fourierdlem60 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem60 40383

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
fourierdlem60.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem60.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem60.altb	⊢ (𝜑 → 𝐴 < 𝐵)
fourierdlem60.f	⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
fourierdlem60.y	⊢ (𝜑 → 𝑌 ∈ (𝐹 lim_ℂ 𝐵))
fourierdlem60.g	⊢ 𝐺 = (ℝ D 𝐹)
fourierdlem60.domg	⊢ (𝜑 → dom 𝐺 = (𝐴(,)𝐵))
fourierdlem60.e	⊢ (𝜑 → 𝐸 ∈ (𝐺 lim_ℂ 𝐵))
fourierdlem60.h	⊢ 𝐻 = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (((𝐹‘(𝐵 + 𝑠)) − 𝑌) / 𝑠))
fourierdlem60.n	⊢ 𝑁 = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌))
fourierdlem60.d	⊢ 𝐷 = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠)

**Assertion**
Ref	Expression
fourierdlem60	⊢ (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 fourierdlem60 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem60.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		fourierdlem60.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3	1, 2	resubcld 10458	. . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ)
4	3	rexrd 10089	. . 3 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ^*)
5		0red 10041	. . 3 ⊢ (𝜑 → 0 ∈ ℝ)
6		fourierdlem60.altb	. . . 4 ⊢ (𝜑 → 𝐴 < 𝐵)
7	1, 2	sublt0d 10653	. . . 4 ⊢ (𝜑 → ((𝐴 − 𝐵) < 0 ↔ 𝐴 < 𝐵))
8	6, 7	mpbird 247	. . 3 ⊢ (𝜑 → (𝐴 − 𝐵) < 0)
9		fourierdlem60.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
10	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
11	1	rexrd 10089	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
12	11	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝐴 ∈ ℝ^*)
13	2	rexrd 10089	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
14	13	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝐵 ∈ ℝ^*)
15	2	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝐵 ∈ ℝ)
16		elioore 12205	. . . . . . . . 9 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) → 𝑠 ∈ ℝ)
17	16	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝑠 ∈ ℝ)
18	15, 17	readdcld 10069	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐵 + 𝑠) ∈ ℝ)
19	2	recnd 10068	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ)
20	1	recnd 10068	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ)
21	19, 20	pncan3d 10395	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 + (𝐴 − 𝐵)) = 𝐴)
22	21	eqcomd 2628	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = (𝐵 + (𝐴 − 𝐵)))
23	22	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝐴 = (𝐵 + (𝐴 − 𝐵)))
24	3	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐴 − 𝐵) ∈ ℝ)
25	4	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐴 − 𝐵) ∈ ℝ^*)
26		0xr 10086	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
27	26	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 0 ∈ ℝ^*)
28		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝑠 ∈ ((𝐴 − 𝐵)(,)0))
29		ioogtlb 39717	. . . . . . . . . 10 ⊢ (((𝐴 − 𝐵) ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐴 − 𝐵) < 𝑠)
30	25, 27, 28, 29	syl3anc 1326	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐴 − 𝐵) < 𝑠)
31	24, 17, 15, 30	ltadd2dd 10196	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐵 + (𝐴 − 𝐵)) < (𝐵 + 𝑠))
32	23, 31	eqbrtrd 4675	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝐴 < (𝐵 + 𝑠))
33		0red 10041	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 0 ∈ ℝ)
34		iooltub 39735	. . . . . . . . . 10 ⊢ (((𝐴 − 𝐵) ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝑠 < 0)
35	25, 27, 28, 34	syl3anc 1326	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝑠 < 0)
36	17, 33, 15, 35	ltadd2dd 10196	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐵 + 𝑠) < (𝐵 + 0))
37	19	addid1d 10236	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 + 0) = 𝐵)
38	37	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐵 + 0) = 𝐵)
39	36, 38	breqtrd 4679	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐵 + 𝑠) < 𝐵)
40	12, 14, 18, 32, 39	eliood 39720	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐵 + 𝑠) ∈ (𝐴(,)𝐵))
41	10, 40	ffvelrnd 6360	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐹‘(𝐵 + 𝑠)) ∈ ℝ)
42		ioossre 12235	. . . . . . . . 9 ⊢ (𝐴(,)𝐵) ⊆ ℝ
43	42	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
44		ax-resscn 9993	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
45	43, 44	syl6ss 3615	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
46		eqid 2622	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
47	46, 11, 2, 6	lptioo2cn 39877	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝐴(,)𝐵)))
48		fourierdlem60.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝐹 lim_ℂ 𝐵))
49	9, 45, 47, 48	limcrecl 39861	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ℝ)
50	49	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝑌 ∈ ℝ)
51	41, 50	resubcld 10458	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((𝐹‘(𝐵 + 𝑠)) − 𝑌) ∈ ℝ)
52		fourierdlem60.n	. . . 4 ⊢ 𝑁 = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌))
53	51, 52	fmptd 6385	. . 3 ⊢ (𝜑 → 𝑁:((𝐴 − 𝐵)(,)0)⟶ℝ)
54		fourierdlem60.d	. . . 4 ⊢ 𝐷 = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠)
55	17, 54	fmptd 6385	. . 3 ⊢ (𝜑 → 𝐷:((𝐴 − 𝐵)(,)0)⟶ℝ)
56	52	oveq2i 6661	. . . . . 6 ⊢ (ℝ D 𝑁) = (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌)))
57	56	a1i 11	. . . . 5 ⊢ (𝜑 → (ℝ D 𝑁) = (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌))))
58	57	dmeqd 5326	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝑁) = dom (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌))))
59		reelprrecn 10028	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
60	59	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
61	41	recnd 10068	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐹‘(𝐵 + 𝑠)) ∈ ℂ)
62		dvfre 23714	. . . . . . . . . . 11 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
63	9, 43, 62	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
64		fourierdlem60.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (ℝ D 𝐹)
65	64	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 = (ℝ D 𝐹))
66	65	feq1d 6030	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺:dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ))
67	63, 66	mpbird 247	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:dom (ℝ D 𝐹)⟶ℝ)
68	67	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝐺:dom (ℝ D 𝐹)⟶ℝ)
69	65	eqcomd 2628	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)
70	69	dmeqd 5326	. . . . . . . . . . 11 ⊢ (𝜑 → dom (ℝ D 𝐹) = dom 𝐺)
71		fourierdlem60.domg	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐺 = (𝐴(,)𝐵))
72	70, 71	eqtr2d 2657	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,)𝐵) = dom (ℝ D 𝐹))
73	72	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐴(,)𝐵) = dom (ℝ D 𝐹))
74	40, 73	eleqtrd 2703	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐵 + 𝑠) ∈ dom (ℝ D 𝐹))
75	68, 74	ffvelrnd 6360	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐺‘(𝐵 + 𝑠)) ∈ ℝ)
76		1red 10055	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 1 ∈ ℝ)
77	9	ffvelrnda 6359	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℝ)
78	77	recnd 10068	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ)
79	72	feq2d 6031	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺:(𝐴(,)𝐵)⟶ℝ ↔ 𝐺:dom (ℝ D 𝐹)⟶ℝ))
80	67, 79	mpbird 247	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:(𝐴(,)𝐵)⟶ℝ)
81	80	ffvelrnda 6359	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑥) ∈ ℝ)
82	19	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝐵 ∈ ℂ)
83	19	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝐵 ∈ ℂ)
84		0red 10041	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 ∈ ℝ)
85	60, 19	dvmptc 23721	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝐵)) = (𝑠 ∈ ℝ ↦ 0))
86		ioossre 12235	. . . . . . . . . . . . 13 ⊢ ((𝐴 − 𝐵)(,)0) ⊆ ℝ
87	86	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 − 𝐵)(,)0) ⊆ ℝ)
88	46	tgioo2 22606	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
89		iooretop 22569	. . . . . . . . . . . . 13 ⊢ ((𝐴 − 𝐵)(,)0) ∈ (topGen‘ran (,))
90	89	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 − 𝐵)(,)0) ∈ (topGen‘ran (,)))
91	60, 83, 84, 85, 87, 88, 46, 90	dvmptres 23726	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝐵)) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 0))
92	17	recnd 10068	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝑠 ∈ ℂ)
93		recn 10026	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℝ → 𝑠 ∈ ℂ)
94	93	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℂ)
95		1red 10055	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
96	60	dvmptid 23720	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑠)) = (𝑠 ∈ ℝ ↦ 1))
97	60, 94, 95, 96, 87, 88, 46, 90	dvmptres 23726	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠)) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1))
98	60, 82, 33, 91, 92, 76, 97	dvmptadd 23723	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐵 + 𝑠))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (0 + 1)))
99		0p1e1 11132	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
100	99	mpteq2i 4741	. . . . . . . . . 10 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (0 + 1)) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1)
101	98, 100	syl6eq 2672	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐵 + 𝑠))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1))
102	9	feqmptd 6249	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)))
103	102	eqcomd 2628	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) = 𝐹)
104	103	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))) = (ℝ D 𝐹))
105	80	feqmptd 6249	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)))
106	104, 69, 105	3eqtrd 2660	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)))
107		fveq2 6191	. . . . . . . . 9 ⊢ (𝑥 = (𝐵 + 𝑠) → (𝐹‘𝑥) = (𝐹‘(𝐵 + 𝑠)))
108		fveq2 6191	. . . . . . . . 9 ⊢ (𝑥 = (𝐵 + 𝑠) → (𝐺‘𝑥) = (𝐺‘(𝐵 + 𝑠)))
109	60, 60, 40, 76, 78, 81, 101, 106, 107, 108	dvmptco 23735	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐹‘(𝐵 + 𝑠)))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐺‘(𝐵 + 𝑠)) · 1)))
110	75	recnd 10068	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐺‘(𝐵 + 𝑠)) ∈ ℂ)
111	110	mulid1d 10057	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((𝐺‘(𝐵 + 𝑠)) · 1) = (𝐺‘(𝐵 + 𝑠)))
112	111	mpteq2dva 4744	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐺‘(𝐵 + 𝑠)) · 1)) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))))
113	109, 112	eqtrd 2656	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐹‘(𝐵 + 𝑠)))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))))
114		limccl 23639	. . . . . . . . 9 ⊢ (𝐹 lim_ℂ 𝐵) ⊆ ℂ
115	114, 48	sseldi 3601	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℂ)
116	115	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝑌 ∈ ℂ)
117	115	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑌 ∈ ℂ)
118	60, 115	dvmptc 23721	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑌)) = (𝑠 ∈ ℝ ↦ 0))
119	60, 117, 84, 118, 87, 88, 46, 90	dvmptres 23726	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑌)) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 0))
120	60, 61, 75, 113, 116, 27, 119	dvmptsub 23730	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐺‘(𝐵 + 𝑠)) − 0)))
121	110	subid1d 10381	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((𝐺‘(𝐵 + 𝑠)) − 0) = (𝐺‘(𝐵 + 𝑠)))
122	121	mpteq2dva 4744	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐺‘(𝐵 + 𝑠)) − 0)) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))))
123	120, 122	eqtrd 2656	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))))
124	123	dmeqd 5326	. . . 4 ⊢ (𝜑 → dom (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌))) = dom (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))))
125	75	ralrimiva 2966	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ ((𝐴 − 𝐵)(,)0)(𝐺‘(𝐵 + 𝑠)) ∈ ℝ)
126		dmmptg 5632	. . . . 5 ⊢ (∀𝑠 ∈ ((𝐴 − 𝐵)(,)0)(𝐺‘(𝐵 + 𝑠)) ∈ ℝ → dom (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))) = ((𝐴 − 𝐵)(,)0))
127	125, 126	syl 17	. . . 4 ⊢ (𝜑 → dom (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))) = ((𝐴 − 𝐵)(,)0))
128	58, 124, 127	3eqtrd 2660	. . 3 ⊢ (𝜑 → dom (ℝ D 𝑁) = ((𝐴 − 𝐵)(,)0))
129	54	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐷 = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠))
130	129	oveq2d 6666	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐷) = (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠)))
131	130, 97	eqtrd 2656	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐷) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1))
132	131	dmeqd 5326	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝐷) = dom (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1))
133	76	ralrimiva 2966	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ ((𝐴 − 𝐵)(,)0)1 ∈ ℝ)
134		dmmptg 5632	. . . . 5 ⊢ (∀𝑠 ∈ ((𝐴 − 𝐵)(,)0)1 ∈ ℝ → dom (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1) = ((𝐴 − 𝐵)(,)0))
135	133, 134	syl 17	. . . 4 ⊢ (𝜑 → dom (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1) = ((𝐴 − 𝐵)(,)0))
136	132, 135	eqtrd 2656	. . 3 ⊢ (𝜑 → dom (ℝ D 𝐷) = ((𝐴 − 𝐵)(,)0))
137		eqid 2622	. . . . 5 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐹‘(𝐵 + 𝑠))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐹‘(𝐵 + 𝑠)))
138		eqid 2622	. . . . 5 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑌) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑌)
139		eqid 2622	. . . . 5 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌)) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌))
140	40	adantrr 753	. . . . . 6 ⊢ ((𝜑 ∧ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐵 + 𝑠) ≠ 𝐵)) → (𝐵 + 𝑠) ∈ (𝐴(,)𝐵))
141		eqid 2622	. . . . . . . 8 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝐵) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝐵)
142		eqid 2622	. . . . . . . 8 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠)
143		eqid 2622	. . . . . . . 8 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐵 + 𝑠)) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐵 + 𝑠))
144	87, 44	syl6ss 3615	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 − 𝐵)(,)0) ⊆ ℂ)
145	5	recnd 10068	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℂ)
146	141, 144, 19, 145	constlimc 39856	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝐵) lim_ℂ 0))
147	144, 142, 145	idlimc 39858	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠) lim_ℂ 0))
148	141, 142, 143, 82, 92, 146, 147	addlimc 39880	. . . . . . 7 ⊢ (𝜑 → (𝐵 + 0) ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐵 + 𝑠)) lim_ℂ 0))
149	37, 148	eqeltrrd 2702	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐵 + 𝑠)) lim_ℂ 0))
150	102	oveq1d 6665	. . . . . . 7 ⊢ (𝜑 → (𝐹 lim_ℂ 𝐵) = ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) lim_ℂ 𝐵))
151	48, 150	eleqtrd 2703	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) lim_ℂ 𝐵))
152		simplrr 801	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐵 + 𝑠) = 𝐵)) ∧ ¬ (𝐹‘(𝐵 + 𝑠)) = 𝑌) → (𝐵 + 𝑠) = 𝐵)
153	18, 39	ltned 10173	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐵 + 𝑠) ≠ 𝐵)
154	153	neneqd 2799	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ¬ (𝐵 + 𝑠) = 𝐵)
155	154	adantrr 753	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐵 + 𝑠) = 𝐵)) → ¬ (𝐵 + 𝑠) = 𝐵)
156	155	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐵 + 𝑠) = 𝐵)) ∧ ¬ (𝐹‘(𝐵 + 𝑠)) = 𝑌) → ¬ (𝐵 + 𝑠) = 𝐵)
157	152, 156	condan 835	. . . . . 6 ⊢ ((𝜑 ∧ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐵 + 𝑠) = 𝐵)) → (𝐹‘(𝐵 + 𝑠)) = 𝑌)
158	140, 78, 149, 151, 107, 157	limcco 23657	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐹‘(𝐵 + 𝑠))) lim_ℂ 0))
159	138, 144, 115, 145	constlimc 39856	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑌) lim_ℂ 0))
160	137, 138, 139, 61, 116, 158, 159	sublimc 39884	. . . 4 ⊢ (𝜑 → (𝑌 − 𝑌) ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌)) lim_ℂ 0))
161	115	subidd 10380	. . . 4 ⊢ (𝜑 → (𝑌 − 𝑌) = 0)
162	52	eqcomi 2631	. . . . . 6 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌)) = 𝑁
163	162	oveq1i 6660	. . . . 5 ⊢ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌)) lim_ℂ 0) = (𝑁 lim_ℂ 0)
164	163	a1i 11	. . . 4 ⊢ (𝜑 → ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌)) lim_ℂ 0) = (𝑁 lim_ℂ 0))
165	160, 161, 164	3eltr3d 2715	. . 3 ⊢ (𝜑 → 0 ∈ (𝑁 lim_ℂ 0))
166	144, 54, 145	idlimc 39858	. . 3 ⊢ (𝜑 → 0 ∈ (𝐷 lim_ℂ 0))
167		ubioo 12207	. . . . 5 ⊢ ¬ 0 ∈ ((𝐴 − 𝐵)(,)0)
168	167	a1i 11	. . . 4 ⊢ (𝜑 → ¬ 0 ∈ ((𝐴 − 𝐵)(,)0))
169		mptresid 5456	. . . . . . 7 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠) = ( I ↾ ((𝐴 − 𝐵)(,)0))
170	129, 169	syl6eq 2672	. . . . . 6 ⊢ (𝜑 → 𝐷 = ( I ↾ ((𝐴 − 𝐵)(,)0)))
171	170	rneqd 5353	. . . . 5 ⊢ (𝜑 → ran 𝐷 = ran ( I ↾ ((𝐴 − 𝐵)(,)0)))
172		rnresi 5479	. . . . 5 ⊢ ran ( I ↾ ((𝐴 − 𝐵)(,)0)) = ((𝐴 − 𝐵)(,)0)
173	171, 172	syl6req 2673	. . . 4 ⊢ (𝜑 → ((𝐴 − 𝐵)(,)0) = ran 𝐷)
174	168, 173	neleqtrd 2722	. . 3 ⊢ (𝜑 → ¬ 0 ∈ ran 𝐷)
175		0ne1 11088	. . . . . 6 ⊢ 0 ≠ 1
176	175	neii 2796	. . . . 5 ⊢ ¬ 0 = 1
177		elsng 4191	. . . . . 6 ⊢ (0 ∈ ℝ → (0 ∈ {1} ↔ 0 = 1))
178	5, 177	syl 17	. . . . 5 ⊢ (𝜑 → (0 ∈ {1} ↔ 0 = 1))
179	176, 178	mtbiri 317	. . . 4 ⊢ (𝜑 → ¬ 0 ∈ {1})
180	131	rneqd 5353	. . . . 5 ⊢ (𝜑 → ran (ℝ D 𝐷) = ran (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1))
181		eqid 2622	. . . . . 6 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1)
182	26	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ^*)
183		ioon0 12201	. . . . . . . 8 ⊢ (((𝐴 − 𝐵) ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (((𝐴 − 𝐵)(,)0) ≠ ∅ ↔ (𝐴 − 𝐵) < 0))
184	4, 182, 183	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → (((𝐴 − 𝐵)(,)0) ≠ ∅ ↔ (𝐴 − 𝐵) < 0))
185	8, 184	mpbird 247	. . . . . 6 ⊢ (𝜑 → ((𝐴 − 𝐵)(,)0) ≠ ∅)
186	181, 76, 185	rnmptc 39353	. . . . 5 ⊢ (𝜑 → ran (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1) = {1})
187	180, 186	eqtr2d 2657	. . . 4 ⊢ (𝜑 → {1} = ran (ℝ D 𝐷))
188	179, 187	neleqtrd 2722	. . 3 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐷))
189	81	recnd 10068	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑥) ∈ ℂ)
190		fourierdlem60.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝐺 lim_ℂ 𝐵))
191	105	oveq1d 6665	. . . . . 6 ⊢ (𝜑 → (𝐺 lim_ℂ 𝐵) = ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)) lim_ℂ 𝐵))
192	190, 191	eleqtrd 2703	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)) lim_ℂ 𝐵))
193		simplrr 801	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐵 + 𝑠) = 𝐵)) ∧ ¬ (𝐺‘(𝐵 + 𝑠)) = 𝐸) → (𝐵 + 𝑠) = 𝐵)
194	155	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐵 + 𝑠) = 𝐵)) ∧ ¬ (𝐺‘(𝐵 + 𝑠)) = 𝐸) → ¬ (𝐵 + 𝑠) = 𝐵)
195	193, 194	condan 835	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐵 + 𝑠) = 𝐵)) → (𝐺‘(𝐵 + 𝑠)) = 𝐸)
196	140, 189, 149, 192, 108, 195	limcco 23657	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))) lim_ℂ 0))
197	110	div1d 10793	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((𝐺‘(𝐵 + 𝑠)) / 1) = (𝐺‘(𝐵 + 𝑠)))
198	56, 123	syl5eq 2668	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D 𝑁) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))))
199	198	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (ℝ D 𝑁) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))))
200	199	fveq1d 6193	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((ℝ D 𝑁)‘𝑠) = ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠)))‘𝑠))
201		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠)))
202	201	fvmpt2 6291	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐺‘(𝐵 + 𝑠)) ∈ ℝ) → ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠)))‘𝑠) = (𝐺‘(𝐵 + 𝑠)))
203	28, 75, 202	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠)))‘𝑠) = (𝐺‘(𝐵 + 𝑠)))
204	200, 203	eqtr2d 2657	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐺‘(𝐵 + 𝑠)) = ((ℝ D 𝑁)‘𝑠))
205	131	fveq1d 6193	. . . . . . . . . 10 ⊢ (𝜑 → ((ℝ D 𝐷)‘𝑠) = ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1)‘𝑠))
206	205	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((ℝ D 𝐷)‘𝑠) = ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1)‘𝑠))
207	181	fvmpt2 6291	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ 1 ∈ ℝ) → ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1)‘𝑠) = 1)
208	28, 76, 207	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1)‘𝑠) = 1)
209	206, 208	eqtr2d 2657	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 1 = ((ℝ D 𝐷)‘𝑠))
210	204, 209	oveq12d 6668	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((𝐺‘(𝐵 + 𝑠)) / 1) = (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠)))
211	197, 210	eqtr3d 2658	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐺‘(𝐵 + 𝑠)) = (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠)))
212	211	mpteq2dva 4744	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠))))
213	212	oveq1d 6665	. . . 4 ⊢ (𝜑 → ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))) lim_ℂ 0) = ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠))) lim_ℂ 0))
214	196, 213	eleqtrd 2703	. . 3 ⊢ (𝜑 → 𝐸 ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠))) lim_ℂ 0))
215	4, 5, 8, 53, 55, 128, 136, 165, 166, 174, 188, 214	lhop2 23778	. 2 ⊢ (𝜑 → 𝐸 ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) lim_ℂ 0))
216	52	fvmpt2 6291	. . . . . . 7 ⊢ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ ((𝐹‘(𝐵 + 𝑠)) − 𝑌) ∈ ℝ) → (𝑁‘𝑠) = ((𝐹‘(𝐵 + 𝑠)) − 𝑌))
217	28, 51, 216	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝑁‘𝑠) = ((𝐹‘(𝐵 + 𝑠)) − 𝑌))
218	54	fvmpt2 6291	. . . . . . 7 ⊢ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐷‘𝑠) = 𝑠)
219	28, 28, 218	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐷‘𝑠) = 𝑠)
220	217, 219	oveq12d 6668	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((𝑁‘𝑠) / (𝐷‘𝑠)) = (((𝐹‘(𝐵 + 𝑠)) − 𝑌) / 𝑠))
221	220	mpteq2dva 4744	. . . 4 ⊢ (𝜑 → (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (((𝐹‘(𝐵 + 𝑠)) − 𝑌) / 𝑠)))
222		fourierdlem60.h	. . . 4 ⊢ 𝐻 = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (((𝐹‘(𝐵 + 𝑠)) − 𝑌) / 𝑠))
223	221, 222	syl6eqr 2674	. . 3 ⊢ (𝜑 → (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) = 𝐻)
224	223	oveq1d 6665	. 2 ⊢ (𝜑 → ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) lim_ℂ 0) = (𝐻 lim_ℂ 0))
225	215, 224	eleqtrd 2703	1 ⊢ (𝜑 → 𝐸 ∈ (𝐻 lim_ℂ 0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ⊆ wss 3574 ∅c0 3915 {csn 4177 {cpr 4179 class class class wbr 4653 ↦ cmpt 4729 I cid 5023 dom cdm 5114 ran crn 5115 ↾ cres 5116 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ℝ^*cxr 10073 < clt 10074 − cmin 10266 / cdiv 10684 (,)cioo 12175 TopOpenctopn 16082 topGenctg 16098 ℂ_fldccnfld 19746 lim_ℂ climc 23626 D cdv 23627

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631

This theorem is referenced by: fourierdlem74 40397

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