fourierdlem60 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem60 47284

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 11751	. . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ)
4	3	rexrd 11366	. . 3 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ^*)
5		0red 11318	. . 3 ⊢ (𝜑 → 0 ∈ ℝ)
6		fourierdlem60.altb	. . . 4 ⊢ (𝜑 → 𝐴 < 𝐵)
7	1, 2	sublt0d 11949	. . . 4 ⊢ (𝜑 → ((𝐴 − 𝐵) < 0 ↔ 𝐴 < 𝐵))
8	6, 7	mpbird 260	. . 3 ⊢ (𝜑 → (𝐴 − 𝐵) < 0)
9		fourierdlem60.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
10	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
11	1	rexrd 11366	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
12	11	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝐴 ∈ ℝ^*)
13	2	rexrd 11366	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
14	13	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝐵 ∈ ℝ^*)
15	2	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝐵 ∈ ℝ)
16		elioore 13513	. . . . . . . . 9 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) → 𝑠 ∈ ℝ)
17	16	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝑠 ∈ ℝ)
18	15, 17	readdcld 11345	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐵 + 𝑠) ∈ ℝ)
19	2	recnd 11344	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ)
20	1	recnd 11344	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ)
21	19, 20	pncan3d 11679	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 + (𝐴 − 𝐵)) = 𝐴)
22	21	eqcomd 2766	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = (𝐵 + (𝐴 − 𝐵)))
23	22	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝐴 = (𝐵 + (𝐴 − 𝐵)))
24	3	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐴 − 𝐵) ∈ ℝ)
25	4	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐴 − 𝐵) ∈ ℝ^*)
26		0xr 11363	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
27	26	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 0 ∈ ℝ^*)
28		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝑠 ∈ ((𝐴 − 𝐵)(,)0))
29	25, 27, 28	ioogtlbd 46670	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐴 − 𝐵) < 𝑠)
30	24, 17, 15, 29	ltadd2dd 11476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐵 + (𝐴 − 𝐵)) < (𝐵 + 𝑠))
31	23, 30	eqbrtrd 5126	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝐴 < (𝐵 + 𝑠))
32		0red 11318	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 0 ∈ ℝ)
33	25, 27, 28	iooltubd 46664	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝑠 < 0)
34	17, 32, 15, 33	ltadd2dd 11476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐵 + 𝑠) < (𝐵 + 0))
35	19	addridd 11517	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 + 0) = 𝐵)
36	35	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐵 + 0) = 𝐵)
37	34, 36	breqtrd 5130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐵 + 𝑠) < 𝐵)
38	12, 14, 18, 31, 37	eliood 46618	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐵 + 𝑠) ∈ (𝐴(,)𝐵))
39	10, 38	ffvelcdmd 7090	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐹‘(𝐵 + 𝑠)) ∈ ℝ)
40		ioossre 13545	. . . . . . . . 9 ⊢ (𝐴(,)𝐵) ⊆ ℝ
41	40	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
42		ax-resscn 11264	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
43	41, 42	sstrdi 3921	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
44		eqid 2760	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	44, 11, 2, 6	lptioo2cn 46763	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝐴(,)𝐵)))
46		fourierdlem60.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝐹 lim_ℂ 𝐵))
47	9, 43, 45, 46	limcrecl 46749	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ℝ)
48	47	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝑌 ∈ ℝ)
49	39, 48	resubcld 11751	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((𝐹‘(𝐵 + 𝑠)) − 𝑌) ∈ ℝ)
50		fourierdlem60.n	. . . 4 ⊢ 𝑁 = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌))
51	49, 50	fmptd 7119	. . 3 ⊢ (𝜑 → 𝑁:((𝐴 − 𝐵)(,)0)⟶ℝ)
52		fourierdlem60.d	. . . 4 ⊢ 𝐷 = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠)
53	17, 52	fmptd 7119	. . 3 ⊢ (𝜑 → 𝐷:((𝐴 − 𝐵)(,)0)⟶ℝ)
54	50	oveq2i 7435	. . . . . 6 ⊢ (ℝ D 𝑁) = (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌)))
55	54	a1i 11	. . . . 5 ⊢ (𝜑 → (ℝ D 𝑁) = (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌))))
56	55	dmeqd 5888	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝑁) = dom (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌))))
57		reelprrecn 11299	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
58	57	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
59	39	recnd 11344	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐹‘(𝐵 + 𝑠)) ∈ ℂ)
60		dvfre 26280	. . . . . . . . . . 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		fourierdlem60.g	. . . . . . . . . . . 12 ⊢ 𝐺 = (ℝ D 𝐹)
63	62	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 = (ℝ D 𝐹))
64	63	feq1d 6696	. . . . . . . . . 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 ∈ ((𝐴 − 𝐵)(,)0)) → 𝐺:dom (ℝ D 𝐹)⟶ℝ)
67	63	eqcomd 2766	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)
68	67	dmeqd 5888	. . . . . . . . . . 11 ⊢ (𝜑 → dom (ℝ D 𝐹) = dom 𝐺)
69		fourierdlem60.domg	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐺 = (𝐴(,)𝐵))
70	68, 69	eqtr2d 2796	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,)𝐵) = dom (ℝ D 𝐹))
71	70	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐴(,)𝐵) = dom (ℝ D 𝐹))
72	38, 71	eleqtrd 2862	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐵 + 𝑠) ∈ dom (ℝ D 𝐹))
73	66, 72	ffvelcdmd 7090	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐺‘(𝐵 + 𝑠)) ∈ ℝ)
74		1red 11316	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 1 ∈ ℝ)
75	9	ffvelcdmda 7089	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℝ)
76	75	recnd 11344	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ)
77	70	feq2d 6698	. . . . . . . . . . 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 7089	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑥) ∈ ℝ)
80	19	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝐵 ∈ ℂ)
81	19	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝐵 ∈ ℂ)
82		0red 11318	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 ∈ ℝ)
83	58, 19	dvmptc 26287	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝐵)) = (𝑠 ∈ ℝ ↦ 0))
84		ioossre 13545	. . . . . . . . . . . . 13 ⊢ ((𝐴 − 𝐵)(,)0) ⊆ ℝ
85	84	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 − 𝐵)(,)0) ⊆ ℝ)
86		tgioo4 25133	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
87		iooretop 25093	. . . . . . . . . . . . 13 ⊢ ((𝐴 − 𝐵)(,)0) ∈ (topGen‘ran (,))
88	87	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 − 𝐵)(,)0) ∈ (topGen‘ran (,)))
89	58, 81, 82, 83, 85, 86, 44, 88	dvmptres 26292	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝐵)) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 0))
90	17	recnd 11344	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝑠 ∈ ℂ)
91		recn 11297	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ℝ → 𝑠 ∈ ℂ)
92	91	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℂ)
93		1red 11316	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
94	58	dvmptid 26286	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑠)) = (𝑠 ∈ ℝ ↦ 1))
95	58, 92, 93, 94, 85, 86, 44, 88	dvmptres 26292	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠)) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1))
96	58, 80, 32, 89, 90, 74, 95	dvmptadd 26289	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐵 + 𝑠))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (0 + 1)))
97		0p1e1 12470	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
98	97	mpteq2i 5200	. . . . . . . . . 10 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (0 + 1)) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1)
99	96, 98	eqtrdi 2811	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐵 + 𝑠))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1))
100	9	feqmptd 6958	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)))
101	100	eqcomd 2766	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) = 𝐹)
102	101	oveq2d 7440	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))) = (ℝ D 𝐹))
103	78	feqmptd 6958	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)))
104	102, 67, 103	3eqtrd 2799	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)))
105		fveq2 6890	. . . . . . . . 9 ⊢ (𝑥 = (𝐵 + 𝑠) → (𝐹‘𝑥) = (𝐹‘(𝐵 + 𝑠)))
106		fveq2 6890	. . . . . . . . 9 ⊢ (𝑥 = (𝐵 + 𝑠) → (𝐺‘𝑥) = (𝐺‘(𝐵 + 𝑠)))
107	58, 58, 38, 74, 76, 79, 99, 104, 105, 106	dvmptco 26301	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐹‘(𝐵 + 𝑠)))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐺‘(𝐵 + 𝑠)) · 1)))
108	73	recnd 11344	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐺‘(𝐵 + 𝑠)) ∈ ℂ)
109	108	mulridd 11333	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((𝐺‘(𝐵 + 𝑠)) · 1) = (𝐺‘(𝐵 + 𝑠)))
110	109	mpteq2dva 5197	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐺‘(𝐵 + 𝑠)) · 1)) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))))
111	107, 110	eqtrd 2795	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐹‘(𝐵 + 𝑠)))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))))
112		limccl 26204	. . . . . . . . 9 ⊢ (𝐹 lim_ℂ 𝐵) ⊆ ℂ
113	112, 46	sselid 3907	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ ℂ)
114	113	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 𝑌 ∈ ℂ)
115	113	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑌 ∈ ℂ)
116	58, 113	dvmptc 26287	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑌)) = (𝑠 ∈ ℝ ↦ 0))
117	58, 115, 82, 116, 85, 86, 44, 88	dvmptres 26292	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑌)) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 0))
118	58, 59, 73, 111, 114, 27, 117	dvmptsub 26296	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐺‘(𝐵 + 𝑠)) − 0)))
119	108	subid1d 11665	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((𝐺‘(𝐵 + 𝑠)) − 0) = (𝐺‘(𝐵 + 𝑠)))
120	119	mpteq2dva 5197	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐺‘(𝐵 + 𝑠)) − 0)) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))))
121	118, 120	eqtrd 2795	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))))
122	121	dmeqd 5888	. . . 4 ⊢ (𝜑 → dom (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌))) = dom (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))))
123	73	ralrimiva 3154	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ ((𝐴 − 𝐵)(,)0)(𝐺‘(𝐵 + 𝑠)) ∈ ℝ)
124		dmmptg 6246	. . . . 5 ⊢ (∀𝑠 ∈ ((𝐴 − 𝐵)(,)0)(𝐺‘(𝐵 + 𝑠)) ∈ ℝ → dom (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))) = ((𝐴 − 𝐵)(,)0))
125	123, 124	syl 18	. . . 4 ⊢ (𝜑 → dom (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))) = ((𝐴 − 𝐵)(,)0))
126	56, 122, 125	3eqtrd 2799	. . 3 ⊢ (𝜑 → dom (ℝ D 𝑁) = ((𝐴 − 𝐵)(,)0))
127	52	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐷 = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠))
128	127	oveq2d 7440	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐷) = (ℝ D (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠)))
129	128, 95	eqtrd 2795	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐷) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1))
130	129	dmeqd 5888	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝐷) = dom (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1))
131	74	ralrimiva 3154	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ ((𝐴 − 𝐵)(,)0)1 ∈ ℝ)
132		dmmptg 6246	. . . . 5 ⊢ (∀𝑠 ∈ ((𝐴 − 𝐵)(,)0)1 ∈ ℝ → dom (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1) = ((𝐴 − 𝐵)(,)0))
133	131, 132	syl 18	. . . 4 ⊢ (𝜑 → dom (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1) = ((𝐴 − 𝐵)(,)0))
134	130, 133	eqtrd 2795	. . 3 ⊢ (𝜑 → dom (ℝ D 𝐷) = ((𝐴 − 𝐵)(,)0))
135		eqid 2760	. . . . 5 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐹‘(𝐵 + 𝑠))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐹‘(𝐵 + 𝑠)))
136		eqid 2760	. . . . 5 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑌) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑌)
137		eqid 2760	. . . . 5 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌)) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌))
138	38	adantrr 730	. . . . . 6 ⊢ ((𝜑 ∧ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐵 + 𝑠) ≠ 𝐵)) → (𝐵 + 𝑠) ∈ (𝐴(,)𝐵))
139		eqid 2760	. . . . . . . 8 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝐵) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝐵)
140		eqid 2760	. . . . . . . 8 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠)
141		eqid 2760	. . . . . . . 8 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐵 + 𝑠)) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐵 + 𝑠))
142	85, 42	sstrdi 3921	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 − 𝐵)(,)0) ⊆ ℂ)
143	5	recnd 11344	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℂ)
144	139, 142, 19, 143	constlimc 46744	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝐵) lim_ℂ 0))
145	142, 140, 143	idlimc 46746	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠) lim_ℂ 0))
146	139, 140, 141, 80, 90, 144, 145	addlimc 46766	. . . . . . 7 ⊢ (𝜑 → (𝐵 + 0) ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐵 + 𝑠)) lim_ℂ 0))
147	35, 146	eqeltrrd 2861	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐵 + 𝑠)) lim_ℂ 0))
148	100	oveq1d 7439	. . . . . . 7 ⊢ (𝜑 → (𝐹 lim_ℂ 𝐵) = ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) lim_ℂ 𝐵))
149	46, 148	eleqtrd 2862	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)) lim_ℂ 𝐵))
150		simplrr 790	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐵 + 𝑠) = 𝐵)) ∧ ¬ (𝐹‘(𝐵 + 𝑠)) = 𝑌) → (𝐵 + 𝑠) = 𝐵)
151	18, 37	ltned 11453	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐵 + 𝑠) ≠ 𝐵)
152	151	neneqd 2960	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ¬ (𝐵 + 𝑠) = 𝐵)
153	152	adantrr 730	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐵 + 𝑠) = 𝐵)) → ¬ (𝐵 + 𝑠) = 𝐵)
154	153	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐵 + 𝑠) = 𝐵)) ∧ ¬ (𝐹‘(𝐵 + 𝑠)) = 𝑌) → ¬ (𝐵 + 𝑠) = 𝐵)
155	150, 154	condan 830	. . . . . 6 ⊢ ((𝜑 ∧ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐵 + 𝑠) = 𝐵)) → (𝐹‘(𝐵 + 𝑠)) = 𝑌)
156	138, 76, 147, 149, 105, 155	limcco 26222	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐹‘(𝐵 + 𝑠))) lim_ℂ 0))
157	136, 142, 113, 143	constlimc 46744	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑌) lim_ℂ 0))
158	135, 136, 137, 59, 114, 156, 157	sublimc 46770	. . . 4 ⊢ (𝜑 → (𝑌 − 𝑌) ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌)) lim_ℂ 0))
159	113	subidd 11664	. . . 4 ⊢ (𝜑 → (𝑌 − 𝑌) = 0)
160	50	eqcomi 2769	. . . . . 6 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌)) = 𝑁
161	160	oveq1i 7434	. . . . 5 ⊢ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌)) lim_ℂ 0) = (𝑁 lim_ℂ 0)
162	161	a1i 11	. . . 4 ⊢ (𝜑 → ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌)) lim_ℂ 0) = (𝑁 lim_ℂ 0))
163	158, 159, 162	3eltr3d 2874	. . 3 ⊢ (𝜑 → 0 ∈ (𝑁 lim_ℂ 0))
164	142, 52, 143	idlimc 46746	. . 3 ⊢ (𝜑 → 0 ∈ (𝐷 lim_ℂ 0))
165		ubioo 13515	. . . . 5 ⊢ ¬ 0 ∈ ((𝐴 − 𝐵)(,)0)
166	165	a1i 11	. . . 4 ⊢ (𝜑 → ¬ 0 ∈ ((𝐴 − 𝐵)(,)0))
167		mptresid 6045	. . . . . . 7 ⊢ ( I ↾ ((𝐴 − 𝐵)(,)0)) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠)
168	127, 167	eqtr4di 2813	. . . . . 6 ⊢ (𝜑 → 𝐷 = ( I ↾ ((𝐴 − 𝐵)(,)0)))
169	168	rneqd 5921	. . . . 5 ⊢ (𝜑 → ran 𝐷 = ran ( I ↾ ((𝐴 − 𝐵)(,)0)))
170		rnresi 6075	. . . . 5 ⊢ ran ( I ↾ ((𝐴 − 𝐵)(,)0)) = ((𝐴 − 𝐵)(,)0)
171	169, 170	eqtr2di 2812	. . . 4 ⊢ (𝜑 → ((𝐴 − 𝐵)(,)0) = ran 𝐷)
172	166, 171	neleqtrd 2882	. . 3 ⊢ (𝜑 → ¬ 0 ∈ ran 𝐷)
173		0ne1 12421	. . . . . 6 ⊢ 0 ≠ 1
174	173	neii 2957	. . . . 5 ⊢ ¬ 0 = 1
175		elsng 4594	. . . . . 6 ⊢ (0 ∈ ℝ → (0 ∈ {1} ↔ 0 = 1))
176	5, 175	syl 18	. . . . 5 ⊢ (𝜑 → (0 ∈ {1} ↔ 0 = 1))
177	174, 176	mtbiri 330	. . . 4 ⊢ (𝜑 → ¬ 0 ∈ {1})
178	129	rneqd 5921	. . . . 5 ⊢ (𝜑 → ran (ℝ D 𝐷) = ran (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1))
179		eqid 2760	. . . . . 6 ⊢ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1)
180	26	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ^*)
181		ioon0 13509	. . . . . . . 8 ⊢ (((𝐴 − 𝐵) ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (((𝐴 − 𝐵)(,)0) ≠ ∅ ↔ (𝐴 − 𝐵) < 0))
182	4, 180, 181	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (((𝐴 − 𝐵)(,)0) ≠ ∅ ↔ (𝐴 − 𝐵) < 0))
183	8, 182	mpbird 260	. . . . . 6 ⊢ (𝜑 → ((𝐴 − 𝐵)(,)0) ≠ ∅)
184	179, 183	rnmptc 7217	. . . . 5 ⊢ (𝜑 → ran (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1) = {1})
185	178, 184	eqtr2d 2796	. . . 4 ⊢ (𝜑 → {1} = ran (ℝ D 𝐷))
186	177, 185	neleqtrd 2882	. . 3 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐷))
187	79	recnd 11344	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑥) ∈ ℂ)
188		fourierdlem60.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝐺 lim_ℂ 𝐵))
189	103	oveq1d 7439	. . . . . 6 ⊢ (𝜑 → (𝐺 lim_ℂ 𝐵) = ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)) lim_ℂ 𝐵))
190	188, 189	eleqtrd 2862	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ((𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑥)) lim_ℂ 𝐵))
191		simplrr 790	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐵 + 𝑠) = 𝐵)) ∧ ¬ (𝐺‘(𝐵 + 𝑠)) = 𝐸) → (𝐵 + 𝑠) = 𝐵)
192	153	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐵 + 𝑠) = 𝐵)) ∧ ¬ (𝐺‘(𝐵 + 𝑠)) = 𝐸) → ¬ (𝐵 + 𝑠) = 𝐵)
193	191, 192	condan 830	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐵 + 𝑠) = 𝐵)) → (𝐺‘(𝐵 + 𝑠)) = 𝐸)
194	138, 187, 147, 190, 106, 193	limcco 26222	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))) lim_ℂ 0))
195	108	div1d 12092	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((𝐺‘(𝐵 + 𝑠)) / 1) = (𝐺‘(𝐵 + 𝑠)))
196	54, 121	eqtrid 2807	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D 𝑁) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))))
197	196	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (ℝ D 𝑁) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))))
198	197	fveq1d 6892	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((ℝ D 𝑁)‘𝑠) = ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠)))‘𝑠))
199		fvmpt4 46358	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ (𝐺‘(𝐵 + 𝑠)) ∈ ℝ) → ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠)))‘𝑠) = (𝐺‘(𝐵 + 𝑠)))
200	28, 73, 199	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠)))‘𝑠) = (𝐺‘(𝐵 + 𝑠)))
201	198, 200	eqtr2d 2796	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐺‘(𝐵 + 𝑠)) = ((ℝ D 𝑁)‘𝑠))
202	129	fveq1d 6892	. . . . . . . . . 10 ⊢ (𝜑 → ((ℝ D 𝐷)‘𝑠) = ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1)‘𝑠))
203	202	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((ℝ D 𝐷)‘𝑠) = ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1)‘𝑠))
204		fvmpt4 46358	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ 1 ∈ ℝ) → ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1)‘𝑠) = 1)
205	28, 74, 204	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 1)‘𝑠) = 1)
206	203, 205	eqtr2d 2796	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → 1 = ((ℝ D 𝐷)‘𝑠))
207	201, 206	oveq12d 7442	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((𝐺‘(𝐵 + 𝑠)) / 1) = (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠)))
208	195, 207	eqtr3d 2797	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐺‘(𝐵 + 𝑠)) = (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠)))
209	208	mpteq2dva 5197	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (𝐺‘(𝐵 + 𝑠))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠))))
210	209	oveq1d 7439	. . . 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))
211	194, 210	eleqtrd 2862	. . 3 ⊢ (𝜑 → 𝐸 ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (((ℝ D 𝑁)‘𝑠) / ((ℝ D 𝐷)‘𝑠))) lim_ℂ 0))
212	4, 5, 8, 51, 53, 126, 134, 163, 164, 172, 186, 211	lhop2 26344	. 2 ⊢ (𝜑 → 𝐸 ∈ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) lim_ℂ 0))
213	50	fvmpt2 7010	. . . . . . 7 ⊢ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ ((𝐹‘(𝐵 + 𝑠)) − 𝑌) ∈ ℝ) → (𝑁‘𝑠) = ((𝐹‘(𝐵 + 𝑠)) − 𝑌))
214	28, 49, 213	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝑁‘𝑠) = ((𝐹‘(𝐵 + 𝑠)) − 𝑌))
215	52	fvmpt2 7010	. . . . . . 7 ⊢ ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐷‘𝑠) = 𝑠)
216	28, 28, 215	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → (𝐷‘𝑠) = 𝑠)
217	214, 216	oveq12d 7442	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ((𝐴 − 𝐵)(,)0)) → ((𝑁‘𝑠) / (𝐷‘𝑠)) = (((𝐹‘(𝐵 + 𝑠)) − 𝑌) / 𝑠))
218	217	mpteq2dva 5197	. . . 4 ⊢ (𝜑 → (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (((𝐹‘(𝐵 + 𝑠)) − 𝑌) / 𝑠)))
219		fourierdlem60.h	. . . 4 ⊢ 𝐻 = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (((𝐹‘(𝐵 + 𝑠)) − 𝑌) / 𝑠))
220	218, 219	eqtr4di 2813	. . 3 ⊢ (𝜑 → (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) = 𝐻)
221	220	oveq1d 7439	. 2 ⊢ (𝜑 → ((𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝑁‘𝑠) / (𝐷‘𝑠))) lim_ℂ 0) = (𝐻 lim_ℂ 0))
222	212, 221	eleqtrd 2862	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 2144 ≠ wne 2955 ∀wral 3076 ⊆ wss 3896 ∅c0 4277 {csn 4584 {cpr 4586 class class class wbr 5102 ↦ cmpt 5185 I cid 5545 dom cdm 5651 ran crn 5652 ↾ cres 5653 ⟶wf 6539 ‘cfv 6543 (class class class)co 7424 ℂcc 11205 ℝcr 11206 0cc0 11207 1c1 11208 + caddc 11210 · cmul 11212 ℝ^*cxr 11349 < clt 11350 − cmin 11548 / cdiv 11980 (,)cioo 13483 TopOpenctopn 17601 topGenctg 17617 ℂ_fldccnfld 21687 lim_ℂ climc 26191 D cdv 26192

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 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285 ax-addf 11286

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 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7697 df-om 7882 df-1st 8005 df-2nd 8006 df-supp 8184 df-frecs 8306 df-wrecs 8337 df-recs 8386 df-rdg 8425 df-1o 8483 df-2o 8484 df-er 8724 df-map 8856 df-pm 8857 df-ixp 8933 df-en 8981 df-dom 8982 df-sdom 8983 df-fin 8984 df-fsupp 9361 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9511 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-dec 12822 df-uz 12973 df-q 13083 df-rp 13128 df-xneg 13248 df-xadd 13249 df-xmul 13250 df-ioo 13487 df-ioc 13488 df-ico 13489 df-icc 13490 df-fz 13647 df-fzo 13796 df-seq 14152 df-exp 14212 df-hash 14482 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-starv 17452 df-sca 17453 df-vsca 17454 df-ip 17455 df-tset 17456 df-ple 17457 df-ds 17459 df-unif 17460 df-hom 17461 df-cco 17462 df-rest 17602 df-topn 17603 df-0g 17621 df-gsum 17622 df-topgen 17623 df-pt 17624 df-prds 17627 df-xrs 17683 df-qtop 17688 df-imas 17689 df-xps 17691 df-mre 17765 df-mrc 17766 df-acs 17768 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-submnd 18988 df-mulg 19287 df-cntz 19540 df-cmn 20005 df-psmet 21679 df-xmet 21680 df-met 21681 df-bl 21682 df-mopn 21683 df-fbas 21684 df-fg 21685 df-cnfld 21688 df-top 23221 df-topon 23238 df-topsp 23260 df-bases 23273 df-cld 23346 df-ntr 23347 df-cls 23348 df-nei 23425 df-lp 23463 df-perf 23464 df-cn 23554 df-cnp 23555 df-haus 23642 df-cmp 23714 df-tx 23890 df-hmeo 24083 df-fil 24174 df-fm 24266 df-flim 24267 df-flf 24268 df-xms 24648 df-ms 24649 df-tms 24650 df-cncf 25208 df-limc 26195 df-dv 26196

This theorem is used by: fourierdlem74 47298

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