sinccvglem - Mathbox for Paul Chapman

	Mathbox for Paul Chapman		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > sinccvglem		Structured version Visualization version GIF version

Theorem sinccvglem 35640

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 2740	. 2 ⊢ (ℤ_≥‘𝑀) = (ℤ_≥‘𝑀)
2		sinccvg.5	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3	2	nnzd 12666	. 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 6617	. . . . 5 ⊢ Fun 𝐻
7		sinccvg.1	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ ∖ {0}))
8		nnex 12299	. . . . . 6 ⊢ ℕ ∈ V
9		fex 7263	. . . . . 6 ⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ ℕ ∈ V) → 𝐹 ∈ V)
10	7, 8, 9	sylancl 585	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
11		cofunexg 7989	. . . . 5 ⊢ ((Fun 𝐻 ∧ 𝐹 ∈ V) → (𝐻 ∘ 𝐹) ∈ V)
12	6, 10, 11	sylancr 586	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ V)
13	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝐹:ℕ⟶(ℝ ∖ {0}))
14		eluznn 12983	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℕ)
15	2, 14	sylan 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℕ)
16	13, 15	ffvelcdmd 7119	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ (ℝ ∖ {0}))
17		eldifsn 4811	. . . . . . 7 ⊢ ((𝐹‘𝑘) ∈ (ℝ ∖ {0}) ↔ ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) ≠ 0))
18	16, 17	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) ≠ 0))
19	18	simpld 494	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)
20	19	recnd 11318	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
21		ax-1cn 11242	. . . . . 6 ⊢ 1 ∈ ℂ
22		sqcl 14168	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥↑2) ∈ ℂ)
23		3cn 12374	. . . . . . . 8 ⊢ 3 ∈ ℂ
24		3ne0 12399	. . . . . . . 8 ⊢ 3 ≠ 0
25		divcl 11955	. . . . . . . 8 ⊢ (((𝑥↑2) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → ((𝑥↑2) / 3) ∈ ℂ)
26	23, 24, 25	mp3an23 1453	. . . . . . 7 ⊢ ((𝑥↑2) ∈ ℂ → ((𝑥↑2) / 3) ∈ ℂ)
27	22, 26	syl 17	. . . . . 6 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) / 3) ∈ ℂ)
28		subcl 11535	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ ((𝑥↑2) / 3) ∈ ℂ) → (1 − ((𝑥↑2) / 3)) ∈ ℂ)
29	21, 27, 28	sylancr 586	. . . . 5 ⊢ (𝑥 ∈ ℂ → (1 − ((𝑥↑2) / 3)) ∈ ℂ)
30	5, 29	fmpti 7146	. . . 4 ⊢ 𝐻:ℂ⟶ℂ
31		eqid 2740	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
32	31	cnfldtopon 24824	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
33	32	a1i 11	. . . . . . . 8 ⊢ (⊤ → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
34		1cnd 11285	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
35	33, 33, 34	cnmptc 23691	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 1) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
36	31	sqcn 24919	. . . . . . . . . 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 24916	. . . . . . . . . . 11 ⊢ ((3 ∈ ℂ ∧ 3 ≠ 0) → (𝑦 ∈ ℂ ↦ (𝑦 / 3)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
39	23, 24, 38	mp2an 691	. . . . . . . . . 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 7455	. . . . . . . . 9 ⊢ (𝑦 = (𝑥↑2) → (𝑦 / 3) = ((𝑥↑2) / 3))
42	33, 37, 33, 40, 41	cnmpt11 23692	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((𝑥↑2) / 3)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
43	31	subcn 24907	. . . . . . . . 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 23695	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (1 − ((𝑥↑2) / 3))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
46	45	mptru 1544	. . . . . 6 ⊢ (𝑥 ∈ ℂ ↦ (1 − ((𝑥↑2) / 3))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
47	31	cncfcn1 24956	. . . . . 6 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
48	46, 5, 47	3eltr4i 2857	. . . . 5 ⊢ 𝐻 ∈ (ℂ–cn→ℂ)
49		cncfi 24939	. . . . 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 1448	. . . 4 ⊢ ((0 ∈ ℂ ∧ 𝑦 ∈ ℝ⁺) → ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℂ ((abs‘(𝑤 − 0)) < 𝑧 → (abs‘((𝐻‘𝑤) − (𝐻‘0))) < 𝑦))
51		fvco3 7021	. . . . . 6 ⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ 𝑘 ∈ ℕ) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘)))
52	7, 51	sylan 579	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘)))
53	15, 52	syldan 590	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘)))
54	1, 4, 12, 3, 20, 30, 50, 53	climcn1lem 15649	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ⇝ (𝐻‘0))
55		0cn 11282	. . . 4 ⊢ 0 ∈ ℂ
56		sq0i 14242	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥↑2) = 0)
57	56	oveq1d 7463	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥↑2) / 3) = (0 / 3))
58	23, 24	div0i 12028	. . . . . . . 8 ⊢ (0 / 3) = 0
59	57, 58	eqtrdi 2796	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝑥↑2) / 3) = 0)
60	59	oveq2d 7464	. . . . . 6 ⊢ (𝑥 = 0 → (1 − ((𝑥↑2) / 3)) = (1 − 0))
61		1m0e1 12414	. . . . . 6 ⊢ (1 − 0) = 1
62	60, 61	eqtrdi 2796	. . . . 5 ⊢ (𝑥 = 0 → (1 − ((𝑥↑2) / 3)) = 1)
63		1ex 11286	. . . . 5 ⊢ 1 ∈ V
64	62, 5, 63	fvmpt 7029	. . . 4 ⊢ (0 ∈ ℂ → (𝐻‘0) = 1)
65	55, 64	ax-mp 5	. . 3 ⊢ (𝐻‘0) = 1
66	54, 65	breqtrdi 5207	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ⇝ 1)
67		sinccvg.3	. . . 4 ⊢ 𝐺 = (𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥))
68	67	funmpt2 6617	. . 3 ⊢ Fun 𝐺
69		cofunexg 7989	. . 3 ⊢ ((Fun 𝐺 ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V)
70	68, 10, 69	sylancr 586	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
71		oveq1 7455	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑘) → (𝑥↑2) = ((𝐹‘𝑘)↑2))
72	71	oveq1d 7463	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → ((𝑥↑2) / 3) = (((𝐹‘𝑘)↑2) / 3))
73	72	oveq2d 7464	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑘) → (1 − ((𝑥↑2) / 3)) = (1 − (((𝐹‘𝑘)↑2) / 3)))
74		ovex 7481	. . . . . 6 ⊢ (1 − (((𝐹‘𝑘)↑2) / 3)) ∈ V
75	73, 5, 74	fvmpt 7029	. . . . 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 2780	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐻 ∘ 𝐹)‘𝑘) = (1 − (((𝐹‘𝑘)↑2) / 3)))
78		1re 11290	. . . 4 ⊢ 1 ∈ ℝ
79	19	resqcld 14175	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐹‘𝑘)↑2) ∈ ℝ)
80		3nn 12372	. . . . 5 ⊢ 3 ∈ ℕ
81		nndivre 12334	. . . . 5 ⊢ ((((𝐹‘𝑘)↑2) ∈ ℝ ∧ 3 ∈ ℕ) → (((𝐹‘𝑘)↑2) / 3) ∈ ℝ)
82	79, 80, 81	sylancl 585	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((𝐹‘𝑘)↑2) / 3) ∈ ℝ)
83		resubcl 11600	. . . 4 ⊢ ((1 ∈ ℝ ∧ (((𝐹‘𝑘)↑2) / 3) ∈ ℝ) → (1 − (((𝐹‘𝑘)↑2) / 3)) ∈ ℝ)
84	78, 82, 83	sylancr 586	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) ∈ ℝ)
85	77, 84	eqeltrd 2844	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐻 ∘ 𝐹)‘𝑘) ∈ ℝ)
86		fvco3 7021	. . . . . 6 ⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ 𝑘 ∈ ℕ) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘)))
87	7, 86	sylan 579	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘)))
88	15, 87	syldan 590	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘)))
89		fveq2 6920	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → (sin‘𝑥) = (sin‘(𝐹‘𝑘)))
90		id 22	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → 𝑥 = (𝐹‘𝑘))
91	89, 90	oveq12d 7466	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑘) → ((sin‘𝑥) / 𝑥) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
92		ovex 7481	. . . . . 6 ⊢ ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) ∈ V
93	91, 67, 92	fvmpt 7029	. . . . 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 2780	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
96	19	resincld 16191	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(𝐹‘𝑘)) ∈ ℝ)
97	18	simprd 495	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ≠ 0)
98	96, 19, 97	redivcld 12122	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) ∈ ℝ)
99	95, 98	eqeltrd 2844	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) ∈ ℝ)
100		1cnd 11285	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 1 ∈ ℂ)
101	82	recnd 11318	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((𝐹‘𝑘)↑2) / 3) ∈ ℂ)
102	20	abscld 15485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
103	102	recnd 11318	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℂ)
104	100, 101, 103	subdird 11747	. . . . . . . 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	mullidd 11308	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 · (abs‘(𝐹‘𝑘))) = (abs‘(𝐹‘𝑘)))
106		df-3 12357	. . . . . . . . . . . . 13 ⊢ 3 = (2 + 1)
107	106	oveq2i 7459	. . . . . . . . . . . 12 ⊢ ((abs‘(𝐹‘𝑘))↑3) = ((abs‘(𝐹‘𝑘))↑(2 + 1))
108		2nn0 12570	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ₀
109		expp1 14119	. . . . . . . . . . . . . 14 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℂ ∧ 2 ∈ ℕ₀) → ((abs‘(𝐹‘𝑘))↑(2 + 1)) = (((abs‘(𝐹‘𝑘))↑2) · (abs‘(𝐹‘𝑘))))
110	103, 108, 109	sylancl 585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘))↑(2 + 1)) = (((abs‘(𝐹‘𝑘))↑2) · (abs‘(𝐹‘𝑘))))
111		absresq 15351	. . . . . . . . . . . . . . 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 7463	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((abs‘(𝐹‘𝑘))↑2) · (abs‘(𝐹‘𝑘))) = (((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))))
114	110, 113	eqtrd 2780	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘))↑(2 + 1)) = (((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))))
115	107, 114	eqtrid 2792	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘))↑3) = (((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))))
116	115	oveq1d 7463	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((abs‘(𝐹‘𝑘))↑3) / 3) = ((((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))) / 3))
117	79	recnd 11318	. . . . . . . . . . 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 12107	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))) / 3) = ((((𝐹‘𝑘)↑2) / 3) · (abs‘(𝐹‘𝑘))))
121	116, 120	eqtr2d 2781	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((((𝐹‘𝑘)↑2) / 3) · (abs‘(𝐹‘𝑘))) = (((abs‘(𝐹‘𝑘))↑3) / 3))
122	105, 121	oveq12d 7466	. . . . . . . 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 2780	. . . . . . 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 15497	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℝ⁺)
125	124	rpgt0d 13102	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 0 < (abs‘(𝐹‘𝑘)))
126		sinccvg.6	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) < 1)
127		ltle 11378	. . . . . . . . . . . 12 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘(𝐹‘𝑘)) < 1 → (abs‘(𝐹‘𝑘)) ≤ 1))
128	102, 78, 127	sylancl 585	. . . . . . . . . . 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 11337	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
131		elioc2 13470	. . . . . . . . . . 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 691	. . . . . . . . . 10 ⊢ ((abs‘(𝐹‘𝑘)) ∈ (0(,]1) ↔ ((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 0 < (abs‘(𝐹‘𝑘)) ∧ (abs‘(𝐹‘𝑘)) ≤ 1))
133	102, 125, 129, 132	syl3anbrc 1343	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ (0(,]1))
134		sin01bnd 16233	. . . . . . . . 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 494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) − (((abs‘(𝐹‘𝑘))↑3) / 3)) < (sin‘(abs‘(𝐹‘𝑘))))
137	123, 136	eqbrtrd 5188	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((1 − (((𝐹‘𝑘)↑2) / 3)) · (abs‘(𝐹‘𝑘))) < (sin‘(abs‘(𝐹‘𝑘))))
138	102	resincld 16191	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(abs‘(𝐹‘𝑘))) ∈ ℝ)
139	84, 138, 124	ltmuldivd 13146	. . . . . 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 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) < ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))))
141		fveq2 6920	. . . . . . . 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 7466	. . . . . . 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 16194	. . . . . . . . . 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 7463	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘-(𝐹‘𝑘)) / -(𝐹‘𝑘)) = (-(sin‘(𝐹‘𝑘)) / -(𝐹‘𝑘)))
148	96	recnd 11318	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(𝐹‘𝑘)) ∈ ℂ)
149	148, 20, 97	div2negd 12085	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (-(sin‘(𝐹‘𝑘)) / -(𝐹‘𝑘)) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
150	147, 149	eqtrd 2780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘-(𝐹‘𝑘)) / -(𝐹‘𝑘)) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
151		fveq2 6920	. . . . . . . . 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 7466	. . . . . . . 8 ⊢ ((abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘) → ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) = ((sin‘-(𝐹‘𝑘)) / -(𝐹‘𝑘)))
154	153	eqeq1d 2742	. . . . . . 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 247	. . . . . 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 15464	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) = (𝐹‘𝑘) ∨ (abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘)))
157	144, 155, 156	mpjaod 859	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
158	140, 157	breqtrd 5192	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) < ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
159	84, 98, 158	ltled 11438	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) ≤ ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
160	159, 77, 95	3brtr4d 5198	. 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 495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(abs‘(𝐹‘𝑘))) < (abs‘(𝐹‘𝑘)))
163	103	mulridd 11307	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) · 1) = (abs‘(𝐹‘𝑘)))
164	162, 163	breqtrrd 5194	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(abs‘(𝐹‘𝑘))) < ((abs‘(𝐹‘𝑘)) · 1))
165	138, 161, 124	ltdivmuld 13150	. . . . . 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 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) < 1)
167	157, 166	eqbrtrrd 5190	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) < 1)
168	98, 161, 167	ltled 11438	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) ≤ 1)
169	95, 168	eqbrtrd 5188	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) ≤ 1)
170	1, 3, 66, 70, 85, 99, 160, 169	climsqz 15687	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ⇝ 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ⊤wtru 1538 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 Vcvv 3488 ∖ cdif 3973 {csn 4648 class class class wbr 5166 ↦ cmpt 5249 ∘ ccom 5704 Fun wfun 6567 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 − cmin 11520 -cneg 11521 / cdiv 11947 ℕcn 12293 2c2 12348 3c3 12349 ℕ₀cn0 12553 ℤ_≥cuz 12903 ℝ⁺crp 13057 (,]cioc 13408 ↑cexp 14112 abscabs 15283 ⇝ cli 15530 sincsin 16111 TopOpenctopn 17481 ℂ_fldccnfld 21387 TopOnctopon 22937 Cn ccn 23253 ×_t ctx 23589 –cn→ccncf 24921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-fac 14323 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cn 23256 df-cnp 23257 df-tx 23591 df-hmeo 23784 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923

This theorem is referenced by: sinccvg 35641

W3C validator