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 36142

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 2763	. 2 ⊢ (ℤ_≥‘𝑀) = (ℤ_≥‘𝑀)
2		sinccvg.5	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3	2	nnzd 12618	. 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 6577	. . . . 5 ⊢ Fun 𝐻
7		sinccvg.1	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ ∖ {0}))
8		nnex 12240	. . . . . 6 ⊢ ℕ ∈ V
9		fex 7226	. . . . . 6 ⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ ℕ ∈ V) → 𝐹 ∈ V)
10	7, 8, 9	sylancl 597	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
11		cofunexg 7947	. . . . 5 ⊢ ((Fun 𝐻 ∧ 𝐹 ∈ V) → (𝐻 ∘ 𝐹) ∈ V)
12	6, 10, 11	sylancr 598	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ V)
13	7	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝐹:ℕ⟶(ℝ ∖ {0}))
14		eluznn 12943	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℕ)
15	2, 14	sylan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℕ)
16	13, 15	ffvelcdmd 7082	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ (ℝ ∖ {0}))
17		eldifsn 4754	. . . . . . 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 499	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ)
20	19	recnd 11238	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
21		ax-1cn 11159	. . . . . 6 ⊢ 1 ∈ ℂ
22		sqcl 14156	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥↑2) ∈ ℂ)
23		3cn 12323	. . . . . . . 8 ⊢ 3 ∈ ℂ
24		3ne0 12351	. . . . . . . 8 ⊢ 3 ≠ 0
25		divcl 11879	. . . . . . . 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 11457	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ ((𝑥↑2) / 3) ∈ ℂ) → (1 − ((𝑥↑2) / 3)) ∈ ℂ)
29	21, 27, 28	sylancr 598	. . . . 5 ⊢ (𝑥 ∈ ℂ → (1 − ((𝑥↑2) / 3)) ∈ ℂ)
30	5, 29	fmpti 7109	. . . 4 ⊢ 𝐻:ℂ⟶ℂ
31		eqid 2763	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
32	31	cnfldtopon 24920	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
33	32	a1i 11	. . . . . . . 8 ⊢ (⊤ → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
34		1cnd 11203	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
35	33, 33, 34	cnmptc 23800	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 1) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
36	31	sqcn 25014	. . . . . . . . . 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 25013	. . . . . . . . . . 11 ⊢ ((3 ∈ ℂ ∧ 3 ≠ 0) → (𝑦 ∈ ℂ ↦ (𝑦 / 3)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
39	23, 24, 38	mp2an 704	. . . . . . . . . 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 7419	. . . . . . . . 9 ⊢ (𝑦 = (𝑥↑2) → (𝑦 / 3) = ((𝑥↑2) / 3))
42	33, 37, 33, 40, 41	cnmpt11 23801	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ ((𝑥↑2) / 3)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
43	31	subcn 25005	. . . . . . . . 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 23804	. . . . . . 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 25051	. . . . . 6 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
48	46, 5, 47	3eltr4i 2876	. . . . 5 ⊢ 𝐻 ∈ (ℂ–cn→ℂ)
49		cncfi 25034	. . . . 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 6983	. . . . . 6 ⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ 𝑘 ∈ ℕ) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘)))
52	7, 51	sylan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘)))
53	15, 52	syldan 602	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐻 ∘ 𝐹)‘𝑘) = (𝐻‘(𝐹‘𝑘)))
54	1, 4, 12, 3, 20, 30, 50, 53	climcn1lem 15656	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ⇝ (𝐻‘0))
55		0cn 11199	. . . 4 ⊢ 0 ∈ ℂ
56		sq0i 14231	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥↑2) = 0)
57	56	oveq1d 7427	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥↑2) / 3) = (0 / 3))
58	23, 24	div0i 11950	. . . . . . . 8 ⊢ (0 / 3) = 0
59	57, 58	eqtrdi 2814	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝑥↑2) / 3) = 0)
60	59	oveq2d 7428	. . . . . 6 ⊢ (𝑥 = 0 → (1 − ((𝑥↑2) / 3)) = (1 − 0))
61		1m0e1 12361	. . . . . 6 ⊢ (1 − 0) = 1
62	60, 61	eqtrdi 2814	. . . . 5 ⊢ (𝑥 = 0 → (1 − ((𝑥↑2) / 3)) = 1)
63		1ex 11204	. . . . 5 ⊢ 1 ∈ V
64	62, 5, 63	fvmpt 6991	. . . 4 ⊢ (0 ∈ ℂ → (𝐻‘0) = 1)
65	55, 64	ax-mp 5	. . 3 ⊢ (𝐻‘0) = 1
66	54, 65	breqtrdi 5153	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ⇝ 1)
67		sinccvg.3	. . . 4 ⊢ 𝐺 = (𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥))
68	67	funmpt2 6577	. . 3 ⊢ Fun 𝐺
69		cofunexg 7947	. . 3 ⊢ ((Fun 𝐺 ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V)
70	68, 10, 69	sylancr 598	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
71		oveq1 7419	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑘) → (𝑥↑2) = ((𝐹‘𝑘)↑2))
72	71	oveq1d 7427	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → ((𝑥↑2) / 3) = (((𝐹‘𝑘)↑2) / 3))
73	72	oveq2d 7428	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑘) → (1 − ((𝑥↑2) / 3)) = (1 − (((𝐹‘𝑘)↑2) / 3)))
74		ovex 7445	. . . . . 6 ⊢ (1 − (((𝐹‘𝑘)↑2) / 3)) ∈ V
75	73, 5, 74	fvmpt 6991	. . . . 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 2798	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐻 ∘ 𝐹)‘𝑘) = (1 − (((𝐹‘𝑘)↑2) / 3)))
78		1re 11209	. . . 4 ⊢ 1 ∈ ℝ
79	19	resqcld 14163	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐹‘𝑘)↑2) ∈ ℝ)
80		3nn 12321	. . . . 5 ⊢ 3 ∈ ℕ
81		nndivre 12278	. . . . 5 ⊢ ((((𝐹‘𝑘)↑2) ∈ ℝ ∧ 3 ∈ ℕ) → (((𝐹‘𝑘)↑2) / 3) ∈ ℝ)
82	79, 80, 81	sylancl 597	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((𝐹‘𝑘)↑2) / 3) ∈ ℝ)
83		resubcl 11523	. . . 4 ⊢ ((1 ∈ ℝ ∧ (((𝐹‘𝑘)↑2) / 3) ∈ ℝ) → (1 − (((𝐹‘𝑘)↑2) / 3)) ∈ ℝ)
84	78, 82, 83	sylancr 598	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) ∈ ℝ)
85	77, 84	eqeltrd 2863	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐻 ∘ 𝐹)‘𝑘) ∈ ℝ)
86		fvco3 6983	. . . . . 6 ⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ 𝑘 ∈ ℕ) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘)))
87	7, 86	sylan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘)))
88	15, 87	syldan 602	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) = (𝐺‘(𝐹‘𝑘)))
89		fveq2 6883	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → (sin‘𝑥) = (sin‘(𝐹‘𝑘)))
90		id 23	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → 𝑥 = (𝐹‘𝑘))
91	89, 90	oveq12d 7430	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑘) → ((sin‘𝑥) / 𝑥) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
92		ovex 7445	. . . . . 6 ⊢ ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) ∈ V
93	91, 67, 92	fvmpt 6991	. . . . 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 2798	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
96	19	resincld 16200	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(𝐹‘𝑘)) ∈ ℝ)
97	18	simprd 500	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ≠ 0)
98	96, 19, 97	redivcld 12044	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) ∈ ℝ)
99	95, 98	eqeltrd 2863	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) ∈ ℝ)
100		1cnd 11203	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 1 ∈ ℂ)
101	82	recnd 11238	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((𝐹‘𝑘)↑2) / 3) ∈ ℂ)
102	20	abscld 15492	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℝ)
103	102	recnd 11238	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℂ)
104	100, 101, 103	subdird 11672	. . . . . . . 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 11228	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 · (abs‘(𝐹‘𝑘))) = (abs‘(𝐹‘𝑘)))
106		df-3 12305	. . . . . . . . . . . . 13 ⊢ 3 = (2 + 1)
107	106	oveq2i 7423	. . . . . . . . . . . 12 ⊢ ((abs‘(𝐹‘𝑘))↑3) = ((abs‘(𝐹‘𝑘))↑(2 + 1))
108		2nn0 12522	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ₀
109		expp1 14106	. . . . . . . . . . . . . 14 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℂ ∧ 2 ∈ ℕ₀) → ((abs‘(𝐹‘𝑘))↑(2 + 1)) = (((abs‘(𝐹‘𝑘))↑2) · (abs‘(𝐹‘𝑘))))
110	103, 108, 109	sylancl 597	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘))↑(2 + 1)) = (((abs‘(𝐹‘𝑘))↑2) · (abs‘(𝐹‘𝑘))))
111		absresq 15355	. . . . . . . . . . . . . . 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 7427	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((abs‘(𝐹‘𝑘))↑2) · (abs‘(𝐹‘𝑘))) = (((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))))
114	110, 113	eqtrd 2798	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘))↑(2 + 1)) = (((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))))
115	107, 114	eqtrid 2810	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘))↑3) = (((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))))
116	115	oveq1d 7427	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (((abs‘(𝐹‘𝑘))↑3) / 3) = ((((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))) / 3))
117	79	recnd 11238	. . . . . . . . . . 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 12029	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((((𝐹‘𝑘)↑2) · (abs‘(𝐹‘𝑘))) / 3) = ((((𝐹‘𝑘)↑2) / 3) · (abs‘(𝐹‘𝑘))))
121	116, 120	eqtr2d 2799	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((((𝐹‘𝑘)↑2) / 3) · (abs‘(𝐹‘𝑘))) = (((abs‘(𝐹‘𝑘))↑3) / 3))
122	105, 121	oveq12d 7430	. . . . . . . 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 2798	. . . . . . 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 15504	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℝ⁺)
125	124	rpgt0d 13064	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 0 < (abs‘(𝐹‘𝑘)))
126		sinccvg.6	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘𝑘)) < 1)
127		ltle 11299	. . . . . . . . . . . 12 ⊢ (((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘(𝐹‘𝑘)) < 1 → (abs‘(𝐹‘𝑘)) ≤ 1))
128	102, 78, 127	sylancl 597	. . . . . . . . . . 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 11257	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
131		elioc2 13437	. . . . . . . . . . 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 704	. . . . . . . . . 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 16242	. . . . . . . . 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 499	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) − (((abs‘(𝐹‘𝑘))↑3) / 3)) < (sin‘(abs‘(𝐹‘𝑘))))
137	123, 136	eqbrtrd 5134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((1 − (((𝐹‘𝑘)↑2) / 3)) · (abs‘(𝐹‘𝑘))) < (sin‘(abs‘(𝐹‘𝑘))))
138	102	resincld 16200	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(abs‘(𝐹‘𝑘))) ∈ ℝ)
139	84, 138, 124	ltmuldivd 13108	. . . . . 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 6883	. . . . . . . 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 7430	. . . . . . 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 16203	. . . . . . . . . 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 7427	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘-(𝐹‘𝑘)) / -(𝐹‘𝑘)) = (-(sin‘(𝐹‘𝑘)) / -(𝐹‘𝑘)))
148	96	recnd 11238	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(𝐹‘𝑘)) ∈ ℂ)
149	148, 20, 97	div2negd 12007	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (-(sin‘(𝐹‘𝑘)) / -(𝐹‘𝑘)) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
150	147, 149	eqtrd 2798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘-(𝐹‘𝑘)) / -(𝐹‘𝑘)) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
151		fveq2 6883	. . . . . . . . 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 7430	. . . . . . . 8 ⊢ ((abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘) → ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) = ((sin‘-(𝐹‘𝑘)) / -(𝐹‘𝑘)))
154	153	eqeq1d 2765	. . . . . . 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 15469	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) = (𝐹‘𝑘) ∨ (abs‘(𝐹‘𝑘)) = -(𝐹‘𝑘)))
157	144, 155, 156	mpjaod 873	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(abs‘(𝐹‘𝑘))) / (abs‘(𝐹‘𝑘))) = ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
158	140, 157	breqtrd 5138	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) < ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
159	84, 98, 158	ltled 11359	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (1 − (((𝐹‘𝑘)↑2) / 3)) ≤ ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)))
160	159, 77, 95	3brtr4d 5144	. 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 500	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(abs‘(𝐹‘𝑘))) < (abs‘(𝐹‘𝑘)))
163	103	mulridd 11227	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) · 1) = (abs‘(𝐹‘𝑘)))
164	162, 163	breqtrrd 5140	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (sin‘(abs‘(𝐹‘𝑘))) < ((abs‘(𝐹‘𝑘)) · 1))
165	138, 161, 124	ltdivmuld 13112	. . . . . 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 5136	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) < 1)
168	98, 161, 167	ltled 11359	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((sin‘(𝐹‘𝑘)) / (𝐹‘𝑘)) ≤ 1)
169	95, 168	eqbrtrd 5134	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝐺 ∘ 𝐹)‘𝑘) ≤ 1)
170	1, 3, 66, 70, 85, 99, 160, 169	climsqz 15694	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ⇝ 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ⊤wtru 1571 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 Vcvv 3455 ∖ cdif 3903 {csn 4590 class class class wbr 5110 ↦ cmpt 5193 ∘ ccom 5667 Fun wfun 6532 ⟶wf 6534 ‘cfv 6538 (class class class)co 7412 ℂcc 11099 ℝcr 11100 0cc0 11101 1c1 11102 + caddc 11104 · cmul 11106 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 − cmin 11442 -cneg 11443 / cdiv 11872 ℕcn 12234 2c2 12296 3c3 12297 ℕ₀cn0 12505 ℤ_≥cuz 12863 ℝ⁺crp 13017 (,]cioc 13374 ↑cexp 14099 abscabs 15287 ⇝ cli 15537 sincsin 16118 TopOpenctopn 17475 ℂ_fldccnfld 21503 TopOnctopon 23048 Cn ccn 23362 ×_t ctx 23698 –cn→ccncf 25016

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8158 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-er 8695 df-map 8827 df-pm 8828 df-ixp 8897 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-fsupp 9323 df-fi 9372 df-sup 9403 df-inf 9404 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-z 12593 df-dec 12713 df-uz 12864 df-q 12974 df-rp 13018 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioc 13378 df-ico 13379 df-icc 13380 df-fz 13537 df-fzo 13685 df-fl 13827 df-seq 14040 df-exp 14100 df-fac 14312 df-hash 14369 df-shft 15106 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-limsup 15524 df-clim 15541 df-rlim 15542 df-sum 15740 df-ef 16122 df-sin 16124 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 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 17476 df-topn 17477 df-0g 17495 df-gsum 17496 df-topgen 17497 df-pt 17498 df-prds 17501 df-xrs 17557 df-qtop 17562 df-imas 17563 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-submnd 18843 df-mulg 19135 df-cntz 19388 df-cmn 19853 df-psmet 21495 df-xmet 21496 df-met 21497 df-bl 21498 df-mopn 21499 df-cnfld 21504 df-top 23032 df-topon 23049 df-topsp 23071 df-bases 23084 df-cn 23365 df-cnp 23366 df-tx 23700 df-hmeo 23893 df-xms 24458 df-ms 24459 df-tms 24460 df-cncf 25018

This theorem is referenced by: sinccvg 36143

W3C validator