fourierdlem61 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem61 43677

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 10977	. . 3 ⊢ (𝜑 → 0 ∈ ℝ)
2		fourierdlem61.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		fourierdlem61.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4	2, 3	resubcld 11401	. . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
5	4	rexrd 11024	. . 3 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ^*)
6		fourierdlem61.altb	. . . 4 ⊢ (𝜑 → 𝐴 < 𝐵)
7	3, 2	posdifd 11560	. . . 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 11024	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
12	11	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐴 ∈ ℝ^*)
13	2	rexrd 11024	. . . . . . . 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 13106	. . . . . . . . 9 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) → 𝑠 ∈ ℝ)
17	16	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑠 ∈ ℝ)
18	15, 17	readdcld 11003	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) ∈ ℝ)
19	3	recnd 11002	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ)
20	19	addid1d 11173	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 + 0) = 𝐴)
21	20	eqcomd 2746	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = (𝐴 + 0))
22	21	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝐴 = (𝐴 + 0))
23		0red 10977	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 0 ∈ ℝ)
24		0xr 11021	. . . . . . . . . . 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 43057	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 0 < 𝑠)
29	23, 17, 15, 28	ltadd2dd 11132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 0) < (𝐴 + 𝑠))
30	22, 29	eqbrtrd 5101	. . . . . . 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 43051	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑠 < (𝐵 − 𝐴))
33	17, 31, 15, 32	ltadd2dd 11132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) < (𝐴 + (𝐵 − 𝐴)))
34	2	recnd 11002	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ)
35	19, 34	pncan3d 11333	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 + (𝐵 − 𝐴)) = 𝐵)
36	35	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + (𝐵 − 𝐴)) = 𝐵)
37	33, 36	breqtrd 5105	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) < 𝐵)
38	12, 14, 18, 30, 37	eliood 43005	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) ∈ (𝐴(,)𝐵))
39	10, 38	ffvelrnd 6957	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐹‘(𝐴 + 𝑠)) ∈ ℝ)
40		ioossre 13137	. . . . . . . . 9 ⊢ (𝐴(,)𝐵) ⊆ ℝ
41	40	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
42		ax-resscn 10927	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
43	41, 42	sstrdi 3938	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
44		eqid 2740	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	44, 13, 3, 6	lptioo1cn 43156	. . . . . . 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 43139	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ℝ)
48	47	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑌 ∈ ℝ)
49	39, 48	resubcld 11401	. . . 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 6983	. . 3 ⊢ (𝜑 → 𝑁:(0(,)(𝐵 − 𝐴))⟶ℝ)
52		fourierdlem61.d	. . . 4 ⊢ 𝐷 = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)
53	17, 52	fmptd 6983	. . 3 ⊢ (𝜑 → 𝐷:(0(,)(𝐵 − 𝐴))⟶ℝ)
54	50	oveq2i 7280	. . . . . 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 5812	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝑁) = dom (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))))
57		reelprrecn 10962	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
58	57	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
59	39	recnd 11002	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐹‘(𝐴 + 𝑠)) ∈ ℂ)
60		dvfre 25111	. . . . . . . . . . 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 6582	. . . . . . . . . 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 2746	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)
68	67	dmeqd 5812	. . . . . . . . . . 11 ⊢ (𝜑 → dom (ℝ D 𝐹) = dom 𝐺)
69		fourierdlem61.domg	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐺 = (𝐴(,)𝐵))
70	68, 69	eqtr2d 2781	. . . . . . . . . 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 2843	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) ∈ dom (ℝ D 𝐹))
73	66, 72	ffvelrnd 6957	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐺‘(𝐴 + 𝑠)) ∈ ℝ)
74		1red 10975	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 1 ∈ ℝ)
75	9	ffvelrnda 6956	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℝ)
76	75	recnd 11002	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ)
77	70	feq2d 6583	. . . . . . . . . . 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	ffvelrnda 6956	. . . . . . . . 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 10977	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 ∈ ℝ)
83	58, 19	dvmptc 25118	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝐴)) = (𝑠 ∈ ℝ ↦ 0))
84		ioossre 13137	. . . . . . . . . . . . 13 ⊢ (0(,)(𝐵 − 𝐴)) ⊆ ℝ
85	84	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (0(,)(𝐵 − 𝐴)) ⊆ ℝ)
86		tgioo4 43080	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
87		iooretop 23925	. . . . . . . . . . . . 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 25123	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝐴)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 0))
90	17	recnd 11002	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 𝑠 ∈ ℂ)
91		recn 10960	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℝ → 𝑠 ∈ ℂ)
92	91	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℂ)
93		1red 10975	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
94	58	dvmptid 25117	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑠)) = (𝑠 ∈ ℝ ↦ 1))
95	58, 92, 93, 94, 85, 86, 44, 88	dvmptres 25123	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
96	58, 80, 23, 89, 90, 74, 95	dvmptadd 25120	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (0 + 1)))
97		0p1e1 12093	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
98	97	mpteq2i 5184	. . . . . . . . . 10 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (0 + 1)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)
99	96, 98	eqtrdi 2796	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
100	9	feqmptd 6832	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)))
101	100	eqcomd 2746	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) = 𝐹)
102	101	oveq2d 7285	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))) = (ℝ D 𝐹))
103	78	feqmptd 6832	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)))
104	102, 67, 103	3eqtrd 2784	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)))
105		fveq2 6769	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 + 𝑠) → (𝐹‘𝑥) = (𝐹‘(𝐴 + 𝑠)))
106		fveq2 6769	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 + 𝑠) → (𝐺‘𝑥) = (𝐺‘(𝐴 + 𝑠)))
107	58, 58, 38, 74, 76, 79, 99, 104, 105, 106	dvmptco 25132	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠)))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐺‘(𝐴 + 𝑠)) · 1)))
108	73	recnd 11002	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐺‘(𝐴 + 𝑠)) ∈ ℂ)
109	108	mulid1d 10991	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝐺‘(𝐴 + 𝑠)) · 1) = (𝐺‘(𝐴 + 𝑠)))
110	109	mpteq2dva 5179	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐺‘(𝐴 + 𝑠)) · 1)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
111	107, 110	eqtrd 2780	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠)))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
112		limccl 25035	. . . . . . . . 9 ⊢ (𝐹 lim_ℂ 𝐴) ⊆ ℂ
113	112, 46	sselid 3924	. . . . . . . 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 25118	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑌)) = (𝑠 ∈ ℝ ↦ 0))
117	58, 115, 82, 116, 85, 86, 44, 88	dvmptres 25123	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑌)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 0))
118	58, 59, 73, 111, 114, 23, 117	dvmptsub 25127	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐺‘(𝐴 + 𝑠)) − 0)))
119	108	subid1d 11319	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝐺‘(𝐴 + 𝑠)) − 0) = (𝐺‘(𝐴 + 𝑠)))
120	119	mpteq2dva 5179	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐺‘(𝐴 + 𝑠)) − 0)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
121	118, 120	eqtrd 2780	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
122	121	dmeqd 5812	. . . 4 ⊢ (𝜑 → dom (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))) = dom (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))))
123	73	ralrimiva 3110	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ (0(,)(𝐵 − 𝐴))(𝐺‘(𝐴 + 𝑠)) ∈ ℝ)
124		dmmptg 6143	. . . . 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 2784	. . 3 ⊢ (𝜑 → dom (ℝ D 𝑁) = (0(,)(𝐵 − 𝐴)))
127	52	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐷 = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠))
128	127	oveq2d 7285	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐷) = (ℝ D (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)))
129	128, 95	eqtrd 2780	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐷) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
130	129	dmeqd 5812	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝐷) = dom (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
131	74	ralrimiva 3110	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ (0(,)(𝐵 − 𝐴))1 ∈ ℝ)
132		dmmptg 6143	. . . . 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 2780	. . 3 ⊢ (𝜑 → dom (ℝ D 𝐷) = (0(,)(𝐵 − 𝐴)))
135		eqid 2740	. . . . 5 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠)))
136		eqid 2740	. . . . 5 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑌) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑌)
137		eqid 2740	. . . . 5 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))
138	38	adantrr 714	. . . . . 6 ⊢ ((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) ≠ 𝐴)) → (𝐴 + 𝑠) ∈ (𝐴(,)𝐵))
139		eqid 2740	. . . . . . . 8 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝐴) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝐴)
140		eqid 2740	. . . . . . . 8 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)
141		eqid 2740	. . . . . . . 8 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠)) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠))
142	85, 42	sstrdi 3938	. . . . . . . . 9 ⊢ (𝜑 → (0(,)(𝐵 − 𝐴)) ⊆ ℂ)
143	1	recnd 11002	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℂ)
144	139, 142, 19, 143	constlimc 43134	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝐴) lim_ℂ 0))
145	142, 140, 143	idlimc 43136	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠) lim_ℂ 0))
146	139, 140, 141, 80, 90, 144, 145	addlimc 43158	. . . . . . 7 ⊢ (𝜑 → (𝐴 + 0) ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠)) lim_ℂ 0))
147	21, 146	eqeltrd 2841	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐴 + 𝑠)) lim_ℂ 0))
148	100	oveq1d 7284	. . . . . . 7 ⊢ (𝜑 → (𝐹 lim_ℂ 𝐴) = ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) lim_ℂ 𝐴))
149	46, 148	eleqtrd 2843	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) lim_ℂ 𝐴))
150		simplrr 775	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) = 𝐴)) ∧ ¬ (𝐹‘(𝐴 + 𝑠)) = 𝑌) → (𝐴 + 𝑠) = 𝐴)
151	15, 30	gtned 11108	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐴 + 𝑠) ≠ 𝐴)
152	151	neneqd 2950	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ¬ (𝐴 + 𝑠) = 𝐴)
153	152	adantrr 714	. . . . . . . 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 815	. . . . . 6 ⊢ ((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) = 𝐴)) → (𝐹‘(𝐴 + 𝑠)) = 𝑌)
156	138, 76, 147, 149, 105, 155	limcco 25053	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐹‘(𝐴 + 𝑠))) lim_ℂ 0))
157	136, 142, 113, 143	constlimc 43134	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑌) lim_ℂ 0))
158	135, 136, 137, 59, 114, 156, 157	sublimc 43162	. . . 4 ⊢ (𝜑 → (𝑌 − 𝑌) ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌)) lim_ℂ 0))
159	113	subidd 11318	. . . 4 ⊢ (𝜑 → (𝑌 − 𝑌) = 0)
160	50	eqcomi 2749	. . . . . 6 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌)) = 𝑁
161	160	oveq1i 7279	. . . . 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 2855	. . 3 ⊢ (𝜑 → 0 ∈ (𝑁 lim_ℂ 0))
164	142, 52, 143	idlimc 43136	. . 3 ⊢ (𝜑 → 0 ∈ (𝐷 lim_ℂ 0))
165		lbioo 13107	. . . . 5 ⊢ ¬ 0 ∈ (0(,)(𝐵 − 𝐴))
166	165	a1i 11	. . . 4 ⊢ (𝜑 → ¬ 0 ∈ (0(,)(𝐵 − 𝐴)))
167		mptresid 5956	. . . . . . 7 ⊢ ( I ↾ (0(,)(𝐵 − 𝐴))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠)
168	127, 167	eqtr4di 2798	. . . . . 6 ⊢ (𝜑 → 𝐷 = ( I ↾ (0(,)(𝐵 − 𝐴))))
169	168	rneqd 5845	. . . . 5 ⊢ (𝜑 → ran 𝐷 = ran ( I ↾ (0(,)(𝐵 − 𝐴))))
170		rnresi 5981	. . . . 5 ⊢ ran ( I ↾ (0(,)(𝐵 − 𝐴))) = (0(,)(𝐵 − 𝐴))
171	169, 170	eqtr2di 2797	. . . 4 ⊢ (𝜑 → (0(,)(𝐵 − 𝐴)) = ran 𝐷)
172	166, 171	neleqtrd 2862	. . 3 ⊢ (𝜑 → ¬ 0 ∈ ran 𝐷)
173		0ne1 12042	. . . . . 6 ⊢ 0 ≠ 1
174	173	neii 2947	. . . . 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 327	. . . 4 ⊢ (𝜑 → ¬ 0 ∈ {1})
178	129	rneqd 5845	. . . . 5 ⊢ (𝜑 → ran (ℝ D 𝐷) = ran (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1))
179		eqid 2740	. . . . . 6 ⊢ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1)
180	24	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ^*)
181		ioon0 13102	. . . . . . . 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 7077	. . . . 5 ⊢ (𝜑 → ran (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 1) = {1})
185	178, 184	eqtr2d 2781	. . . 4 ⊢ (𝜑 → {1} = ran (ℝ D 𝐷))
186	177, 185	neleqtrd 2862	. . 3 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐷))
187	79	recnd 11002	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑥) ∈ ℂ)
188		fourierdlem61.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝐺 lim_ℂ 𝐴))
189	103	oveq1d 7284	. . . . . 6 ⊢ (𝜑 → (𝐺 lim_ℂ 𝐴) = ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)) lim_ℂ 𝐴))
190	188, 189	eleqtrd 2843	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)) lim_ℂ 𝐴))
191		simplrr 775	. . . . . 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 815	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ∧ (𝐴 + 𝑠) = 𝐴)) → (𝐺‘(𝐴 + 𝑠)) = 𝐸)
194	138, 187, 147, 190, 106, 193	limcco 25053	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))) lim_ℂ 0))
195	108	div1d 11741	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝐺‘(𝐴 + 𝑠)) / 1) = (𝐺‘(𝐴 + 𝑠)))
196	54, 121	eqtrid 2792	. . . . . . . . . . 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 6771	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((ℝ D 𝑁)‘𝑠) = ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠)))‘𝑠))
199		fvmpt4 42750	. . . . . . . . . 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 2781	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐺‘(𝐴 + 𝑠)) = ((ℝ D 𝑁)‘𝑠))
202	129	fveq1d 6771	. . . . . . . . . 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 42750	. . . . . . . . . 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 2781	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → 1 = ((ℝ D 𝐷)‘𝑠))
207	201, 206	oveq12d 7287	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝐺‘(𝐴 + 𝑠)) / 1) = (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠)))
208	195, 207	eqtr3d 2782	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → (𝐺‘(𝐴 + 𝑠)) = (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠)))
209	208	mpteq2dva 5179	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (𝐺‘(𝐴 + 𝑠))) = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠))))
210	209	oveq1d 7284	. . . 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 2843	. . 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 25174	. 2 ⊢ (𝜑 → 𝐸 ∈ ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) lim_ℂ 0))
213	50	fvmpt2 6881	. . . . . . 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 6881	. . . . . . 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 7287	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0(,)(𝐵 − 𝐴))) → ((𝑁‘𝑠) / (𝐷‘𝑠)) = (((𝐹‘(𝐴 + 𝑠)) − 𝑌) / 𝑠))
218	217	mpteq2dva 5179	. . . 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 2798	. . 3 ⊢ (𝜑 → (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) = 𝐻)
221	220	oveq1d 7284	. 2 ⊢ (𝜑 → ((𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) lim_ℂ 0) = (𝐻 lim_ℂ 0))
222	212, 221	eleqtrd 2843	1 ⊢ (𝜑 → 𝐸 ∈ (𝐻 lim_ℂ 0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ∀wral 3066 ⊆ wss 3892 ∅c0 4262 {csn 4567 {cpr 4569 class class class wbr 5079 ↦ cmpt 5162 I cid 5488 dom cdm 5589 ran crn 5590 ↾ cres 5591 ⟶wf 6427 ‘cfv 6431 (class class class)co 7269 ℂcc 10868 ℝcr 10869 0cc0 10870 1c1 10871 + caddc 10873 · cmul 10875 ℝ^*cxr 11007 < clt 11008 − cmin 11203 / cdiv 11630 (,)cioo 13076 TopOpenctopn 17128 topGenctg 17144 ℂ_fldccnfld 20593 lim_ℂ climc 25022 D cdv 25023

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948 ax-addf 10949 ax-mulf 10950

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7705 df-1st 7822 df-2nd 7823 df-supp 7967 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-2o 8287 df-er 8479 df-map 8598 df-pm 8599 df-ixp 8667 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-fsupp 9105 df-fi 9146 df-sup 9177 df-inf 9178 df-oi 9245 df-card 9696 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12435 df-uz 12580 df-q 12686 df-rp 12728 df-xneg 12845 df-xadd 12846 df-xmul 12847 df-ioo 13080 df-ioc 13081 df-ico 13082 df-icc 13083 df-fz 13237 df-fzo 13380 df-seq 13718 df-exp 13779 df-hash 14041 df-cj 14806 df-re 14807 df-im 14808 df-sqrt 14942 df-abs 14943 df-struct 16844 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-ress 16938 df-plusg 16971 df-mulr 16972 df-starv 16973 df-sca 16974 df-vsca 16975 df-ip 16976 df-tset 16977 df-ple 16978 df-ds 16980 df-unif 16981 df-hom 16982 df-cco 16983 df-rest 17129 df-topn 17130 df-0g 17148 df-gsum 17149 df-topgen 17150 df-pt 17151 df-prds 17154 df-xrs 17209 df-qtop 17214 df-imas 17215 df-xps 17217 df-mre 17291 df-mrc 17292 df-acs 17294 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-submnd 18427 df-mulg 18697 df-cntz 18919 df-cmn 19384 df-psmet 20585 df-xmet 20586 df-met 20587 df-bl 20588 df-mopn 20589 df-fbas 20590 df-fg 20591 df-cnfld 20594 df-top 22039 df-topon 22056 df-topsp 22078 df-bases 22092 df-cld 22166 df-ntr 22167 df-cls 22168 df-nei 22245 df-lp 22283 df-perf 22284 df-cn 22374 df-cnp 22375 df-haus 22462 df-cmp 22534 df-tx 22709 df-hmeo 22902 df-fil 22993 df-fm 23085 df-flim 23086 df-flf 23087 df-xms 23469 df-ms 23470 df-tms 23471 df-cncf 24037 df-limc 25026 df-dv 25027

This theorem is referenced by: fourierdlem75 43691

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