fourierdlem68 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem68 46129

Description: The derivative of 𝑂 is bounded on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem68.f	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
fourierdlem68.xre	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem68.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem68.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem68.altb	⊢ (𝜑 → 𝐴 < 𝐵)
fourierdlem68.ab	⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))
fourierdlem68.n0	⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
fourierdlem68.fdv	⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))):((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℝ)
fourierdlem68.d	⊢ (𝜑 → 𝐷 ∈ ℝ)
fourierdlem68.fbd	⊢ ((𝜑 ∧ 𝑡 ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) → (abs‘(𝐹‘𝑡)) ≤ 𝐷)
fourierdlem68.e	⊢ (𝜑 → 𝐸 ∈ ℝ)
fourierdlem68.fdvbd	⊢ ((𝜑 ∧ 𝑡 ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘𝑡)) ≤ 𝐸)
fourierdlem68.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
fourierdlem68.o	⊢ 𝑂 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))

**Assertion**
Ref	Expression
fourierdlem68	⊢ (𝜑 → (dom (ℝ D 𝑂) = (𝐴(,)𝐵) ∧ ∃𝑏 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑂)(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑏))

Distinct variable groups: 𝐴,𝑏,𝑠 𝑡,𝐴,𝑠 𝐵,𝑏,𝑠 𝑡,𝐵 𝐶,𝑏,𝑠 𝐷,𝑏,𝑠 𝑡,𝐷 𝐸,𝑏,𝑠 𝑡,𝐸 𝐹,𝑏,𝑠 𝑡,𝐹 𝑋,𝑏,𝑠 𝑡,𝑋 𝜑,𝑏,𝑠 𝜑,𝑡 Allowed substitution hints: 𝐶(𝑡) 𝑂(𝑡,𝑠,𝑏)

Proof of Theorem fourierdlem68 Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem68.f	. . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
2		fourierdlem68.xre	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℝ)
3		fourierdlem68.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		fourierdlem68.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5		fourierdlem68.fdv	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))):((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℝ)
6		ioossicc 13469	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
7		fourierdlem68.ab	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))
8	6, 7	sstrid 4006	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (-π[,]π))
9		fourierdlem68.n0	. . . . . . 7 ⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))
10	6	sseli 3990	. . . . . . 7 ⊢ (0 ∈ (𝐴(,)𝐵) → 0 ∈ (𝐴[,]𝐵))
11	9, 10	nsyl 140	. . . . . 6 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
12		fourierdlem68.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ)
13		fourierdlem68.o	. . . . . 6 ⊢ 𝑂 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))
14	1, 2, 3, 4, 5, 8, 11, 12, 13	fourierdlem57 46118	. . . . 5 ⊢ ((𝜑 → ((ℝ D 𝑂):(𝐴(,)𝐵)⟶ℝ ∧ (ℝ D 𝑂) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠)) · (2 · (sin‘(𝑠 / 2)))) − ((cos‘(𝑠 / 2)) · ((𝐹‘(𝑋 + 𝑠)) − 𝐶))) / ((2 · (sin‘(𝑠 / 2)))↑2))))) ∧ (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (2 · (sin‘(𝑠 / 2))))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (cos‘(𝑠 / 2))))
15	14	simpli 483	. . . 4 ⊢ (𝜑 → ((ℝ D 𝑂):(𝐴(,)𝐵)⟶ℝ ∧ (ℝ D 𝑂) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠)) · (2 · (sin‘(𝑠 / 2)))) − ((cos‘(𝑠 / 2)) · ((𝐹‘(𝑋 + 𝑠)) − 𝐶))) / ((2 · (sin‘(𝑠 / 2)))↑2)))))
16	15	simpld 494	. . 3 ⊢ (𝜑 → (ℝ D 𝑂):(𝐴(,)𝐵)⟶ℝ)
17	16	fdmd 6746	. 2 ⊢ (𝜑 → dom (ℝ D 𝑂) = (𝐴(,)𝐵))
18		eqid 2734	. . . . . 6 ⊢ (𝑡 ∈ (𝐴[,]𝐵) ↦ (2 · (sin‘(𝑡 / 2)))) = (𝑡 ∈ (𝐴[,]𝐵) ↦ (2 · (sin‘(𝑡 / 2))))
19		fourierdlem68.altb	. . . . . . 7 ⊢ (𝜑 → 𝐴 < 𝐵)
20	3, 4, 19	ltled 11406	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
21		2re 12337	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
22	21	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 2 ∈ ℝ)
23	3, 4	iccssred 13470	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
24	23	sselda 3994	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ℝ)
25	24	rehalfcld 12510	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 / 2) ∈ ℝ)
26	25	resincld 16175	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (sin‘(𝑡 / 2)) ∈ ℝ)
27	22, 26	remulcld 11288	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (2 · (sin‘(𝑡 / 2))) ∈ ℝ)
28		2cnd 12341	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 2 ∈ ℂ)
29	26	recnd 11286	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (sin‘(𝑡 / 2)) ∈ ℂ)
30		2ne0 12367	. . . . . . . . . . 11 ⊢ 2 ≠ 0
31	30	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 2 ≠ 0)
32	7	sselda 3994	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (-π[,]π))
33		eqcom 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 0 ↔ 0 = 𝑡)
34	33	biimpi 216	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 0 → 0 = 𝑡)
35	34	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ (𝐴[,]𝐵) ∧ 𝑡 = 0) → 0 = 𝑡)
36		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ (𝐴[,]𝐵) ∧ 𝑡 = 0) → 𝑡 ∈ (𝐴[,]𝐵))
37	35, 36	eqeltrd 2838	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ (𝐴[,]𝐵) ∧ 𝑡 = 0) → 0 ∈ (𝐴[,]𝐵))
38	37	adantll 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ 𝑡 = 0) → 0 ∈ (𝐴[,]𝐵))
39	9	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ 𝑡 = 0) → ¬ 0 ∈ (𝐴[,]𝐵))
40	38, 39	pm2.65da 817	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → ¬ 𝑡 = 0)
41	40	neqned 2944	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ≠ 0)
42		fourierdlem44 46106	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (-π[,]π) ∧ 𝑡 ≠ 0) → (sin‘(𝑡 / 2)) ≠ 0)
43	32, 41, 42	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (sin‘(𝑡 / 2)) ≠ 0)
44	28, 29, 31, 43	mulne0d 11912	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (2 · (sin‘(𝑡 / 2))) ≠ 0)
45		eldifsn 4790	. . . . . . . . 9 ⊢ ((2 · (sin‘(𝑡 / 2))) ∈ (ℝ ∖ {0}) ↔ ((2 · (sin‘(𝑡 / 2))) ∈ ℝ ∧ (2 · (sin‘(𝑡 / 2))) ≠ 0))
46	27, 44, 45	sylanbrc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (2 · (sin‘(𝑡 / 2))) ∈ (ℝ ∖ {0}))
47	46, 18	fmptd 7133	. . . . . . 7 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (2 · (sin‘(𝑡 / 2)))):(𝐴[,]𝐵)⟶(ℝ ∖ {0}))
48		difss 4145	. . . . . . . . . 10 ⊢ (ℝ ∖ {0}) ⊆ ℝ
49		ax-resscn 11209	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
50	48, 49	sstri 4004	. . . . . . . . 9 ⊢ (ℝ ∖ {0}) ⊆ ℂ
51	50	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (ℝ ∖ {0}) ⊆ ℂ)
52	23, 49	sstrdi 4007	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
53		2cnd 12341	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℂ)
54		ssid 4017	. . . . . . . . . . 11 ⊢ ℂ ⊆ ℂ
55	54	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℂ ⊆ ℂ)
56	52, 53, 55	constcncfg 45827	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ 2) ∈ ((𝐴[,]𝐵)–cn→ℂ))
57		sincn 26502	. . . . . . . . . . 11 ⊢ sin ∈ (ℂ–cn→ℂ)
58	57	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → sin ∈ (ℂ–cn→ℂ))
59	52, 55	idcncfg 45828	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ 𝑡) ∈ ((𝐴[,]𝐵)–cn→ℂ))
60		eldifsn 4790	. . . . . . . . . . . . . 14 ⊢ (2 ∈ (ℂ ∖ {0}) ↔ (2 ∈ ℂ ∧ 2 ≠ 0))
61	28, 31, 60	sylanbrc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 2 ∈ (ℂ ∖ {0}))
62		eqid 2734	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ (𝐴[,]𝐵) ↦ 2) = (𝑡 ∈ (𝐴[,]𝐵) ↦ 2)
63	61, 62	fmptd 7133	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ 2):(𝐴[,]𝐵)⟶(ℂ ∖ {0}))
64		difssd 4146	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ)
65		cncfcdm 24937	. . . . . . . . . . . . 13 ⊢ (((ℂ ∖ {0}) ⊆ ℂ ∧ (𝑡 ∈ (𝐴[,]𝐵) ↦ 2) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → ((𝑡 ∈ (𝐴[,]𝐵) ↦ 2) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})) ↔ (𝑡 ∈ (𝐴[,]𝐵) ↦ 2):(𝐴[,]𝐵)⟶(ℂ ∖ {0})))
66	64, 56, 65	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑡 ∈ (𝐴[,]𝐵) ↦ 2) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})) ↔ (𝑡 ∈ (𝐴[,]𝐵) ↦ 2):(𝐴[,]𝐵)⟶(ℂ ∖ {0})))
67	63, 66	mpbird 257	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ 2) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})))
68	59, 67	divcncf 25495	. . . . . . . . . 10 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (𝑡 / 2)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
69	58, 68	cncfmpt1f 24953	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (sin‘(𝑡 / 2))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
70	56, 69	mulcncf 25493	. . . . . . . 8 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (2 · (sin‘(𝑡 / 2)))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
71		cncfcdm 24937	. . . . . . . 8 ⊢ (((ℝ ∖ {0}) ⊆ ℂ ∧ (𝑡 ∈ (𝐴[,]𝐵) ↦ (2 · (sin‘(𝑡 / 2)))) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → ((𝑡 ∈ (𝐴[,]𝐵) ↦ (2 · (sin‘(𝑡 / 2)))) ∈ ((𝐴[,]𝐵)–cn→(ℝ ∖ {0})) ↔ (𝑡 ∈ (𝐴[,]𝐵) ↦ (2 · (sin‘(𝑡 / 2)))):(𝐴[,]𝐵)⟶(ℝ ∖ {0})))
72	51, 70, 71	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((𝑡 ∈ (𝐴[,]𝐵) ↦ (2 · (sin‘(𝑡 / 2)))) ∈ ((𝐴[,]𝐵)–cn→(ℝ ∖ {0})) ↔ (𝑡 ∈ (𝐴[,]𝐵) ↦ (2 · (sin‘(𝑡 / 2)))):(𝐴[,]𝐵)⟶(ℝ ∖ {0})))
73	47, 72	mpbird 257	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ (𝐴[,]𝐵) ↦ (2 · (sin‘(𝑡 / 2)))) ∈ ((𝐴[,]𝐵)–cn→(ℝ ∖ {0})))
74	18, 3, 4, 20, 73	cncficcgt0 45843	. . . . 5 ⊢ (𝜑 → ∃𝑐 ∈ ℝ⁺ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2)))))
75		reelprrecn 11244	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
76	75	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) → ℝ ∈ {ℝ, ℂ})
77	1	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐹:ℝ⟶ℝ)
78	2	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑋 ∈ ℝ)
79		elioore 13413	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
80	79	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
81	78, 80	readdcld 11287	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ℝ)
82	77, 81	ffvelcdmd 7104	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℝ)
83	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℝ)
84	82, 83	resubcld 11688	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℝ)
85	84	recnd 11286	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℂ)
86	85	3ad2antl1 1184	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝐹‘(𝑋 + 𝑠)) − 𝐶) ∈ ℂ)
87	75	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
88	82	recnd 11286	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘(𝑋 + 𝑠)) ∈ ℂ)
89	5	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))):((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℝ)
90	2, 3	readdcld 11287	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 + 𝐴) ∈ ℝ)
91	90	rexrd 11308	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 + 𝐴) ∈ ℝ^*)
92	91	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝐴) ∈ ℝ^*)
93	2, 4	readdcld 11287	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 + 𝐵) ∈ ℝ)
94	93	rexrd 11308	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 + 𝐵) ∈ ℝ^*)
95	94	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝐵) ∈ ℝ^*)
96	3	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
97	96	rexrd 11308	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
98	4	rexrd 11308	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
99	98	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ^*)
100		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (𝐴(,)𝐵))
101		ioogtlb 45447	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑠)
102	97, 99, 100, 101	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑠)
103	96, 80, 78, 102	ltadd2dd 11417	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝐴) < (𝑋 + 𝑠))
104	4	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ)
105		iooltub 45462	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < 𝐵)
106	97, 99, 100, 105	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 < 𝐵)
107	80, 104, 78, 106	ltadd2dd 11417	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) < (𝑋 + 𝐵))
108	92, 95, 81, 103, 107	eliood 45450	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑋 + 𝑠) ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))
109	89, 108	ffvelcdmd 7104	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠)) ∈ ℝ)
110		eqid 2734	. . . . . . . . . . 11 ⊢ (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) = (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))
111	1, 2, 3, 4, 110, 5	fourierdlem28 46090	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠)))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠))))
112	83	recnd 11286	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℂ)
113		0red 11261	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
114		iooretop 24801	. . . . . . . . . . . . 13 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
115		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
116	115	tgioo2 24838	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
117	114, 116	eleqtri 2836	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ)
118	117	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
119	12	recnd 11286	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℂ)
120	87, 118, 119	dvmptconst 45870	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝐶)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 0))
121	87, 88, 109, 111, 112, 113, 120	dvmptsub 26019	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠)) − 0)))
122	109	recnd 11286	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠)) ∈ ℂ)
123	122	subid1d 11606	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠)) − 0) = ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠)))
124	123	mpteq2dva 5247	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠)) − 0)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠))))
125	121, 124	eqtrd 2774	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠))))
126	125	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) − 𝐶))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠))))
127	122	3ad2antl1 1184	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠)) ∈ ℂ)
128		2cnd 12341	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 2 ∈ ℂ)
129	79	recnd 11286	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℂ)
130	129	halfcld 12508	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (𝑠 / 2) ∈ ℂ)
131	130	sincld 16162	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (sin‘(𝑠 / 2)) ∈ ℂ)
132	128, 131	mulcld 11278	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
133	132	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (2 · (sin‘(𝑠 / 2))) ∈ ℂ)
134		fourierdlem68.e	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ)
135	134	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) → 𝐸 ∈ ℝ)
136		1re 11258	. . . . . . . . 9 ⊢ 1 ∈ ℝ
137	21, 136	remulcli 11274	. . . . . . . 8 ⊢ (2 · 1) ∈ ℝ
138	137	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) → (2 · 1) ∈ ℝ)
139		1red 11259	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) → 1 ∈ ℝ)
140		fourierdlem68.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℝ)
141	119	abscld 15471	. . . . . . . . 9 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ)
142	140, 141	readdcld 11287	. . . . . . . 8 ⊢ (𝜑 → (𝐷 + (abs‘𝐶)) ∈ ℝ)
143	142	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) → (𝐷 + (abs‘𝐶)) ∈ ℝ)
144		simpl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝜑)
145	144, 108	jca 511	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝜑 ∧ (𝑋 + 𝑠) ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))
146		eleq1 2826	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑋 + 𝑠) → (𝑡 ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) ↔ (𝑋 + 𝑠) ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))
147	146	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑡 = (𝑋 + 𝑠) → ((𝜑 ∧ 𝑡 ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) ↔ (𝜑 ∧ (𝑋 + 𝑠) ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))))
148		fveq2 6906	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑋 + 𝑠) → ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘𝑡) = ((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠)))
149	148	fveq2d 6910	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑋 + 𝑠) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘𝑡)) = (abs‘((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠))))
150	149	breq1d 5157	. . . . . . . . . . 11 ⊢ (𝑡 = (𝑋 + 𝑠) → ((abs‘((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘𝑡)) ≤ 𝐸 ↔ (abs‘((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠))) ≤ 𝐸))
151	147, 150	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑡 = (𝑋 + 𝑠) → (((𝜑 ∧ 𝑡 ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘𝑡)) ≤ 𝐸) ↔ ((𝜑 ∧ (𝑋 + 𝑠) ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠))) ≤ 𝐸)))
152		fourierdlem68.fdvbd	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘𝑡)) ≤ 𝐸)
153	151, 152	vtoclg 3553	. . . . . . . . 9 ⊢ ((𝑋 + 𝑠) ∈ ℝ → ((𝜑 ∧ (𝑋 + 𝑠) ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠))) ≤ 𝐸))
154	81, 145, 153	sylc 65	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠))) ≤ 𝐸)
155	154	3ad2antl1 1184	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠))) ≤ 𝐸)
156	128, 131	absmuld 15489	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (abs‘(2 · (sin‘(𝑠 / 2)))) = ((abs‘2) · (abs‘(sin‘(𝑠 / 2)))))
157		0le2 12365	. . . . . . . . . . . 12 ⊢ 0 ≤ 2
158		absid 15331	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2)
159	21, 157, 158	mp2an 692	. . . . . . . . . . 11 ⊢ (abs‘2) = 2
160	159	oveq1i 7440	. . . . . . . . . 10 ⊢ ((abs‘2) · (abs‘(sin‘(𝑠 / 2)))) = (2 · (abs‘(sin‘(𝑠 / 2))))
161	131	abscld 15471	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (abs‘(sin‘(𝑠 / 2))) ∈ ℝ)
162		1red 11259	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 1 ∈ ℝ)
163	21	a1i 11	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 2 ∈ ℝ)
164	157	a1i 11	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 0 ≤ 2)
165	79	rehalfcld 12510	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (𝑠 / 2) ∈ ℝ)
166		abssinbd 45245	. . . . . . . . . . . 12 ⊢ ((𝑠 / 2) ∈ ℝ → (abs‘(sin‘(𝑠 / 2))) ≤ 1)
167	165, 166	syl 17	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (abs‘(sin‘(𝑠 / 2))) ≤ 1)
168	161, 162, 163, 164, 167	lemul2ad 12205	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (2 · (abs‘(sin‘(𝑠 / 2)))) ≤ (2 · 1))
169	160, 168	eqbrtrid 5182	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → ((abs‘2) · (abs‘(sin‘(𝑠 / 2)))) ≤ (2 · 1))
170	156, 169	eqbrtrd 5169	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (abs‘(2 · (sin‘(𝑠 / 2)))) ≤ (2 · 1))
171	170	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (abs‘(2 · (sin‘(𝑠 / 2)))) ≤ (2 · 1))
172		abscosbd 45228	. . . . . . . . 9 ⊢ ((𝑠 / 2) ∈ ℝ → (abs‘(cos‘(𝑠 / 2))) ≤ 1)
173	100, 165, 172	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (abs‘(cos‘(𝑠 / 2))) ≤ 1)
174	173	3ad2antl1 1184	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (abs‘(cos‘(𝑠 / 2))) ≤ 1)
175	85	abscld 15471	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (abs‘((𝐹‘(𝑋 + 𝑠)) − 𝐶)) ∈ ℝ)
176	88	abscld 15471	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘(𝑋 + 𝑠))) ∈ ℝ)
177	112	abscld 15471	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (abs‘𝐶) ∈ ℝ)
178	176, 177	readdcld 11287	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((abs‘(𝐹‘(𝑋 + 𝑠))) + (abs‘𝐶)) ∈ ℝ)
179	140	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝐷 ∈ ℝ)
180	179, 177	readdcld 11287	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐷 + (abs‘𝐶)) ∈ ℝ)
181	88, 112	abs2dif2d 15493	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (abs‘((𝐹‘(𝑋 + 𝑠)) − 𝐶)) ≤ ((abs‘(𝐹‘(𝑋 + 𝑠))) + (abs‘𝐶)))
182		fveq2 6906	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑋 + 𝑠) → (𝐹‘𝑡) = (𝐹‘(𝑋 + 𝑠)))
183	182	fveq2d 6910	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑋 + 𝑠) → (abs‘(𝐹‘𝑡)) = (abs‘(𝐹‘(𝑋 + 𝑠))))
184	183	breq1d 5157	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑋 + 𝑠) → ((abs‘(𝐹‘𝑡)) ≤ 𝐷 ↔ (abs‘(𝐹‘(𝑋 + 𝑠))) ≤ 𝐷))
185	147, 184	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑋 + 𝑠) → (((𝜑 ∧ 𝑡 ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) → (abs‘(𝐹‘𝑡)) ≤ 𝐷) ↔ ((𝜑 ∧ (𝑋 + 𝑠) ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) → (abs‘(𝐹‘(𝑋 + 𝑠))) ≤ 𝐷)))
186		fourierdlem68.fbd	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) → (abs‘(𝐹‘𝑡)) ≤ 𝐷)
187	185, 186	vtoclg 3553	. . . . . . . . . . 11 ⊢ ((𝑋 + 𝑠) ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)) → ((𝜑 ∧ (𝑋 + 𝑠) ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) → (abs‘(𝐹‘(𝑋 + 𝑠))) ≤ 𝐷))
188	108, 145, 187	sylc 65	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘(𝑋 + 𝑠))) ≤ 𝐷)
189	176, 179, 177, 188	leadd1dd 11874	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((abs‘(𝐹‘(𝑋 + 𝑠))) + (abs‘𝐶)) ≤ (𝐷 + (abs‘𝐶)))
190	175, 178, 180, 181, 189	letrd 11415	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (abs‘((𝐹‘(𝑋 + 𝑠)) − 𝐶)) ≤ (𝐷 + (abs‘𝐶)))
191	190	3ad2antl1 1184	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (abs‘((𝐹‘(𝑋 + 𝑠)) − 𝐶)) ≤ (𝐷 + (abs‘𝐶)))
192	14	simpri 485	. . . . . . . 8 ⊢ (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (2 · (sin‘(𝑠 / 2))))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (cos‘(𝑠 / 2)))
193	192	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (2 · (sin‘(𝑠 / 2))))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (cos‘(𝑠 / 2))))
194	130	coscld 16163	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (cos‘(𝑠 / 2)) ∈ ℂ)
195	194	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (cos‘(𝑠 / 2)) ∈ ℂ)
196		simp2 1136	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) → 𝑐 ∈ ℝ⁺)
197		oveq1 7437	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑠 → (𝑡 / 2) = (𝑠 / 2))
198	197	fveq2d 6910	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑠 → (sin‘(𝑡 / 2)) = (sin‘(𝑠 / 2)))
199	198	oveq2d 7446	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑠 → (2 · (sin‘(𝑡 / 2))) = (2 · (sin‘(𝑠 / 2))))
200	199	fveq2d 6910	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑠 → (abs‘(2 · (sin‘(𝑡 / 2)))) = (abs‘(2 · (sin‘(𝑠 / 2)))))
201	200	breq2d 5159	. . . . . . . . . 10 ⊢ (𝑡 = 𝑠 → (𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2)))) ↔ 𝑐 ≤ (abs‘(2 · (sin‘(𝑠 / 2))))))
202	201	cbvralvw 3234	. . . . . . . . 9 ⊢ (∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2)))) ↔ ∀𝑠 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑠 / 2)))))
203		nfv 1911	. . . . . . . . . . 11 ⊢ Ⅎ𝑠𝜑
204		nfra1 3281	. . . . . . . . . . 11 ⊢ Ⅎ𝑠∀𝑠 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑠 / 2))))
205	203, 204	nfan 1896	. . . . . . . . . 10 ⊢ Ⅎ𝑠(𝜑 ∧ ∀𝑠 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑠 / 2)))))
206		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑠 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑠 / 2))))) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ∀𝑠 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑠 / 2)))))
207	6, 100	sselid 3992	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (𝐴[,]𝐵))
208	207	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑠 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑠 / 2))))) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (𝐴[,]𝐵))
209		rspa 3245	. . . . . . . . . . . 12 ⊢ ((∀𝑠 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑠 / 2)))) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑐 ≤ (abs‘(2 · (sin‘(𝑠 / 2)))))
210	206, 208, 209	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑠 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑠 / 2))))) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑐 ≤ (abs‘(2 · (sin‘(𝑠 / 2)))))
211	210	ex 412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑠 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑠 / 2))))) → (𝑠 ∈ (𝐴(,)𝐵) → 𝑐 ≤ (abs‘(2 · (sin‘(𝑠 / 2))))))
212	205, 211	ralrimi 3254	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑠 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑠 / 2))))) → ∀𝑠 ∈ (𝐴(,)𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑠 / 2)))))
213	202, 212	sylan2b 594	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) → ∀𝑠 ∈ (𝐴(,)𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑠 / 2)))))
214	213	3adant2 1130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) → ∀𝑠 ∈ (𝐴(,)𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑠 / 2)))))
215		eqid 2734	. . . . . . 7 ⊢ (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))) = (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))
216	76, 86, 126, 127, 133, 135, 138, 139, 143, 155, 171, 174, 191, 193, 195, 196, 214, 215	dvdivbd 45878	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ⁺ ∧ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2))))) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ (𝐴(,)𝐵)(abs‘((ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))‘𝑠)) ≤ 𝑏)
217	216	rexlimdv3a 3156	. . . . 5 ⊢ (𝜑 → (∃𝑐 ∈ ℝ⁺ ∀𝑡 ∈ (𝐴[,]𝐵)𝑐 ≤ (abs‘(2 · (sin‘(𝑡 / 2)))) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ (𝐴(,)𝐵)(abs‘((ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))‘𝑠)) ≤ 𝑏))
218	74, 217	mpd 15	. . . 4 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ (𝐴(,)𝐵)(abs‘((ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))‘𝑠)) ≤ 𝑏)
219		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑠ℝ
220		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑠 D
221		nfmpt1 5255	. . . . . . . . . 10 ⊢ Ⅎ𝑠(𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))
222	13, 221	nfcxfr 2900	. . . . . . . . 9 ⊢ Ⅎ𝑠𝑂
223	219, 220, 222	nfov 7460	. . . . . . . 8 ⊢ Ⅎ𝑠(ℝ D 𝑂)
224	223	nfdm 5964	. . . . . . 7 ⊢ Ⅎ𝑠dom (ℝ D 𝑂)
225		nfcv 2902	. . . . . . 7 ⊢ Ⅎ𝑠(𝐴(,)𝐵)
226	224, 225	raleqf 3350	. . . . . 6 ⊢ (dom (ℝ D 𝑂) = (𝐴(,)𝐵) → (∀𝑠 ∈ dom (ℝ D 𝑂)(abs‘((ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))‘𝑠)) ≤ 𝑏 ↔ ∀𝑠 ∈ (𝐴(,)𝐵)(abs‘((ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))‘𝑠)) ≤ 𝑏))
227	17, 226	syl 17	. . . . 5 ⊢ (𝜑 → (∀𝑠 ∈ dom (ℝ D 𝑂)(abs‘((ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))‘𝑠)) ≤ 𝑏 ↔ ∀𝑠 ∈ (𝐴(,)𝐵)(abs‘((ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))‘𝑠)) ≤ 𝑏))
228	227	rexbidv 3176	. . . 4 ⊢ (𝜑 → (∃𝑏 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑂)(abs‘((ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))‘𝑠)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑠 ∈ (𝐴(,)𝐵)(abs‘((ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))‘𝑠)) ≤ 𝑏))
229	218, 228	mpbird 257	. . 3 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑂)(abs‘((ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))‘𝑠)) ≤ 𝑏)
230	13	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑂 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))
231	230	oveq2d 7446	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝑂) = (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))))
232	231	fveq1d 6908	. . . . . 6 ⊢ (𝜑 → ((ℝ D 𝑂)‘𝑠) = ((ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))‘𝑠))
233	232	fveq2d 6910	. . . . 5 ⊢ (𝜑 → (abs‘((ℝ D 𝑂)‘𝑠)) = (abs‘((ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))‘𝑠)))
234	233	breq1d 5157	. . . 4 ⊢ (𝜑 → ((abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑏 ↔ (abs‘((ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))‘𝑠)) ≤ 𝑏))
235	234	rexralbidv 3220	. . 3 ⊢ (𝜑 → (∃𝑏 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑂)(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑂)(abs‘((ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))))‘𝑠)) ≤ 𝑏))
236	229, 235	mpbird 257	. 2 ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑂)(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑏)
237	17, 236	jca 511	1 ⊢ (𝜑 → (dom (ℝ D 𝑂) = (𝐴(,)𝐵) ∧ ∃𝑏 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑂)(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑏))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ∖ cdif 3959 ⊆ wss 3962 {csn 4630 {cpr 4632 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5688 ran crn 5689 ↾ cres 5690 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 ℝ^*cxr 11291 < clt 11292 ≤ cle 11293 − cmin 11489 -cneg 11490 / cdiv 11917 2c2 12318 ℝ⁺crp 13031 (,)cioo 13383 [,]cicc 13386 ↑cexp 14098 abscabs 15269 sincsin 16095 cosccos 16096 πcpi 16098 ↾_t crest 17466 TopOpenctopn 17467 topGenctg 17483 ℂ_fldccnfld 21381 –cn→ccncf 24915 D cdv 25912

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-t1 23337 df-haus 23338 df-cmp 23410 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-limc 25915 df-dv 25916

This theorem is referenced by: fourierdlem80 46141

W3C validator