))))
210	209	oveq1d 7171	. . . 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 2915	. . 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 24611	. 2 ⊢ (𝜑 → 𝐸 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) lim_ℂ 0))
213	50	fvmpt2 6779	. . . . . . 7 ⊢ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ ((𝐹‘(𝐴 + 𝑠)) − 𝑌) ∈ ℝ) → (𝑁‘𝑠) = ((𝐹‘(𝐴 + 𝑠)) − 𝑌))
214	27, 49, 213	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝑁‘𝑠) = ((𝐹‘(𝐴 + 𝑠)) − 𝑌))
215	52	fvmpt2 6779	. . . . . . 7 ⊢ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐷‘𝑠) = 𝑠)
216	27, 27, 215	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐷‘𝑠) = 𝑠)
217	214, 216	oveq12d 7174	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝑁‘𝑠) / (𝐷‘𝑠)) = (((𝐹‘(𝐴 + 𝑠)) − 𝑌) / 𝑠))
218	217	mpteq2dva 5161	. . . 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	syl6eqr 2874	. . 3 ⊢ (𝜑 → (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) = 𝐻)
221	220	oveq1d 7171	. 2 ⊢ (𝜑 → ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) lim_ℂ 0) = (𝐻 lim_ℂ 0))
222	212, 221	eleqtrd 2915	1 ⊢ (𝜑 → 𝐸 ∈ (𝐻 lim_ℂ 0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ⊆ wss 3936 ∅c0 4291 {csn 4567 {cpr 4569 class class class wbr 5066 ↦ cmpt 5146 I cid 5459 dom cdm 5555 ran crn 5556 ↾ cres 5557 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 ℝ^*cxr 10674 < clt 10675 − cmin 10870 / cdiv 11297 (,)cioo 12739 TopOpenctopn 16695 topGenctg 16711 ℂ_fldccnfld 20545 lim_ℂ climc 24460 D cdv 24461

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465

This theorem is referenced by: fourierdlem75 42515

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