sinccvglem - Mathbox for Paul Chapman

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Theorem sinccvglem 36184

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 2766	. 2 ⊢ (ℤ_≥‘𝑀) = (ℤ_≥‘𝑀)
2		sinccvg.5	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3	2	nnzd 12635	. 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 6582	. . . . 5 ⊢ Fun 𝐻
7		sinccvg.1	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ ∖ {0}))
8		nnex 12257	. . . . . 6 ⊢ ℕ ∈ V
9		fex 7231	. . . . . 6 ⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ ℕ ∈ V) → 𝐹 ∈ V)
10	7, 8, 9	sylancl 598	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
11		cofunexg 7955	. . . . 5 ⊢ ((Fun 𝐻 ∧ 𝐹 ∈ V) → (𝐻 ∘ 𝐹) ∈ V)
12	6, 10, 11	sylancr 599	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ V)
13	7	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝐹:ℕ⟶(ℝ ∖ {0}))
14		eluznn 12960	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℕ)
15	2, 14	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℕ)
16	13, 15	ffvelcdmd 7087	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ (ℝ ∖ {0}))
17		eldifsn 4758	. . . . . . 7 ⊢ ((𝐹‘𝑘) ∈ (ℝ ∖ {0}) ↔ ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) ≠ 0))
18	16, 17	sylib 221	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) ≠ 0))
19	18	simpld 500	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)
20	19	recnd 11255	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
21		ax-1cn 11176	. . . . . 6 ⊢ 1 ∈ ℂ
22		sqcl 14174	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥↑2) ∈ ℂ)
23		3cn 12340	. . . . . . . 8 ⊢ 3 ∈ ℂ
24		3ne0 12368	. . . . . . . 8 ⊢ 3 ≠ 0
25		divcl 11896	. . . . . . . 8 ⊢ (((𝑥↑2) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → ((𝑥↑2) / 3) ∈ ℂ)
26	23, 24, 25	mp3an23 1482	. . . . . . 7 ⊢ ((𝑥↑2) ∈ ℂ → ((𝑥↑2) / 3) ∈ ℂ)
27	22, 26	syl 18	. . . . . 6 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) / 3) ∈ ℂ)
28		subcl 11474	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ ((𝑥↑2) / 3) ∈ ℂ) → (1 − ((𝑥↑2) / 3)) ∈ ℂ)
29	21, 27, 28	sylancr 599	. . . . 5 ⊢ (𝑥 ∈ ℂ → (1 − ((𝑥↑2) / 3)) ∈ ℂ)
30	5, 29	fmpti 7114	. . . 4 ⊢ 𝐻:ℂ⟶ℂ
31		eqid 2766	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
32	31	cnfldtopon 24976	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
33	32	a1i 11	. . . . . . . 8 ⊢ (⊤ → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
34		1cnd 11220	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
35	33, 33, 34	cnmptc 23856	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 1) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
36	31	sqcn 25070	. . . . . . . . . 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 25069	. . . . . . . . . . 11 ⊢ ((3 ∈ ℂ ∧ 3 ≠ 0) → (𝑦 ∈ ℂ ↦ (𝑦 / 3)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
39	23, 24, 38	mp2an 705	. . . . . . . . . 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 7430	. . . . . . . . 9 ⊢ (𝑦 = (𝑥↑2) → (𝑦 / 3) = ((𝑥↑2) / 3))
42	33, 37, 33, 40, 41	cnmpt11 23857	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((𝑥↑2) / 3)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
43	31	subcn 25061	. . . . . . . . 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 23860	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (1 − ((𝑥↑2) / 3))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
46	45	mptru 1577	. . . . . 6 ⊢ (𝑥 ∈ ℂ ↦ (1 − ((𝑥↑2) / 3))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
47	31	cncfcn1 25107	. . . . . 6 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
48	46, 5, 47	3eltr4i 2879	. . . . 5 ⊢ 𝐻 ∈ (ℂ–cn→ℂ)
49		cncfi 25090	. . . . 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 1477	. . . 4 ⊢ ((0 ∈ ℂ ∧ 𝑦 ∈ ℝ⁺) → ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℂ ((abs‘(𝑤 − 0)) < 𝑧 → (abs‘((𝐻‘𝑤) − (𝐻‘0))) < 𝑦))
51		fvco3 6988	. . . . . 6 ⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ 𝑘 ∈ ℕ) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘)))
52	7, 51	sylan 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘)))
53	15, 52	syldan 603	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘)))
54	1, 4, 12, 3, 20, 30, 50, 53	climcn1lem 15680	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ⇝ (𝐻‘0))
55		0cn 11216	. . . 4 ⊢ 0 ∈ ℂ
56		sq0i 14249	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥↑2) = 0)
57	56	oveq1d 7438	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥↑2) / 3) = (0 / 3))
58	23, 24	div0i 11967	. . . . . . . 8 ⊢ (0 / 3) = 0
59	57, 58	eqtrdi 2817	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝑥↑2) / 3) = 0)
60	59	oveq2d 7439	. . . . . 6 ⊢ (𝑥 = 0 → (1 − ((𝑥↑2) / 3)) = (1 − 0))
61		1m0e1 12378	. . . . . 6 ⊢ (1 − 0) = 1
62	60, 61	eqtrdi 2817	. . . . 5 ⊢ (𝑥 = 0 → (1 − ((𝑥↑2) / 3)) = 1)
63		1ex 11221	. . . . 5 ⊢ 1 ∈ V
64	62, 5, 63	fvmpt 6996	. . . 4 ⊢ (0 ∈ ℂ → (𝐻‘0) = 1)
65	55, 64	ax-mp 5	. . 3 ⊢ (𝐻‘0) = 1
66	54, 65	breqtrdi 5157	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ⇝ 1)
67		sinccvg.3	. . . 4 ⊢ 𝐺 = (𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥))
68	67	funmpt2 6582	. . 3 ⊢ Fun 𝐺
69		cofunexg 7955	. . 3 ⊢ ((Fun 𝐺 ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V)
70	68, 10, 69	sylancr 599	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
71		oveq1 7430	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑘) → (𝑥↑2) = ((𝐹‘𝑘)↑2))
72	71	oveq1d 7438	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → ((𝑥↑2) / 3) = (((𝐹‘𝑘)↑2) / 3))
73	72	oveq2d 7439	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑘) → (1 − ((𝑥↑2) / 3)) = (1 − (((𝐹‘𝑘)↑2) / 3)))
74		ovex 7456	. . . . . 6 ⊢ (1 − (((𝐹‘𝑘)↑2) / 3)) ∈ V
75	73, 5, 74	fvmpt 6996	. . . . 5 ⊢ ((𝐹‘𝑘) ∈ ℂ → (𝐻‘(𝐹‘𝑘)) = (1 − (((𝐹‘𝑘)↑2) / 3)))
76	20, 75	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐻‘(𝐹‘𝑘)) = (1 − (((𝐹‘𝑘)↑2) / 3)))
77	53, 76	eqtrd 2801	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐻 ∘ 𝐹)‘𝑘) = (1 − (((𝐹‘𝑘)↑2) / 3)))
78		1re 11226	. . . 4 ⊢ 1 ∈ ℝ
79	19	resqcld 14181	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐹‘𝑘)↑2) ∈ ℝ)
80		3nn 12338	. . . . 5 ⊢ 3 ∈ ℕ
81		nndivre 12295	. . . . 5 ⊢ ((((𝐹‘𝑘)↑2) ∈ ℝ ∧ 3 ∈ ℕ) → (((𝐹‘𝑘)↑2) / 3) ∈ ℝ)
82	79, 80, 81	sylancl 598	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((𝐹‘𝑘)↑2) / 3) ∈ ℝ)
83		resubcl 11540	. . . 4 ⊢ ((1 ∈ ℝ ∧ (((𝐹‘𝑘)↑2) / 3) ∈ ℝ) → (1 − (((𝐹‘𝑘)↑2) / 3)) ∈ ℝ)
84	78, 82, 83	sylancr 599	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) ∈ ℝ)
85	77, 84	eqeltrd 2866	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐻 ∘ 𝐹)‘𝑘) ∈ ℝ)
86		fvco3 6988	. . . . . 6 ⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ 𝑘 ∈ ℕ) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘)))
87	7, 86	sylan 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘)))
88	15, 87	syldan 603	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘)))
89		fveq2 6888	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → (sin‘𝑥) = (sin‘(𝐹‘𝑘)))
90		id 23	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → 𝑥 = (𝐹‘𝑘))
91	89, 90	oveq12d 7441	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑘) → ((sin‘𝑥) / 𝑥) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
92		ovex 7456	. . . . . 6 ⊢ ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) ∈ V
93	91, 67, 92	fvmpt 6996	. . . . 5 ⊢ ((𝐹‘𝑘) ∈ (ℝ ∖ {0}) → (𝐺‘(𝐹‘𝑘)) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
94	16, 93	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐺‘(𝐹‘𝑘)) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
95	88, 94	eqtrd 2801	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
96	19	resincld 16224	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(𝐹‘𝑘)) ∈ ℝ)
97	18	simprd 501	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ≠ 0)
98	96, 19, 97	redivcld 12061	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) ∈ ℝ)
99	95, 98	eqeltrd 2866	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) ∈ ℝ)
100		1cnd 11220	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 1 ∈ ℂ)
101	82	recnd 11255	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((𝐹‘𝑘)↑2) / 3) ∈ ℂ)
102	20	abscld 15516	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
103	102	recnd 11255	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℂ)
104	100, 101, 103	subdird 11689	. . . . . . . 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 11245	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 · (abs‘(𝐹‘𝑘))) = (abs‘(𝐹‘𝑘)))
106		df-3 12322	. . . . . . . . . . . . 13 ⊢ 3 = (2 + 1)
107	106	oveq2i 7434	. . . . . . . . . . . 12 ⊢ ((abs‘(𝐹‘𝑘))↑3) = ((abs‘(𝐹‘𝑘))↑(2 + 1))
108		2nn0 12539	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ₀
109		expp1 14124	. . . . . . . . . . . . . 14 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℂ ∧ 2 ∈ ℕ₀) → ((abs‘(𝐹‘𝑘))↑(2 + 1)) = (((abs‘(𝐹‘𝑘))↑2) · (abs‘(𝐹‘𝑘))))
110	103, 108, 109	sylancl 598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘))↑(2 + 1)) = (((abs‘(𝐹‘𝑘))↑2) · (abs‘(𝐹‘𝑘))))
111		absresq 15379	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) ∈ ℝ → ((abs‘(𝐹‘𝑘))↑2) = ((𝐹‘𝑘)↑2))
112	19, 111	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘))↑2) = ((𝐹‘𝑘)↑2))
113	112	oveq1d 7438	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((abs‘(𝐹‘𝑘))↑2) · (abs‘(𝐹‘𝑘))) = (((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))))
114	110, 113	eqtrd 2801	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘))↑(2 + 1)) = (((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))))
115	107, 114	eqtrid 2813	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘))↑3) = (((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))))
116	115	oveq1d 7438	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((abs‘(𝐹‘𝑘))↑3) / 3) = ((((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))) / 3))
117	79	recnd 11255	. . . . . . . . . . 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 12046	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))) / 3) = ((((𝐹‘𝑘)↑2) / 3) · (abs‘(𝐹‘𝑘))))
121	116, 120	eqtr2d 2802	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((((𝐹‘𝑘)↑2) / 3) · (abs‘(𝐹‘𝑘))) = (((abs‘(𝐹‘𝑘))↑3) / 3))
122	105, 121	oveq12d 7441	. . . . . . . 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 2801	. . . . . . 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 15528	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℝ⁺)
125	124	rpgt0d 13081	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 0 < (abs‘(𝐹‘𝑘)))
126		sinccvg.6	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) < 1)
127		ltle 11316	. . . . . . . . . . . 12 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘(𝐹‘𝑘)) < 1 → (abs‘(𝐹‘𝑘)) ≤ 1))
128	102, 78, 127	sylancl 598	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) < 1 → (abs‘(𝐹‘𝑘)) ≤ 1))
129	126, 128	mpd 16	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ≤ 1)
130		0xr 11274	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
131		elioc2 13454	. . . . . . . . . . 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 705	. . . . . . . . . 10 ⊢ ((abs‘(𝐹‘𝑘)) ∈ (0(,]1) ↔ ((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 0 < (abs‘(𝐹‘𝑘)) ∧ (abs‘(𝐹‘𝑘)) ≤ 1))
133	102, 125, 129, 132	syl3anbrc 1362	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ (0(,]1))
134		sin01bnd 16266	. . . . . . . . 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 18	. . . . . . . 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 500	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) − (((abs‘(𝐹‘𝑘))↑3) / 3)) < (sin‘(abs‘(𝐹‘𝑘))))
137	123, 136	eqbrtrd 5138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((1 − (((𝐹‘𝑘)↑2) / 3)) · (abs‘(𝐹‘𝑘))) < (sin‘(abs‘(𝐹‘𝑘))))
138	102	resincld 16224	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(abs‘(𝐹‘𝑘))) ∈ ℝ)
139	84, 138, 124	ltmuldivd 13125	. . . . . 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 235	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) < ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))))
141		fveq2 6888	. . . . . . . 8 ⊢ ((abs‘(𝐹‘𝑘)) = (𝐹‘𝑘) → (sin‘(abs‘(𝐹‘𝑘))) = (sin‘(𝐹‘𝑘)))
142		id 23	. . . . . . . 8 ⊢ ((abs‘(𝐹‘𝑘)) = (𝐹‘𝑘) → (abs‘(𝐹‘𝑘)) = (𝐹‘𝑘))
143	141, 142	oveq12d 7441	. . . . . . 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 16227	. . . . . . . . . 10 ⊢ ((𝐹‘𝑘) ∈ ℂ → (sin‘-(𝐹‘𝑘)) = -(sin‘(𝐹‘𝑘)))
146	20, 145	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘-(𝐹‘𝑘)) = -(sin‘(𝐹‘𝑘)))
147	146	oveq1d 7438	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘-(𝐹‘𝑘)) / -(𝐹‘𝑘)) = (-(sin‘(𝐹‘𝑘)) / -(𝐹‘𝑘)))
148	96	recnd 11255	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(𝐹‘𝑘)) ∈ ℂ)
149	148, 20, 97	div2negd 12024	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (-(sin‘(𝐹‘𝑘)) / -(𝐹‘𝑘)) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
150	147, 149	eqtrd 2801	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘-(𝐹‘𝑘)) / -(𝐹‘𝑘)) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
151		fveq2 6888	. . . . . . . . 9 ⊢ ((abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘) → (sin‘(abs‘(𝐹‘𝑘))) = (sin‘-(𝐹‘𝑘)))
152		id 23	. . . . . . . . 9 ⊢ ((abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘) → (abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘))
153	151, 152	oveq12d 7441	. . . . . . . 8 ⊢ ((abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘) → ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) = ((sin‘-(𝐹‘𝑘)) / -(𝐹‘𝑘)))
154	153	eqeq1d 2768	. . . . . . 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 250	. . . . . 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 15493	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) = (𝐹‘𝑘) ∨ (abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘)))
157	144, 155, 156	mpjaod 874	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
158	140, 157	breqtrd 5142	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) < ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
159	84, 98, 158	ltled 11376	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) ≤ ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
160	159, 77, 95	3brtr4d 5148	. 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 501	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(abs‘(𝐹‘𝑘))) < (abs‘(𝐹‘𝑘)))
163	103	mulridd 11244	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) · 1) = (abs‘(𝐹‘𝑘)))
164	162, 163	breqtrrd 5144	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(abs‘(𝐹‘𝑘))) < ((abs‘(𝐹‘𝑘)) · 1))
165	138, 161, 124	ltdivmuld 13129	. . . . . 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 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) < 1)
167	157, 166	eqbrtrrd 5140	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) < 1)
168	98, 161, 167	ltled 11376	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) ≤ 1)
169	95, 168	eqbrtrd 5138	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) ≤ 1)
170	1, 3, 66, 70, 85, 99, 160, 169	climsqz 15718	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ⇝ 1)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ⊤wtru 1571 ∈ wcel 2146 ≠ wne 2961 ∀wral 3082 ∃wrex 3092 Vcvv 3458 ∖ cdif 3905 {csn 4594 class class class wbr 5114 ↦ cmpt 5197 ∘ ccom 5670 Fun wfun 6537 ⟶wf 6539 ‘cfv 6543 (class class class)co 7423 ℂcc 11116 ℝcr 11117 0cc0 11118 1c1 11119 + caddc 11121 · cmul 11123 ℝ^*cxr 11260 < clt 11261 ≤ cle 11262 − cmin 11459 -cneg 11460 / cdiv 11889 ℕcn 12251 2c2 12313 3c3 12314 ℕ₀cn0 12522 ℤ_≥cuz 12880 ℝ⁺crp 13034 (,]cioc 13391 ↑cexp 14117 abscabs 15311 ⇝ cli 15561 sincsin 16142 TopOpenctopn 17499 ℂ_fldccnfld 21559 TopOnctopon 23104 Cn ccn 23418 ×_t ctx 23754 –cn→ccncf 25072

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-inf2 9620 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 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 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-pm 8836 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioc 13395 df-ico 13396 df-icc 13397 df-fz 13554 df-fzo 13702 df-fl 13845 df-seq 14058 df-exp 14118 df-fac 14330 df-hash 14387 df-shft 15130 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-limsup 15548 df-clim 15565 df-rlim 15566 df-sum 15764 df-ef 16146 df-sin 16148 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-starv 17350 df-sca 17351 df-vsca 17352 df-ip 17353 df-tset 17354 df-ple 17355 df-ds 17357 df-unif 17358 df-hom 17359 df-cco 17360 df-rest 17500 df-topn 17501 df-0g 17519 df-gsum 17520 df-topgen 17521 df-pt 17522 df-prds 17525 df-xrs 17581 df-qtop 17586 df-imas 17587 df-xps 17589 df-mre 17663 df-mrc 17664 df-acs 17666 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-submnd 18873 df-mulg 19165 df-cntz 19418 df-cmn 19883 df-psmet 21551 df-xmet 21552 df-met 21553 df-bl 21554 df-mopn 21555 df-cnfld 21560 df-top 23088 df-topon 23105 df-topsp 23127 df-bases 23140 df-cn 23421 df-cnp 23422 df-tx 23756 df-hmeo 23949 df-xms 24514 df-ms 24515 df-tms 24516 df-cncf 25074

This theorem is used by: sinccvg 36185

W3C validator