sinccvglem - Mathbox for Paul Chapman

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Theorem sinccvglem 35866

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 2736	. 2 ⊢ (ℤ_≥‘𝑀) = (ℤ_≥‘𝑀)
2		sinccvg.5	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3	2	nnzd 12514	. 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 6531	. . . . 5 ⊢ Fun 𝐻
7		sinccvg.1	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ ∖ {0}))
8		nnex 12151	. . . . . 6 ⊢ ℕ ∈ V
9		fex 7172	. . . . . 6 ⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ ℕ ∈ V) → 𝐹 ∈ V)
10	7, 8, 9	sylancl 586	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
11		cofunexg 7893	. . . . 5 ⊢ ((Fun 𝐻 ∧ 𝐹 ∈ V) → (𝐻 ∘ 𝐹) ∈ V)
12	6, 10, 11	sylancr 587	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ V)
13	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝐹:ℕ⟶(ℝ ∖ {0}))
14		eluznn 12831	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℕ)
15	2, 14	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℕ)
16	13, 15	ffvelcdmd 7030	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ (ℝ ∖ {0}))
17		eldifsn 4742	. . . . . . 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 11160	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
21		ax-1cn 11084	. . . . . 6 ⊢ 1 ∈ ℂ
22		sqcl 14041	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥↑2) ∈ ℂ)
23		3cn 12226	. . . . . . . 8 ⊢ 3 ∈ ℂ
24		3ne0 12251	. . . . . . . 8 ⊢ 3 ≠ 0
25		divcl 11802	. . . . . . . 8 ⊢ (((𝑥↑2) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → ((𝑥↑2) / 3) ∈ ℂ)
26	23, 24, 25	mp3an23 1455	. . . . . . 7 ⊢ ((𝑥↑2) ∈ ℂ → ((𝑥↑2) / 3) ∈ ℂ)
27	22, 26	syl 17	. . . . . 6 ⊢ (𝑥 ∈ ℂ → ((𝑥↑2) / 3) ∈ ℂ)
28		subcl 11379	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ ((𝑥↑2) / 3) ∈ ℂ) → (1 − ((𝑥↑2) / 3)) ∈ ℂ)
29	21, 27, 28	sylancr 587	. . . . 5 ⊢ (𝑥 ∈ ℂ → (1 − ((𝑥↑2) / 3)) ∈ ℂ)
30	5, 29	fmpti 7057	. . . 4 ⊢ 𝐻:ℂ⟶ℂ
31		eqid 2736	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
32	31	cnfldtopon 24726	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
33	32	a1i 11	. . . . . . . 8 ⊢ (⊤ → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
34		1cnd 11127	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
35	33, 33, 34	cnmptc 23606	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 1) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
36	31	sqcn 24823	. . . . . . . . . 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 24820	. . . . . . . . . . 11 ⊢ ((3 ∈ ℂ ∧ 3 ≠ 0) → (𝑦 ∈ ℂ ↦ (𝑦 / 3)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
39	23, 24, 38	mp2an 692	. . . . . . . . . 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 7365	. . . . . . . . 9 ⊢ (𝑦 = (𝑥↑2) → (𝑦 / 3) = ((𝑥↑2) / 3))
42	33, 37, 33, 40, 41	cnmpt11 23607	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((𝑥↑2) / 3)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
43	31	subcn 24811	. . . . . . . . 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 23610	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (1 − ((𝑥↑2) / 3))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
46	45	mptru 1548	. . . . . 6 ⊢ (𝑥 ∈ ℂ ↦ (1 − ((𝑥↑2) / 3))) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
47	31	cncfcn1 24860	. . . . . 6 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
48	46, 5, 47	3eltr4i 2849	. . . . 5 ⊢ 𝐻 ∈ (ℂ–cn→ℂ)
49		cncfi 24843	. . . . 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 1450	. . . 4 ⊢ ((0 ∈ ℂ ∧ 𝑦 ∈ ℝ⁺) → ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ ℂ ((abs‘(𝑤 − 0)) < 𝑧 → (abs‘((𝐻‘𝑤) − (𝐻‘0))) < 𝑦))
51		fvco3 6933	. . . . . 6 ⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ 𝑘 ∈ ℕ) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘)))
52	7, 51	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘)))
53	15, 52	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘)))
54	1, 4, 12, 3, 20, 30, 50, 53	climcn1lem 15526	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ⇝ (𝐻‘0))
55		0cn 11124	. . . 4 ⊢ 0 ∈ ℂ
56		sq0i 14116	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥↑2) = 0)
57	56	oveq1d 7373	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥↑2) / 3) = (0 / 3))
58	23, 24	div0i 11875	. . . . . . . 8 ⊢ (0 / 3) = 0
59	57, 58	eqtrdi 2787	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝑥↑2) / 3) = 0)
60	59	oveq2d 7374	. . . . . 6 ⊢ (𝑥 = 0 → (1 − ((𝑥↑2) / 3)) = (1 − 0))
61		1m0e1 12261	. . . . . 6 ⊢ (1 − 0) = 1
62	60, 61	eqtrdi 2787	. . . . 5 ⊢ (𝑥 = 0 → (1 − ((𝑥↑2) / 3)) = 1)
63		1ex 11128	. . . . 5 ⊢ 1 ∈ V
64	62, 5, 63	fvmpt 6941	. . . 4 ⊢ (0 ∈ ℂ → (𝐻‘0) = 1)
65	55, 64	ax-mp 5	. . 3 ⊢ (𝐻‘0) = 1
66	54, 65	breqtrdi 5139	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ⇝ 1)
67		sinccvg.3	. . . 4 ⊢ 𝐺 = (𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥))
68	67	funmpt2 6531	. . 3 ⊢ Fun 𝐺
69		cofunexg 7893	. . 3 ⊢ ((Fun 𝐺 ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V)
70	68, 10, 69	sylancr 587	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
71		oveq1 7365	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑘) → (𝑥↑2) = ((𝐹‘𝑘)↑2))
72	71	oveq1d 7373	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → ((𝑥↑2) / 3) = (((𝐹‘𝑘)↑2) / 3))
73	72	oveq2d 7374	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑘) → (1 − ((𝑥↑2) / 3)) = (1 − (((𝐹‘𝑘)↑2) / 3)))
74		ovex 7391	. . . . . 6 ⊢ (1 − (((𝐹‘𝑘)↑2) / 3)) ∈ V
75	73, 5, 74	fvmpt 6941	. . . . 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 2771	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐻 ∘ 𝐹)‘𝑘) = (1 − (((𝐹‘𝑘)↑2) / 3)))
78		1re 11132	. . . 4 ⊢ 1 ∈ ℝ
79	19	resqcld 14048	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐹‘𝑘)↑2) ∈ ℝ)
80		3nn 12224	. . . . 5 ⊢ 3 ∈ ℕ
81		nndivre 12186	. . . . 5 ⊢ ((((𝐹‘𝑘)↑2) ∈ ℝ ∧ 3 ∈ ℕ) → (((𝐹‘𝑘)↑2) / 3) ∈ ℝ)
82	79, 80, 81	sylancl 586	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((𝐹‘𝑘)↑2) / 3) ∈ ℝ)
83		resubcl 11445	. . . 4 ⊢ ((1 ∈ ℝ ∧ (((𝐹‘𝑘)↑2) / 3) ∈ ℝ) → (1 − (((𝐹‘𝑘)↑2) / 3)) ∈ ℝ)
84	78, 82, 83	sylancr 587	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) ∈ ℝ)
85	77, 84	eqeltrd 2836	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐻 ∘ 𝐹)‘𝑘) ∈ ℝ)
86		fvco3 6933	. . . . . 6 ⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ 𝑘 ∈ ℕ) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘)))
87	7, 86	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘)))
88	15, 87	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘)))
89		fveq2 6834	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → (sin‘𝑥) = (sin‘(𝐹‘𝑘)))
90		id 22	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → 𝑥 = (𝐹‘𝑘))
91	89, 90	oveq12d 7376	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑘) → ((sin‘𝑥) / 𝑥) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
92		ovex 7391	. . . . . 6 ⊢ ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) ∈ V
93	91, 67, 92	fvmpt 6941	. . . . 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 2771	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
96	19	resincld 16068	. . . 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 11969	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) ∈ ℝ)
99	95, 98	eqeltrd 2836	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) ∈ ℝ)
100		1cnd 11127	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 1 ∈ ℂ)
101	82	recnd 11160	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((𝐹‘𝑘)↑2) / 3) ∈ ℂ)
102	20	abscld 15362	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
103	102	recnd 11160	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℂ)
104	100, 101, 103	subdird 11594	. . . . . . . 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 11150	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 · (abs‘(𝐹‘𝑘))) = (abs‘(𝐹‘𝑘)))
106		df-3 12209	. . . . . . . . . . . . 13 ⊢ 3 = (2 + 1)
107	106	oveq2i 7369	. . . . . . . . . . . 12 ⊢ ((abs‘(𝐹‘𝑘))↑3) = ((abs‘(𝐹‘𝑘))↑(2 + 1))
108		2nn0 12418	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ₀
109		expp1 13991	. . . . . . . . . . . . . 14 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℂ ∧ 2 ∈ ℕ₀) → ((abs‘(𝐹‘𝑘))↑(2 + 1)) = (((abs‘(𝐹‘𝑘))↑2) · (abs‘(𝐹‘𝑘))))
110	103, 108, 109	sylancl 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘))↑(2 + 1)) = (((abs‘(𝐹‘𝑘))↑2) · (abs‘(𝐹‘𝑘))))
111		absresq 15225	. . . . . . . . . . . . . . 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 7373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((abs‘(𝐹‘𝑘))↑2) · (abs‘(𝐹‘𝑘))) = (((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))))
114	110, 113	eqtrd 2771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘))↑(2 + 1)) = (((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))))
115	107, 114	eqtrid 2783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘))↑3) = (((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))))
116	115	oveq1d 7373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((abs‘(𝐹‘𝑘))↑3) / 3) = ((((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))) / 3))
117	79	recnd 11160	. . . . . . . . . . 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 11954	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))) / 3) = ((((𝐹‘𝑘)↑2) / 3) · (abs‘(𝐹‘𝑘))))
121	116, 120	eqtr2d 2772	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((((𝐹‘𝑘)↑2) / 3) · (abs‘(𝐹‘𝑘))) = (((abs‘(𝐹‘𝑘))↑3) / 3))
122	105, 121	oveq12d 7376	. . . . . . . 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 2771	. . . . . . 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 15374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℝ⁺)
125	124	rpgt0d 12952	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 0 < (abs‘(𝐹‘𝑘)))
126		sinccvg.6	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) < 1)
127		ltle 11221	. . . . . . . . . . . 12 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘(𝐹‘𝑘)) < 1 → (abs‘(𝐹‘𝑘)) ≤ 1))
128	102, 78, 127	sylancl 586	. . . . . . . . . . 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 11179	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
131		elioc2 13325	. . . . . . . . . . 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 692	. . . . . . . . . 10 ⊢ ((abs‘(𝐹‘𝑘)) ∈ (0(,]1) ↔ ((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 0 < (abs‘(𝐹‘𝑘)) ∧ (abs‘(𝐹‘𝑘)) ≤ 1))
133	102, 125, 129, 132	syl3anbrc 1344	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ (0(,]1))
134		sin01bnd 16110	. . . . . . . . 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 5120	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((1 − (((𝐹‘𝑘)↑2) / 3)) · (abs‘(𝐹‘𝑘))) < (sin‘(abs‘(𝐹‘𝑘))))
138	102	resincld 16068	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(abs‘(𝐹‘𝑘))) ∈ ℝ)
139	84, 138, 124	ltmuldivd 12996	. . . . . 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 6834	. . . . . . . 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 7376	. . . . . . 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 16071	. . . . . . . . . 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 7373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘-(𝐹‘𝑘)) / -(𝐹‘𝑘)) = (-(sin‘(𝐹‘𝑘)) / -(𝐹‘𝑘)))
148	96	recnd 11160	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(𝐹‘𝑘)) ∈ ℂ)
149	148, 20, 97	div2negd 11932	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (-(sin‘(𝐹‘𝑘)) / -(𝐹‘𝑘)) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
150	147, 149	eqtrd 2771	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘-(𝐹‘𝑘)) / -(𝐹‘𝑘)) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
151		fveq2 6834	. . . . . . . . 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 7376	. . . . . . . 8 ⊢ ((abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘) → ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) = ((sin‘-(𝐹‘𝑘)) / -(𝐹‘𝑘)))
154	153	eqeq1d 2738	. . . . . . 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 15339	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) = (𝐹‘𝑘) ∨ (abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘)))
157	144, 155, 156	mpjaod 860	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
158	140, 157	breqtrd 5124	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) < ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
159	84, 98, 158	ltled 11281	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) ≤ ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
160	159, 77, 95	3brtr4d 5130	. 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 11149	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) · 1) = (abs‘(𝐹‘𝑘)))
164	162, 163	breqtrrd 5126	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(abs‘(𝐹‘𝑘))) < ((abs‘(𝐹‘𝑘)) · 1))
165	138, 161, 124	ltdivmuld 13000	. . . . . 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 5122	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) < 1)
168	98, 161, 167	ltled 11281	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) ≤ 1)
169	95, 168	eqbrtrd 5120	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) ≤ 1)
170	1, 3, 66, 70, 85, 99, 160, 169	climsqz 15564	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ⇝ 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ⊤wtru 1542 ∈ wcel 2113 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 Vcvv 3440 ∖ cdif 3898 {csn 4580 class class class wbr 5098 ↦ cmpt 5179 ∘ ccom 5628 Fun wfun 6486 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 ℝcr 11025 0cc0 11026 1c1 11027 + caddc 11029 · cmul 11031 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167 − cmin 11364 -cneg 11365 / cdiv 11794 ℕcn 12145 2c2 12200 3c3 12201 ℕ₀cn0 12401 ℤ_≥cuz 12751 ℝ⁺crp 12905 (,]cioc 13262 ↑cexp 13984 abscabs 15157 ⇝ cli 15407 sincsin 15986 TopOpenctopn 17341 ℂ_fldccnfld 21309 TopOnctopon 22854 Cn ccn 23168 ×_t ctx 23504 –cn→ccncf 24825

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-seq 13925 df-exp 13985 df-fac 14197 df-hash 14254 df-shft 14990 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15610 df-ef 15990 df-sin 15992 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cn 23171 df-cnp 23172 df-tx 23506 df-hmeo 23699 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827

This theorem is referenced by: sinccvg 35867

W3C validator