sinccvglem - Mathbox for Paul Chapman

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Theorem sinccvglem 32167

Description: ((sin‘𝑥) / 𝑥) ⇝ 1 as (real) 𝑥 ⇝ 0. (Contributed by Paul Chapman, 10-Nov-2012.) (Revised by Mario Carneiro, 21-May-2014.)

**Hypotheses**
Ref	Expression
sinccvg.1	⊢ (𝜑 → 𝐹:ℕ⟶(ℝ ∖ {0}))
sinccvg.2	⊢ (𝜑 → 𝐹 ⇝ 0)
sinccvg.3	⊢ 𝐺 = (𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥))
sinccvg.4	⊢ 𝐻 = (𝑥 ∈ ℂ ↦ (1 − ((𝑥↑2) / 3)))
sinccvg.5	⊢ (𝜑 → 𝑀 ∈ ℕ)
sinccvg.6	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) < 1)

**Assertion**
Ref	Expression
sinccvglem	⊢ (𝜑 → (𝐺 ∘ 𝐹) ⇝ 1)

Distinct variable groups: 𝑥,𝑘,𝐹 𝑘,𝐻 𝑘,𝑀 𝜑,𝑘 𝑘,𝐺 Allowed substitution hints: 𝜑(𝑥) 𝐺(𝑥) 𝐻(𝑥) 𝑀(𝑥)

Proof of Theorem sinccvglem Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2778	. 2 ⊢ (ℤ_≥‘𝑀) = (ℤ_≥‘𝑀)
2		sinccvg.5	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3	2	nnzd 11837	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4		sinccvg.2	. . . 4 ⊢ (𝜑 → 𝐹 ⇝ 0)
5		sinccvg.4	. . . . . 6 ⊢ 𝐻 = (𝑥 ∈ ℂ ↦ (1 − ((𝑥↑2) / 3)))
6	5	funmpt2 6176	. . . . 5 ⊢ Fun 𝐻
7		sinccvg.1	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ ∖ {0}))
8		nnex 11385	. . . . . 6 ⊢ ℕ ∈ V
9		fex 6763	. . . . . 6 ⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ ℕ ∈ V) → 𝐹 ∈ V)
10	7, 8, 9	sylancl 580	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
11		cofunexg 7411	. . . . 5 ⊢ ((Fun 𝐻 ∧ 𝐹 ∈ V) → (𝐻 ∘ 𝐹) ∈ V)
12	6, 10, 11	sylancr 581	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ V)
13	7	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝐹:ℕ⟶(ℝ ∖ {0}))
14		eluznn 12069	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℕ)
15	2, 14	sylan 575	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℕ)
16	13, 15	ffvelrnd 6626	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ (ℝ ∖ {0}))
17		eldifsn 4550	. . . . . . 7 ⊢ ((𝐹‘𝑘) ∈ (ℝ ∖ {0}) ↔ ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) ≠ 0))
18	16, 17	sylib 210	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) ≠ 0))
19	18	simpld 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)
20	19	recnd 10407	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
21		ax-1cn 10332	. . . . . 6 ⊢ 1 ∈ ℂ
22		sqcl 13247	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥↑2) ∈ ℂ)
23		3cn 11460	. . . . . . . 8 ⊢ 3 ∈ ℂ
24		3ne0 11492	. . . . . . . 8 ⊢ 3 ≠ 0
25		divcl 11041	. . . . . . . 8 ⊢ (((𝑥↑2) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → ((𝑥↑2) / 3) ∈ ℂ)
26	23, 24, 25	mp3an23 1526	. . . . . . 7 ⊢ ((𝑥↑2) ∈ ℂ → ((𝑥↑2) / 3) ∈ ℂ)
27	22, 26	syl 17	. . . . . 6 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) / 3) ∈ ℂ)
28		subcl 10623	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ ((𝑥↑2) / 3) ∈ ℂ) → (1 − ((𝑥↑2) / 3)) ∈ ℂ)
29	21, 27, 28	sylancr 581	. . . . 5 ⊢ (𝑥 ∈ ℂ → (1 − ((𝑥↑2) / 3)) ∈ ℂ)
30	5, 29	fmpti 6648	. . . 4 ⊢ 𝐻:ℂ⟶ℂ
31		eqid 2778	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
32	31	cnfldtopon 22998	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
33	32	a1i 11	. . . . . . . 8 ⊢ (⊤ → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
34		1cnd 10373	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
35	33, 33, 34	cnmptc 21878	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 1) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
36	31	sqcn 23089	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑2)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
37	36	a1i 11	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (𝑥↑2)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
38	31	divccn 23088	. . . . . . . . . . 11 ⊢ ((3 ∈ ℂ ∧ 3 ≠ 0) → (𝑦 ∈ ℂ ↦ (𝑦 / 3)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
39	23, 24, 38	mp2an 682	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℂ ↦ (𝑦 / 3)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
40	39	a1i 11	. . . . . . . . 9 ⊢ (⊤ → (𝑦 ∈ ℂ ↦ (𝑦 / 3)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
41		oveq1 6931	. . . . . . . . 9 ⊢ (𝑦 = (𝑥↑2) → (𝑦 / 3) = ((𝑥↑2) / 3))
42	33, 37, 33, 40, 41	cnmpt11 21879	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((𝑥↑2) / 3)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
43	31	subcn 23081	. . . . . . . . 9 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
44	43	a1i 11	. . . . . . . 8 ⊢ (⊤ → − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
45	33, 35, 42, 44	cnmpt12f 21882	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (1 − ((𝑥↑2) / 3))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
46	45	mptru 1609	. . . . . 6 ⊢ (𝑥 ∈ ℂ ↦ (1 − ((𝑥↑2) / 3))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
47	31	cncfcn1 23125	. . . . . 6 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
48	46, 5, 47	3eltr4i 2872	. . . . 5 ⊢ 𝐻 ∈ (ℂ–cn→ℂ)
49		cncfi 23109	. . . . 5 ⊢ ((𝐻 ∈ (ℂ–cn→ℂ) ∧ 0 ∈ ℂ ∧ 𝑦 ∈ ℝ⁺) → ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℂ ((abs‘(𝑤 − 0)) < 𝑧 → (abs‘((𝐻‘𝑤) − (𝐻‘0))) < 𝑦))
50	48, 49	mp3an1 1521	. . . 4 ⊢ ((0 ∈ ℂ ∧ 𝑦 ∈ ℝ⁺) → ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℂ ((abs‘(𝑤 − 0)) < 𝑧 → (abs‘((𝐻‘𝑤) − (𝐻‘0))) < 𝑦))
51		fvco3 6537	. . . . . 6 ⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ 𝑘 ∈ ℕ) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘)))
52	7, 51	sylan 575	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘)))
53	15, 52	syldan 585	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘)))
54	1, 4, 12, 3, 20, 30, 50, 53	climcn1lem 14745	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ⇝ (𝐻‘0))
55		0cn 10370	. . . 4 ⊢ 0 ∈ ℂ
56		sq0i 13279	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥↑2) = 0)
57	56	oveq1d 6939	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥↑2) / 3) = (0 / 3))
58	23, 24	div0i 11111	. . . . . . . 8 ⊢ (0 / 3) = 0
59	57, 58	syl6eq 2830	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝑥↑2) / 3) = 0)
60	59	oveq2d 6940	. . . . . 6 ⊢ (𝑥 = 0 → (1 − ((𝑥↑2) / 3)) = (1 − 0))
61		1m0e1 11507	. . . . . 6 ⊢ (1 − 0) = 1
62	60, 61	syl6eq 2830	. . . . 5 ⊢ (𝑥 = 0 → (1 − ((𝑥↑2) / 3)) = 1)
63		1ex 10374	. . . . 5 ⊢ 1 ∈ V
64	62, 5, 63	fvmpt 6544	. . . 4 ⊢ (0 ∈ ℂ → (𝐻‘0) = 1)
65	55, 64	ax-mp 5	. . 3 ⊢ (𝐻‘0) = 1
66	54, 65	syl6breq 4929	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ⇝ 1)
67		sinccvg.3	. . . 4 ⊢ 𝐺 = (𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥))
68	67	funmpt2 6176	. . 3 ⊢ Fun 𝐺
69		cofunexg 7411	. . 3 ⊢ ((Fun 𝐺 ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V)
70	68, 10, 69	sylancr 581	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
71		oveq1 6931	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑘) → (𝑥↑2) = ((𝐹‘𝑘)↑2))
72	71	oveq1d 6939	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → ((𝑥↑2) / 3) = (((𝐹‘𝑘)↑2) / 3))
73	72	oveq2d 6940	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑘) → (1 − ((𝑥↑2) / 3)) = (1 − (((𝐹‘𝑘)↑2) / 3)))
74		ovex 6956	. . . . . 6 ⊢ (1 − (((𝐹‘𝑘)↑2) / 3)) ∈ V
75	73, 5, 74	fvmpt 6544	. . . . 5 ⊢ ((𝐹‘𝑘) ∈ ℂ → (𝐻‘(𝐹‘𝑘)) = (1 − (((𝐹‘𝑘)↑2) / 3)))
76	20, 75	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐻‘(𝐹‘𝑘)) = (1 − (((𝐹‘𝑘)↑2) / 3)))
77	53, 76	eqtrd 2814	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐻 ∘ 𝐹)‘𝑘) = (1 − (((𝐹‘𝑘)↑2) / 3)))
78		1re 10378	. . . 4 ⊢ 1 ∈ ℝ
79	19	resqcld 13360	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐹‘𝑘)↑2) ∈ ℝ)
80		3nn 11458	. . . . 5 ⊢ 3 ∈ ℕ
81		nndivre 11420	. . . . 5 ⊢ ((((𝐹‘𝑘)↑2) ∈ ℝ ∧ 3 ∈ ℕ) → (((𝐹‘𝑘)↑2) / 3) ∈ ℝ)
82	79, 80, 81	sylancl 580	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((𝐹‘𝑘)↑2) / 3) ∈ ℝ)
83		resubcl 10689	. . . 4 ⊢ ((1 ∈ ℝ ∧ (((𝐹‘𝑘)↑2) / 3) ∈ ℝ) → (1 − (((𝐹‘𝑘)↑2) / 3)) ∈ ℝ)
84	78, 82, 83	sylancr 581	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) ∈ ℝ)
85	77, 84	eqeltrd 2859	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐻 ∘ 𝐹)‘𝑘) ∈ ℝ)
86		fvco3 6537	. . . . . 6 ⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ 𝑘 ∈ ℕ) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘)))
87	7, 86	sylan 575	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘)))
88	15, 87	syldan 585	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘)))
89		fveq2 6448	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → (sin‘𝑥) = (sin‘(𝐹‘𝑘)))
90		id 22	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → 𝑥 = (𝐹‘𝑘))
91	89, 90	oveq12d 6942	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑘) → ((sin‘𝑥) / 𝑥) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
92		ovex 6956	. . . . . 6 ⊢ ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) ∈ V
93	91, 67, 92	fvmpt 6544	. . . . 5 ⊢ ((𝐹‘𝑘) ∈ (ℝ ∖ {0}) → (𝐺‘(𝐹‘𝑘)) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
94	16, 93	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐺‘(𝐹‘𝑘)) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
95	88, 94	eqtrd 2814	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
96	19	resincld 15279	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(𝐹‘𝑘)) ∈ ℝ)
97	18	simprd 491	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ≠ 0)
98	96, 19, 97	redivcld 11205	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) ∈ ℝ)
99	95, 98	eqeltrd 2859	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) ∈ ℝ)
100		1cnd 10373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 1 ∈ ℂ)
101	82	recnd 10407	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((𝐹‘𝑘)↑2) / 3) ∈ ℂ)
102	20	abscld 14587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
103	102	recnd 10407	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℂ)
104	100, 101, 103	subdird 10834	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((1 − (((𝐹‘𝑘)↑2) / 3)) · (abs‘(𝐹‘𝑘))) = ((1 · (abs‘(𝐹‘𝑘))) − ((((𝐹‘𝑘)↑2) / 3) · (abs‘(𝐹‘𝑘)))))
105	103	mulid2d 10397	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 · (abs‘(𝐹‘𝑘))) = (abs‘(𝐹‘𝑘)))
106		df-3 11443	. . . . . . . . . . . . 13 ⊢ 3 = (2 + 1)
107	106	oveq2i 6935	. . . . . . . . . . . 12 ⊢ ((abs‘(𝐹‘𝑘))↑3) = ((abs‘(𝐹‘𝑘))↑(2 + 1))
108		2nn0 11665	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ₀
109		expp1 13189	. . . . . . . . . . . . . 14 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℂ ∧ 2 ∈ ℕ₀) → ((abs‘(𝐹‘𝑘))↑(2 + 1)) = (((abs‘(𝐹‘𝑘))↑2) · (abs‘(𝐹‘𝑘))))
110	103, 108, 109	sylancl 580	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘))↑(2 + 1)) = (((abs‘(𝐹‘𝑘))↑2) · (abs‘(𝐹‘𝑘))))
111		absresq 14453	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) ∈ ℝ → ((abs‘(𝐹‘𝑘))↑2) = ((𝐹‘𝑘)↑2))
112	19, 111	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘))↑2) = ((𝐹‘𝑘)↑2))
113	112	oveq1d 6939	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((abs‘(𝐹‘𝑘))↑2) · (abs‘(𝐹‘𝑘))) = (((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))))
114	110, 113	eqtrd 2814	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘))↑(2 + 1)) = (((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))))
115	107, 114	syl5eq 2826	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘))↑3) = (((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))))
116	115	oveq1d 6939	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((abs‘(𝐹‘𝑘))↑3) / 3) = ((((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))) / 3))
117	79	recnd 10407	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐹‘𝑘)↑2) ∈ ℂ)
118	23	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 3 ∈ ℂ)
119	24	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 3 ≠ 0)
120	117, 103, 118, 119	div23d 11190	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))) / 3) = ((((𝐹‘𝑘)↑2) / 3) · (abs‘(𝐹‘𝑘))))
121	116, 120	eqtr2d 2815	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((((𝐹‘𝑘)↑2) / 3) · (abs‘(𝐹‘𝑘))) = (((abs‘(𝐹‘𝑘))↑3) / 3))
122	105, 121	oveq12d 6942	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((1 · (abs‘(𝐹‘𝑘))) − ((((𝐹‘𝑘)↑2) / 3) · (abs‘(𝐹‘𝑘)))) = ((abs‘(𝐹‘𝑘)) − (((abs‘(𝐹‘𝑘))↑3) / 3)))
123	104, 122	eqtrd 2814	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((1 − (((𝐹‘𝑘)↑2) / 3)) · (abs‘(𝐹‘𝑘))) = ((abs‘(𝐹‘𝑘)) − (((abs‘(𝐹‘𝑘))↑3) / 3)))
124	20, 97	absrpcld 14599	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℝ⁺)
125	124	rpgt0d 12188	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 0 < (abs‘(𝐹‘𝑘)))
126		sinccvg.6	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) < 1)
127		ltle 10467	. . . . . . . . . . . 12 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘(𝐹‘𝑘)) < 1 → (abs‘(𝐹‘𝑘)) ≤ 1))
128	102, 78, 127	sylancl 580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) < 1 → (abs‘(𝐹‘𝑘)) ≤ 1))
129	126, 128	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ≤ 1)
130		0xr 10425	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
131		elioc2 12552	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → ((abs‘(𝐹‘𝑘)) ∈ (0(,]1) ↔ ((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 0 < (abs‘(𝐹‘𝑘)) ∧ (abs‘(𝐹‘𝑘)) ≤ 1)))
132	130, 78, 131	mp2an 682	. . . . . . . . . 10 ⊢ ((abs‘(𝐹‘𝑘)) ∈ (0(,]1) ↔ ((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 0 < (abs‘(𝐹‘𝑘)) ∧ (abs‘(𝐹‘𝑘)) ≤ 1))
133	102, 125, 129, 132	syl3anbrc 1400	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ (0(,]1))
134		sin01bnd 15321	. . . . . . . . 9 ⊢ ((abs‘(𝐹‘𝑘)) ∈ (0(,]1) → (((abs‘(𝐹‘𝑘)) − (((abs‘(𝐹‘𝑘))↑3) / 3)) < (sin‘(abs‘(𝐹‘𝑘))) ∧ (sin‘(abs‘(𝐹‘𝑘))) < (abs‘(𝐹‘𝑘))))
135	133, 134	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((abs‘(𝐹‘𝑘)) − (((abs‘(𝐹‘𝑘))↑3) / 3)) < (sin‘(abs‘(𝐹‘𝑘))) ∧ (sin‘(abs‘(𝐹‘𝑘))) < (abs‘(𝐹‘𝑘))))
136	135	simpld 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) − (((abs‘(𝐹‘𝑘))↑3) / 3)) < (sin‘(abs‘(𝐹‘𝑘))))
137	123, 136	eqbrtrd 4910	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((1 − (((𝐹‘𝑘)↑2) / 3)) · (abs‘(𝐹‘𝑘))) < (sin‘(abs‘(𝐹‘𝑘))))
138	102	resincld 15279	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(abs‘(𝐹‘𝑘))) ∈ ℝ)
139	84, 138, 124	ltmuldivd 12232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((1 − (((𝐹‘𝑘)↑2) / 3)) · (abs‘(𝐹‘𝑘))) < (sin‘(abs‘(𝐹‘𝑘))) ↔ (1 − (((𝐹‘𝑘)↑2) / 3)) < ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘)))))
140	137, 139	mpbid 224	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) < ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))))
141		fveq2 6448	. . . . . . . 8 ⊢ ((abs‘(𝐹‘𝑘)) = (𝐹‘𝑘) → (sin‘(abs‘(𝐹‘𝑘))) = (sin‘(𝐹‘𝑘)))
142		id 22	. . . . . . . 8 ⊢ ((abs‘(𝐹‘𝑘)) = (𝐹‘𝑘) → (abs‘(𝐹‘𝑘)) = (𝐹‘𝑘))
143	141, 142	oveq12d 6942	. . . . . . 7 ⊢ ((abs‘(𝐹‘𝑘)) = (𝐹‘𝑘) → ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
144	143	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) = (𝐹‘𝑘) → ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘))))
145		sinneg 15282	. . . . . . . . . 10 ⊢ ((𝐹‘𝑘) ∈ ℂ → (sin‘-(𝐹‘𝑘)) = -(sin‘(𝐹‘𝑘)))
146	20, 145	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘-(𝐹‘𝑘)) = -(sin‘(𝐹‘𝑘)))
147	146	oveq1d 6939	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘-(𝐹‘𝑘)) / -(𝐹‘𝑘)) = (-(sin‘(𝐹‘𝑘)) / -(𝐹‘𝑘)))
148	96	recnd 10407	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(𝐹‘𝑘)) ∈ ℂ)
149	148, 20, 97	div2negd 11168	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (-(sin‘(𝐹‘𝑘)) / -(𝐹‘𝑘)) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
150	147, 149	eqtrd 2814	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘-(𝐹‘𝑘)) / -(𝐹‘𝑘)) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
151		fveq2 6448	. . . . . . . . 9 ⊢ ((abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘) → (sin‘(abs‘(𝐹‘𝑘))) = (sin‘-(𝐹‘𝑘)))
152		id 22	. . . . . . . . 9 ⊢ ((abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘) → (abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘))
153	151, 152	oveq12d 6942	. . . . . . . 8 ⊢ ((abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘) → ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) = ((sin‘-(𝐹‘𝑘)) / -(𝐹‘𝑘)))
154	153	eqeq1d 2780	. . . . . . 7 ⊢ ((abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘) → (((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) ↔ ((sin‘-(𝐹‘𝑘)) / -(𝐹‘𝑘)) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘))))
155	150, 154	syl5ibrcom 239	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘) → ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘))))
156	19	absord 14566	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) = (𝐹‘𝑘) ∨ (abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘)))
157	144, 155, 156	mpjaod 849	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
158	140, 157	breqtrd 4914	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) < ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
159	84, 98, 158	ltled 10526	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) ≤ ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
160	159, 77, 95	3brtr4d 4920	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐻 ∘ 𝐹)‘𝑘) ≤ ((𝐺 ∘ 𝐹)‘𝑘))
161	78	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 1 ∈ ℝ)
162	135	simprd 491	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(abs‘(𝐹‘𝑘))) < (abs‘(𝐹‘𝑘)))
163	103	mulid1d 10396	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) · 1) = (abs‘(𝐹‘𝑘)))
164	162, 163	breqtrrd 4916	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(abs‘(𝐹‘𝑘))) < ((abs‘(𝐹‘𝑘)) · 1))
165	138, 161, 124	ltdivmuld 12236	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) < 1 ↔ (sin‘(abs‘(𝐹‘𝑘))) < ((abs‘(𝐹‘𝑘)) · 1)))
166	164, 165	mpbird 249	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) < 1)
167	157, 166	eqbrtrrd 4912	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) < 1)
168	98, 161, 167	ltled 10526	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) ≤ 1)
169	95, 168	eqbrtrd 4910	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) ≤ 1)
170	1, 3, 66, 70, 85, 99, 160, 169	climsqz 14783	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ⇝ 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ⊤wtru 1602 ∈ wcel 2107 ≠ wne 2969 ∀wral 3090 ∃wrex 3091 Vcvv 3398 ∖ cdif 3789 {csn 4398 class class class wbr 4888 ↦ cmpt 4967 ∘ ccom 5361 Fun wfun 6131 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 ℝcr 10273 0cc0 10274 1c1 10275 + caddc 10277 · cmul 10279 ℝ^*cxr 10412 < clt 10413 ≤ cle 10414 − cmin 10608 -cneg 10609 / cdiv 11034 ℕcn 11378 2c2 11434 3c3 11435 ℕ₀cn0 11646 ℤ_≥cuz 11996 ℝ⁺crp 12141 (,]cioc 12492 ↑cexp 13182 abscabs 14385 ⇝ cli 14627 sincsin 15200 TopOpenctopn 16472 ℂ_fldccnfld 20146 TopOnctopon 21126 Cn ccn 21440 ×_t ctx 21776 –cn→ccncf 23091

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-q 12100 df-rp 12142 df-xneg 12261 df-xadd 12262 df-xmul 12263 df-ioc 12496 df-ico 12497 df-icc 12498 df-fz 12648 df-fzo 12789 df-fl 12916 df-seq 13124 df-exp 13183 df-fac 13383 df-hash 13440 df-shft 14218 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-limsup 14614 df-clim 14631 df-rlim 14632 df-sum 14829 df-ef 15204 df-sin 15206 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-starv 16357 df-sca 16358 df-vsca 16359 df-ip 16360 df-tset 16361 df-ple 16362 df-ds 16364 df-unif 16365 df-hom 16366 df-cco 16367 df-rest 16473 df-topn 16474 df-0g 16492 df-gsum 16493 df-topgen 16494 df-pt 16495 df-prds 16498 df-xrs 16552 df-qtop 16557 df-imas 16558 df-xps 16560 df-mre 16636 df-mrc 16637 df-acs 16639 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-submnd 17726 df-mulg 17932 df-cntz 18137 df-cmn 18585 df-psmet 20138 df-xmet 20139 df-met 20140 df-bl 20141 df-mopn 20142 df-cnfld 20147 df-top 21110 df-topon 21127 df-topsp 21149 df-bases 21162 df-cn 21443 df-cnp 21444 df-tx 21778 df-hmeo 21971 df-xms 22537 df-ms 22538 df-tms 22539 df-cncf 23093

This theorem is referenced by: sinccvg 32168

W3C validator