fourierdlem61 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem61 47316

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 11321	. . 3 ⊢ (𝜑 → 0 ∈ ℝ)
2		fourierdlem61.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		fourierdlem61.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4	2, 3	resubcld 11754	. . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
5	4	rexrd 11369	. . 3 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ^*)
6		fourierdlem61.altb	. . . 4 ⊢ (𝜑 → 𝐴 < 𝐵)
7	3, 2	posdifd 11913	. . . 4 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
8	6, 7	mpbid 235	. . 3 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
9		fourierdlem61.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
10	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
11	3	rexrd 11369	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
12	11	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐴 ∈ ℝ^*)
13	2	rexrd 11369	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
14	13	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐵 ∈ ℝ^*)
15	3	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐴 ∈ ℝ)
16		elioore 13516	. . . . . . . . 9 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) → 𝑠 ∈ ℝ)
17	16	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑠 ∈ ℝ)
18	15, 17	readdcld 11348	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) ∈ ℝ)
19	3	recnd 11347	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ)
20	19	addridd 11520	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 + 0) = 𝐴)
21	20	eqcomd 2769	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = (𝐴 + 0))
22	21	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐴 = (𝐴 + 0))
23		0red 11321	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 0 ∈ ℝ)
24		0xr 11366	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
25	24	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 0 ∈ ℝ^*)
26	5	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐵 − 𝐴) ∈ ℝ^*)
27		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑠 ∈ (0(,)(𝐵 − 𝐴)))
28	25, 26, 27	ioogtlbd 46701	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 0 < 𝑠)
29	23, 17, 15, 28	ltadd2dd 11479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 0) < (𝐴 + 𝑠))
30	22, 29	eqbrtrd 5129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐴 < (𝐴 + 𝑠))
31	4	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐵 − 𝐴) ∈ ℝ)
32	25, 26, 27	iooltubd 46695	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑠 < (𝐵 − 𝐴))
33	17, 31, 15, 32	ltadd2dd 11479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) < (𝐴 + (𝐵 − 𝐴)))
34	2	recnd 11347	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ)
35	19, 34	pncan3d 11682	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 + (𝐵 − 𝐴)) = 𝐵)
36	35	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + (𝐵 − 𝐴)) = 𝐵)
37	33, 36	breqtrd 5133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) < 𝐵)
38	12, 14, 18, 30, 37	eliood 46649	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) ∈ (𝐴(,)𝐵))
39	10, 38	ffvelcdmd 7093	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐹‘(𝐴 + 𝑠)) ∈ ℝ)
40		ioossre 13548	. . . . . . . . 9 ⊢ (𝐴(,)𝐵) ⊆ ℝ
41	40	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
42		ax-resscn 11267	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
43	41, 42	sstrdi 3924	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
44		eqid 2763	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	44, 13, 3, 6	lptioo1cn 46795	. . . . . . 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 46780	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ℝ)
48	47	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑌 ∈ ℝ)
49	39, 48	resubcld 11754	. . . 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 7122	. . 3 ⊢ (𝜑 → 𝑁:(0(,)(𝐵 − 𝐴))⟶ℝ)
52		fourierdlem61.d	. . . 4 ⊢ 𝐷 = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)
53	17, 52	fmptd 7122	. . 3 ⊢ (𝜑 → 𝐷:(0(,)(𝐵 − 𝐴))⟶ℝ)
54	50	oveq2i 7438	. . . . . 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 5891	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝑁) = dom (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))))
57		reelprrecn 11302	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
58	57	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
59	39	recnd 11347	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐹‘(𝐴 + 𝑠)) ∈ ℂ)
60		dvfre 26283	. . . . . . . . . . 11 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
61	9, 41, 60	syl2anc 596	. . . . . . . . . 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 6699	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺:dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ))
65	61, 64	mpbird 260	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:dom (ℝ D 𝐹)⟶ℝ)
66	65	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐺:dom (ℝ D 𝐹)⟶ℝ)
67	63	eqcomd 2769	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)
68	67	dmeqd 5891	. . . . . . . . . . 11 ⊢ (𝜑 → dom (ℝ D 𝐹) = dom 𝐺)
69		fourierdlem61.domg	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐺 = (𝐴(,)𝐵))
70	68, 69	eqtr2d 2799	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,)𝐵) = dom (ℝ D 𝐹))
71	70	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴(,)𝐵) = dom (ℝ D 𝐹))
72	38, 71	eleqtrd 2865	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) ∈ dom (ℝ D 𝐹))
73	66, 72	ffvelcdmd 7093	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐺‘(𝐴 + 𝑠)) ∈ ℝ)
74		1red 11319	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 1 ∈ ℝ)
75	9	ffvelcdmda 7092	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℝ)
76	75	recnd 11347	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ)
77	70	feq2d 6701	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺:(𝐴(,)𝐵)⟶ℝ ↔ 𝐺:dom (ℝ D 𝐹)⟶ℝ))
78	65, 77	mpbird 260	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:(𝐴(,)𝐵)⟶ℝ)
79	78	ffvelcdmda 7092	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑥) ∈ ℝ)
80	19	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐴 ∈ ℂ)
81	19	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝐴 ∈ ℂ)
82		0red 11321	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 ∈ ℝ)
83	58, 19	dvmptc 26290	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝐴)) = (𝑠 ∈ ℝ ↦ 0))
84		ioossre 13548	. . . . . . . . . . . . 13 ⊢ (0(,)(𝐵 − 𝐴)) ⊆ ℝ
85	84	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (0(,)(𝐵 − 𝐴)) ⊆ ℝ)
86		tgioo4 25136	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
87		iooretop 25096	. . . . . . . . . . . . 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 26295	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝐴)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 0))
90	17	recnd 11347	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑠 ∈ ℂ)
91		recn 11300	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℝ → 𝑠 ∈ ℂ)
92	91	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℂ)
93		1red 11319	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
94	58	dvmptid 26289	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑠)) = (𝑠 ∈ ℝ ↦ 1))
95	58, 92, 93, 94, 85, 86, 44, 88	dvmptres 26295	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
96	58, 80, 23, 89, 90, 74, 95	dvmptadd 26292	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (0 + 1)))
97		0p1e1 12473	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
98	97	mpteq2i 5203	. . . . . . . . . 10 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (0 + 1)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)
99	96, 98	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
100	9	feqmptd 6961	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)))
101	100	eqcomd 2769	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) = 𝐹)
102	101	oveq2d 7443	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))) = (ℝ D 𝐹))
103	78	feqmptd 6961	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)))
104	102, 67, 103	3eqtrd 2802	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)))
105		fveq2 6893	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 + 𝑠) → (𝐹‘𝑥) = (𝐹‘(𝐴 + 𝑠)))
106		fveq2 6893	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 + 𝑠) → (𝐺‘𝑥) = (𝐺‘(𝐴 + 𝑠)))
107	58, 58, 38, 74, 76, 79, 99, 104, 105, 106	dvmptco 26304	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠)))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐺‘(𝐴 + 𝑠)) · 1)))
108	73	recnd 11347	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐺‘(𝐴 + 𝑠)) ∈ ℂ)
109	108	mulridd 11336	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝐺‘(𝐴 + 𝑠)) · 1) = (𝐺‘(𝐴 + 𝑠)))
110	109	mpteq2dva 5200	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐺‘(𝐴 + 𝑠)) · 1)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
111	107, 110	eqtrd 2798	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠)))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
112		limccl 26207	. . . . . . . . 9 ⊢ (𝐹 lim_ℂ 𝐴) ⊆ ℂ
113	112, 46	sselid 3910	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℂ)
114	113	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑌 ∈ ℂ)
115	113	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑌 ∈ ℂ)
116	58, 113	dvmptc 26290	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑌)) = (𝑠 ∈ ℝ ↦ 0))
117	58, 115, 82, 116, 85, 86, 44, 88	dvmptres 26295	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑌)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 0))
118	58, 59, 73, 111, 114, 23, 117	dvmptsub 26299	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐺‘(𝐴 + 𝑠)) − 0)))
119	108	subid1d 11668	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝐺‘(𝐴 + 𝑠)) − 0) = (𝐺‘(𝐴 + 𝑠)))
120	119	mpteq2dva 5200	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐺‘(𝐴 + 𝑠)) − 0)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
121	118, 120	eqtrd 2798	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
122	121	dmeqd 5891	. . . 4 ⊢ (𝜑 → dom (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))) = dom (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
123	73	ralrimiva 3157	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ (0(,)(𝐵 − 𝐴))(𝐺‘(𝐴 + 𝑠)) ∈ ℝ)
124		dmmptg 6249	. . . . 5 ⊢ (∀𝑠 ∈ (0(,)(𝐵 − 𝐴))(𝐺‘(𝐴 + 𝑠)) ∈ ℝ → dom (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))) = (0(,)(𝐵 − 𝐴)))
125	123, 124	syl 18	. . . 4 ⊢ (𝜑 → dom (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))) = (0(,)(𝐵 − 𝐴)))
126	56, 122, 125	3eqtrd 2802	. . 3 ⊢ (𝜑 → dom (ℝ D 𝑁) = (0(,)(𝐵 − 𝐴)))
127	52	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐷 = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠))
128	127	oveq2d 7443	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐷) = (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)))
129	128, 95	eqtrd 2798	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐷) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
130	129	dmeqd 5891	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝐷) = dom (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
131	74	ralrimiva 3157	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ (0(,)(𝐵 − 𝐴))1 ∈ ℝ)
132		dmmptg 6249	. . . . 5 ⊢ (∀𝑠 ∈ (0(,)(𝐵 − 𝐴))1 ∈ ℝ → dom (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1) = (0(,)(𝐵 − 𝐴)))
133	131, 132	syl 18	. . . 4 ⊢ (𝜑 → dom (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1) = (0(,)(𝐵 − 𝐴)))
134	130, 133	eqtrd 2798	. . 3 ⊢ (𝜑 → dom (ℝ D 𝐷) = (0(,)(𝐵 − 𝐴)))
135		eqid 2763	. . . . 5 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠)))
136		eqid 2763	. . . . 5 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑌) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑌)
137		eqid 2763	. . . . 5 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))
138	38	adantrr 730	. . . . . 6 ⊢ ((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) ≠ 𝐴)) → (𝐴 + 𝑠) ∈ (𝐴(,)𝐵))
139		eqid 2763	. . . . . . . 8 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝐴) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝐴)
140		eqid 2763	. . . . . . . 8 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)
141		eqid 2763	. . . . . . . 8 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠))
142	85, 42	sstrdi 3924	. . . . . . . . 9 ⊢ (𝜑 → (0(,)(𝐵 − 𝐴)) ⊆ ℂ)
143	1	recnd 11347	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℂ)
144	139, 142, 19, 143	constlimc 46775	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝐴) lim_ℂ 0))
145	142, 140, 143	idlimc 46777	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠) lim_ℂ 0))
146	139, 140, 141, 80, 90, 144, 145	addlimc 46797	. . . . . . 7 ⊢ (𝜑 → (𝐴 + 0) ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠)) lim_ℂ 0))
147	21, 146	eqeltrd 2863	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠)) lim_ℂ 0))
148	100	oveq1d 7442	. . . . . . 7 ⊢ (𝜑 → (𝐹 lim_ℂ 𝐴) = ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) lim_ℂ 𝐴))
149	46, 148	eleqtrd 2865	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) lim_ℂ 𝐴))
150		simplrr 790	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) = 𝐴)) ∧ ¬ (𝐹‘(𝐴 + 𝑠)) = 𝑌) → (𝐴 + 𝑠) = 𝐴)
151	15, 30	gtned 11455	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) ≠ 𝐴)
152	151	neneqd 2963	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ¬ (𝐴 + 𝑠) = 𝐴)
153	152	adantrr 730	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) = 𝐴)) → ¬ (𝐴 + 𝑠) = 𝐴)
154	153	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) = 𝐴)) ∧ ¬ (𝐹‘(𝐴 + 𝑠)) = 𝑌) → ¬ (𝐴 + 𝑠) = 𝐴)
155	150, 154	condan 830	. . . . . 6 ⊢ ((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) = 𝐴)) → (𝐹‘(𝐴 + 𝑠)) = 𝑌)
156	138, 76, 147, 149, 105, 155	limcco 26225	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠))) lim_ℂ 0))
157	136, 142, 113, 143	constlimc 46775	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑌) lim_ℂ 0))
158	135, 136, 137, 59, 114, 156, 157	sublimc 46801	. . . 4 ⊢ (𝜑 → (𝑌 − 𝑌) ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌)) lim_ℂ 0))
159	113	subidd 11667	. . . 4 ⊢ (𝜑 → (𝑌 − 𝑌) = 0)
160	50	eqcomi 2772	. . . . . 6 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌)) = 𝑁
161	160	oveq1i 7437	. . . . 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 2877	. . 3 ⊢ (𝜑 → 0 ∈ (𝑁 lim_ℂ 0))
164	142, 52, 143	idlimc 46777	. . 3 ⊢ (𝜑 → 0 ∈ (𝐷 lim_ℂ 0))
165		lbioo 13517	. . . . 5 ⊢ ¬ 0 ∈ (0(,)(𝐵 − 𝐴))
166	165	a1i 11	. . . 4 ⊢ (𝜑 → ¬ 0 ∈ (0(,)(𝐵 − 𝐴)))
167		mptresid 6048	. . . . . . 7 ⊢ ( I ↾ (0(,)(𝐵 − 𝐴))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)
168	127, 167	eqtr4di 2816	. . . . . 6 ⊢ (𝜑 → 𝐷 = ( I ↾ (0(,)(𝐵 − 𝐴))))
169	168	rneqd 5924	. . . . 5 ⊢ (𝜑 → ran 𝐷 = ran ( I ↾ (0(,)(𝐵 − 𝐴))))
170		rnresi 6078	. . . . 5 ⊢ ran ( I ↾ (0(,)(𝐵 − 𝐴))) = (0(,)(𝐵 − 𝐴))
171	169, 170	eqtr2di 2815	. . . 4 ⊢ (𝜑 → (0(,)(𝐵 − 𝐴)) = ran 𝐷)
172	166, 171	neleqtrd 2885	. . 3 ⊢ (𝜑 → ¬ 0 ∈ ran 𝐷)
173		0ne1 12424	. . . . . 6 ⊢ 0 ≠ 1
174	173	neii 2960	. . . . 5 ⊢ ¬ 0 = 1
175		elsng 4597	. . . . . 6 ⊢ (0 ∈ ℝ → (0 ∈ {1} ↔ 0 = 1))
176	1, 175	syl 18	. . . . 5 ⊢ (𝜑 → (0 ∈ {1} ↔ 0 = 1))
177	174, 176	mtbiri 330	. . . 4 ⊢ (𝜑 → ¬ 0 ∈ {1})
178	129	rneqd 5924	. . . . 5 ⊢ (𝜑 → ran (ℝ D 𝐷) = ran (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
179		eqid 2763	. . . . . 6 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)
180	24	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ^*)
181		ioon0 13512	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ (𝐵 − 𝐴) ∈ ℝ^*) → ((0(,)(𝐵 − 𝐴)) ≠ ∅ ↔ 0 < (𝐵 − 𝐴)))
182	180, 5, 181	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → ((0(,)(𝐵 − 𝐴)) ≠ ∅ ↔ 0 < (𝐵 − 𝐴)))
183	8, 182	mpbird 260	. . . . . 6 ⊢ (𝜑 → (0(,)(𝐵 − 𝐴)) ≠ ∅)
184	179, 183	rnmptc 7220	. . . . 5 ⊢ (𝜑 → ran (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1) = {1})
185	178, 184	eqtr2d 2799	. . . 4 ⊢ (𝜑 → {1} = ran (ℝ D 𝐷))
186	177, 185	neleqtrd 2885	. . 3 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐷))
187	79	recnd 11347	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑥) ∈ ℂ)
188		fourierdlem61.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝐺 lim_ℂ 𝐴))
189	103	oveq1d 7442	. . . . . 6 ⊢ (𝜑 → (𝐺 lim_ℂ 𝐴) = ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)) lim_ℂ 𝐴))
190	188, 189	eleqtrd 2865	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)) lim_ℂ 𝐴))
191		simplrr 790	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) = 𝐴)) ∧ ¬ (𝐺‘(𝐴 + 𝑠)) = 𝐸) → (𝐴 + 𝑠) = 𝐴)
192	153	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) = 𝐴)) ∧ ¬ (𝐺‘(𝐴 + 𝑠)) = 𝐸) → ¬ (𝐴 + 𝑠) = 𝐴)
193	191, 192	condan 830	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) = 𝐴)) → (𝐺‘(𝐴 + 𝑠)) = 𝐸)
194	138, 187, 147, 190, 106, 193	limcco 26225	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))) lim_ℂ 0))
195	108	div1d 12095	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝐺‘(𝐴 + 𝑠)) / 1) = (𝐺‘(𝐴 + 𝑠)))
196	54, 121	eqtrid 2810	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D 𝑁) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
197	196	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (ℝ D 𝑁) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
198	197	fveq1d 6895	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((ℝ D 𝑁)‘𝑠) = ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠)))‘𝑠))
199		fvmpt4 46389	. . . . . . . . . 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 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠)))‘𝑠) = (𝐺‘(𝐴 + 𝑠)))
201	198, 200	eqtr2d 2799	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐺‘(𝐴 + 𝑠)) = ((ℝ D 𝑁)‘𝑠))
202	129	fveq1d 6895	. . . . . . . . . 10 ⊢ (𝜑 → ((ℝ D 𝐷)‘𝑠) = ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)‘𝑠))
203	202	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((ℝ D 𝐷)‘𝑠) = ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)‘𝑠))
204		fvmpt4 46389	. . . . . . . . . 10 ⊢ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ 1 ∈ ℝ) → ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)‘𝑠) = 1)
205	27, 74, 204	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)‘𝑠) = 1)
206	203, 205	eqtr2d 2799	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 1 = ((ℝ D 𝐷)‘𝑠))
207	201, 206	oveq12d 7445	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝐺‘(𝐴 + 𝑠)) / 1) = (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠)))
208	195, 207	eqtr3d 2800	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐺‘(𝐴 + 𝑠)) = (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠)))
209	208	mpteq2dva 5200	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠))))
210	209	oveq1d 7442	. . . 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 2865	. . 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 26346	. 2 ⊢ (𝜑 → 𝐸 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) lim_ℂ 0))
213	50	fvmpt2 7013	. . . . . . 7 ⊢ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ ((𝐹‘(𝐴 + 𝑠)) − 𝑌) ∈ ℝ) → (𝑁‘𝑠) = ((𝐹‘(𝐴 + 𝑠)) − 𝑌))
214	27, 49, 213	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝑁‘𝑠) = ((𝐹‘(𝐴 + 𝑠)) − 𝑌))
215	52	fvmpt2 7013	. . . . . . 7 ⊢ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐷‘𝑠) = 𝑠)
216	27, 27, 215	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐷‘𝑠) = 𝑠)
217	214, 216	oveq12d 7445	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝑁‘𝑠) / (𝐷‘𝑠)) = (((𝐹‘(𝐴 + 𝑠)) − 𝑌) / 𝑠))
218	217	mpteq2dva 5200	. . . 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 2816	. . 3 ⊢ (𝜑 → (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) = 𝐻)
221	220	oveq1d 7442	. 2 ⊢ (𝜑 → ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) lim_ℂ 0) = (𝐻 lim_ℂ 0))
222	212, 221	eleqtrd 2865	1 ⊢ (𝜑 → 𝐸 ∈ (𝐻 lim_ℂ 0))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2146 ≠ wne 2958 ∀wral 3079 ⊆ wss 3899 ∅c0 4280 {csn 4587 {cpr 4589 class class class wbr 5105 ↦ cmpt 5188 I cid 5548 dom cdm 5654 ran crn 5655 ↾ cres 5656 ⟶wf 6542 ‘cfv 6546 (class class class)co 7427 ℂcc 11208 ℝcr 11209 0cc0 11210 1c1 11211 + caddc 11213 · cmul 11215 ℝ^*cxr 11352 < clt 11353 − cmin 11551 / cdiv 11983 (,)cioo 13486 TopOpenctopn 17604 topGenctg 17620 ℂ_fldccnfld 21690 lim_ℂ climc 26194 D cdv 26195

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-pre-sup 11288 ax-addf 11289

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4838 df-uni 4870 df-int 4910 df-iun 4955 df-iin 4956 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-se 5608 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-of 7700 df-om 7885 df-1st 8008 df-2nd 8009 df-supp 8188 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-1o 8487 df-2o 8488 df-er 8728 df-map 8860 df-pm 8861 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9365 df-fi 9413 df-sup 9444 df-inf 9445 df-oi 9514 df-card 10030 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-div 11984 df-nn 12346 df-2 12415 df-3 12416 df-4 12417 df-5 12418 df-6 12419 df-7 12420 df-8 12421 df-9 12422 df-n0 12617 df-z 12704 df-dec 12825 df-uz 12976 df-q 13086 df-rp 13131 df-xneg 13251 df-xadd 13252 df-xmul 13253 df-ioo 13490 df-ioc 13491 df-ico 13492 df-icc 13493 df-fz 13650 df-fzo 13799 df-seq 14155 df-exp 14215 df-hash 14485 df-cj 15278 df-re 15279 df-im 15280 df-sqrt 15414 df-abs 15415 df-struct 17337 df-sets 17354 df-slot 17372 df-ndx 17384 df-base 17400 df-ress 17421 df-plusg 17453 df-mulr 17454 df-starv 17455 df-sca 17456 df-vsca 17457 df-ip 17458 df-tset 17459 df-ple 17460 df-ds 17462 df-unif 17463 df-hom 17464 df-cco 17465 df-rest 17605 df-topn 17606 df-0g 17624 df-gsum 17625 df-topgen 17626 df-pt 17627 df-prds 17630 df-xrs 17686 df-qtop 17691 df-imas 17692 df-xps 17694 df-mre 17768 df-mrc 17769 df-acs 17771 df-mgm 18828 df-sgrp 18920 df-mnd 18936 df-submnd 18991 df-mulg 19290 df-cntz 19543 df-cmn 20008 df-psmet 21682 df-xmet 21683 df-met 21684 df-bl 21685 df-mopn 21686 df-fbas 21687 df-fg 21688 df-cnfld 21691 df-top 23224 df-topon 23241 df-topsp 23263 df-bases 23276 df-cld 23349 df-ntr 23350 df-cls 23351 df-nei 23428 df-lp 23466 df-perf 23467 df-cn 23557 df-cnp 23558 df-haus 23645 df-cmp 23717 df-tx 23893 df-hmeo 24086 df-fil 24177 df-fm 24269 df-flim 24270 df-flf 24271 df-xms 24651 df-ms 24652 df-tms 24653 df-cncf 25211 df-limc 26198 df-dv 26199

This theorem is used by: fourierdlem75 47330

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