cos01bnd - Intuitionistic Logic Explorer

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Theorem cos01bnd 11721

Description: Bounds on the cosine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

**Assertion**
Ref	Expression
cos01bnd	⊢ (𝐴 ∈ (0(,]1) → ((1 − (2 · ((𝐴↑2) / 3))) < (cos‘𝐴) ∧ (cos‘𝐴) < (1 − ((𝐴↑2) / 3))))

Proof of Theorem cos01bnd Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0xr 7966	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
2		1re 7919	. . . . . . . . 9 ⊢ 1 ∈ ℝ
3		elioc2 9893	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → (𝐴 ∈ (0(,]1) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 1)))
4	1, 2, 3	mp2an 424	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,]1) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 1))
5	4	simp1bi 1007	. . . . . . 7 ⊢ (𝐴 ∈ (0(,]1) → 𝐴 ∈ ℝ)
6		eqid 2170	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))
7	6	recos4p 11682	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (cos‘𝐴) = ((1 − ((𝐴↑2) / 2)) + (ℜ‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘))))
8	5, 7	syl 14	. . . . . 6 ⊢ (𝐴 ∈ (0(,]1) → (cos‘𝐴) = ((1 − ((𝐴↑2) / 2)) + (ℜ‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘))))
9	8	eqcomd 2176	. . . . 5 ⊢ (𝐴 ∈ (0(,]1) → ((1 − ((𝐴↑2) / 2)) + (ℜ‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘))) = (cos‘𝐴))
10	5	recoscld 11687	. . . . . . 7 ⊢ (𝐴 ∈ (0(,]1) → (cos‘𝐴) ∈ ℝ)
11	10	recnd 7948	. . . . . 6 ⊢ (𝐴 ∈ (0(,]1) → (cos‘𝐴) ∈ ℂ)
12	5	resqcld 10635	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,]1) → (𝐴↑2) ∈ ℝ)
13	12	rehalfcld 9124	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴↑2) / 2) ∈ ℝ)
14		resubcl 8183	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ ((𝐴↑2) / 2) ∈ ℝ) → (1 − ((𝐴↑2) / 2)) ∈ ℝ)
15	2, 13, 14	sylancr 412	. . . . . . 7 ⊢ (𝐴 ∈ (0(,]1) → (1 − ((𝐴↑2) / 2)) ∈ ℝ)
16	15	recnd 7948	. . . . . 6 ⊢ (𝐴 ∈ (0(,]1) → (1 − ((𝐴↑2) / 2)) ∈ ℂ)
17		ax-icn 7869	. . . . . . . . . 10 ⊢ i ∈ ℂ
18	5	recnd 7948	. . . . . . . . . 10 ⊢ (𝐴 ∈ (0(,]1) → 𝐴 ∈ ℂ)
19		mulcl 7901	. . . . . . . . . 10 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
20	17, 18, 19	sylancr 412	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,]1) → (i · 𝐴) ∈ ℂ)
21		4nn0 9154	. . . . . . . . 9 ⊢ 4 ∈ ℕ₀
22	6	eftlcl 11651	. . . . . . . . 9 ⊢ (((i · 𝐴) ∈ ℂ ∧ 4 ∈ ℕ₀) → Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘) ∈ ℂ)
23	20, 21, 22	sylancl 411	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,]1) → Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘) ∈ ℂ)
24	23	recld 10902	. . . . . . 7 ⊢ (𝐴 ∈ (0(,]1) → (ℜ‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘)) ∈ ℝ)
25	24	recnd 7948	. . . . . 6 ⊢ (𝐴 ∈ (0(,]1) → (ℜ‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘)) ∈ ℂ)
26	11, 16, 25	subaddd 8248	. . . . 5 ⊢ (𝐴 ∈ (0(,]1) → (((cos‘𝐴) − (1 − ((𝐴↑2) / 2))) = (ℜ‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘)) ↔ ((1 − ((𝐴↑2) / 2)) + (ℜ‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘))) = (cos‘𝐴)))
27	9, 26	mpbird 166	. . . 4 ⊢ (𝐴 ∈ (0(,]1) → ((cos‘𝐴) − (1 − ((𝐴↑2) / 2))) = (ℜ‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘)))
28	27	fveq2d 5500	. . 3 ⊢ (𝐴 ∈ (0(,]1) → (abs‘((cos‘𝐴) − (1 − ((𝐴↑2) / 2)))) = (abs‘(ℜ‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘))))
29	25	abscld 11145	. . . 4 ⊢ (𝐴 ∈ (0(,]1) → (abs‘(ℜ‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘))) ∈ ℝ)
30	23	abscld 11145	. . . 4 ⊢ (𝐴 ∈ (0(,]1) → (abs‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘)) ∈ ℝ)
31		6nn 9043	. . . . 5 ⊢ 6 ∈ ℕ
32		nndivre 8914	. . . . 5 ⊢ (((𝐴↑2) ∈ ℝ ∧ 6 ∈ ℕ) → ((𝐴↑2) / 6) ∈ ℝ)
33	12, 31, 32	sylancl 411	. . . 4 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴↑2) / 6) ∈ ℝ)
34		absrele 11047	. . . . 5 ⊢ (Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘) ∈ ℂ → (abs‘(ℜ‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘))) ≤ (abs‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘)))
35	23, 34	syl 14	. . . 4 ⊢ (𝐴 ∈ (0(,]1) → (abs‘(ℜ‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘))) ≤ (abs‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘)))
36		reexpcl 10493	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 4 ∈ ℕ₀) → (𝐴↑4) ∈ ℝ)
37	5, 21, 36	sylancl 411	. . . . . 6 ⊢ (𝐴 ∈ (0(,]1) → (𝐴↑4) ∈ ℝ)
38		nndivre 8914	. . . . . 6 ⊢ (((𝐴↑4) ∈ ℝ ∧ 6 ∈ ℕ) → ((𝐴↑4) / 6) ∈ ℝ)
39	37, 31, 38	sylancl 411	. . . . 5 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴↑4) / 6) ∈ ℝ)
40	6	ef01bndlem 11719	. . . . 5 ⊢ (𝐴 ∈ (0(,]1) → (abs‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘)) < ((𝐴↑4) / 6))
41		2nn0 9152	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
42	41	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ (0(,]1) → 2 ∈ ℕ₀)
43		4z 9242	. . . . . . . . 9 ⊢ 4 ∈ ℤ
44		2re 8948	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
45		4re 8955	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
46		2lt4 9051	. . . . . . . . . 10 ⊢ 2 < 4
47	44, 45, 46	ltleii 8022	. . . . . . . . 9 ⊢ 2 ≤ 4
48		2z 9240	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
49	48	eluz1i 9494	. . . . . . . . 9 ⊢ (4 ∈ (ℤ_≥‘2) ↔ (4 ∈ ℤ ∧ 2 ≤ 4))
50	43, 47, 49	mpbir2an 937	. . . . . . . 8 ⊢ 4 ∈ (ℤ_≥‘2)
51	50	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ (0(,]1) → 4 ∈ (ℤ_≥‘2))
52	4	simp2bi 1008	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,]1) → 0 < 𝐴)
53		0re 7920	. . . . . . . . 9 ⊢ 0 ∈ ℝ
54		ltle 8007	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 → 0 ≤ 𝐴))
55	53, 5, 54	sylancr 412	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,]1) → (0 < 𝐴 → 0 ≤ 𝐴))
56	52, 55	mpd 13	. . . . . . 7 ⊢ (𝐴 ∈ (0(,]1) → 0 ≤ 𝐴)
57	4	simp3bi 1009	. . . . . . 7 ⊢ (𝐴 ∈ (0(,]1) → 𝐴 ≤ 1)
58	5, 42, 51, 56, 57	leexp2rd 10639	. . . . . 6 ⊢ (𝐴 ∈ (0(,]1) → (𝐴↑4) ≤ (𝐴↑2))
59		6re 8959	. . . . . . . 8 ⊢ 6 ∈ ℝ
60	59	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ (0(,]1) → 6 ∈ ℝ)
61		6pos 8979	. . . . . . . 8 ⊢ 0 < 6
62	61	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ (0(,]1) → 0 < 6)
63		lediv1 8785	. . . . . . 7 ⊢ (((𝐴↑4) ∈ ℝ ∧ (𝐴↑2) ∈ ℝ ∧ (6 ∈ ℝ ∧ 0 < 6)) → ((𝐴↑4) ≤ (𝐴↑2) ↔ ((𝐴↑4) / 6) ≤ ((𝐴↑2) / 6)))
64	37, 12, 60, 62, 63	syl112anc 1237	. . . . . 6 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴↑4) ≤ (𝐴↑2) ↔ ((𝐴↑4) / 6) ≤ ((𝐴↑2) / 6)))
65	58, 64	mpbid 146	. . . . 5 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴↑4) / 6) ≤ ((𝐴↑2) / 6))
66	30, 39, 33, 40, 65	ltletrd 8342	. . . 4 ⊢ (𝐴 ∈ (0(,]1) → (abs‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘)) < ((𝐴↑2) / 6))
67	29, 30, 33, 35, 66	lelttrd 8044	. . 3 ⊢ (𝐴 ∈ (0(,]1) → (abs‘(ℜ‘Σ𝑘 ∈ (ℤ_≥‘4)((𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))‘𝑘))) < ((𝐴↑2) / 6))
68	28, 67	eqbrtrd 4011	. 2 ⊢ (𝐴 ∈ (0(,]1) → (abs‘((cos‘𝐴) − (1 − ((𝐴↑2) / 2)))) < ((𝐴↑2) / 6))
69	10, 15, 33	absdifltd 11142	. . 3 ⊢ (𝐴 ∈ (0(,]1) → ((abs‘((cos‘𝐴) − (1 − ((𝐴↑2) / 2)))) < ((𝐴↑2) / 6) ↔ (((1 − ((𝐴↑2) / 2)) − ((𝐴↑2) / 6)) < (cos‘𝐴) ∧ (cos‘𝐴) < ((1 − ((𝐴↑2) / 2)) + ((𝐴↑2) / 6)))))
70		1cnd 7936	. . . . . . 7 ⊢ (𝐴 ∈ (0(,]1) → 1 ∈ ℂ)
71	13	recnd 7948	. . . . . . 7 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴↑2) / 2) ∈ ℂ)
72	33	recnd 7948	. . . . . . 7 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴↑2) / 6) ∈ ℂ)
73	70, 71, 72	subsub4d 8261	. . . . . 6 ⊢ (𝐴 ∈ (0(,]1) → ((1 − ((𝐴↑2) / 2)) − ((𝐴↑2) / 6)) = (1 − (((𝐴↑2) / 2) + ((𝐴↑2) / 6))))
74		halfpm6th 9098	. . . . . . . . . . 11 ⊢ (((1 / 2) − (1 / 6)) = (1 / 3) ∧ ((1 / 2) + (1 / 6)) = (2 / 3))
75	74	simpri 112	. . . . . . . . . 10 ⊢ ((1 / 2) + (1 / 6)) = (2 / 3)
76	75	oveq2i 5864	. . . . . . . . 9 ⊢ ((𝐴↑2) · ((1 / 2) + (1 / 6))) = ((𝐴↑2) · (2 / 3))
77	12	recnd 7948	. . . . . . . . . 10 ⊢ (𝐴 ∈ (0(,]1) → (𝐴↑2) ∈ ℂ)
78		2cn 8949	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
79		2ap0 8971	. . . . . . . . . . . 12 ⊢ 2 # 0
80	78, 79	recclapi 8659	. . . . . . . . . . 11 ⊢ (1 / 2) ∈ ℂ
81		6cn 8960	. . . . . . . . . . . 12 ⊢ 6 ∈ ℂ
82	31	nnap0i 8909	. . . . . . . . . . . 12 ⊢ 6 # 0
83	81, 82	recclapi 8659	. . . . . . . . . . 11 ⊢ (1 / 6) ∈ ℂ
84		adddi 7906	. . . . . . . . . . 11 ⊢ (((𝐴↑2) ∈ ℂ ∧ (1 / 2) ∈ ℂ ∧ (1 / 6) ∈ ℂ) → ((𝐴↑2) · ((1 / 2) + (1 / 6))) = (((𝐴↑2) · (1 / 2)) + ((𝐴↑2) · (1 / 6))))
85	80, 83, 84	mp3an23 1324	. . . . . . . . . 10 ⊢ ((𝐴↑2) ∈ ℂ → ((𝐴↑2) · ((1 / 2) + (1 / 6))) = (((𝐴↑2) · (1 / 2)) + ((𝐴↑2) · (1 / 6))))
86	77, 85	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴↑2) · ((1 / 2) + (1 / 6))) = (((𝐴↑2) · (1 / 2)) + ((𝐴↑2) · (1 / 6))))
87	76, 86	eqtr3id 2217	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴↑2) · (2 / 3)) = (((𝐴↑2) · (1 / 2)) + ((𝐴↑2) · (1 / 6))))
88		3cn 8953	. . . . . . . . . . 11 ⊢ 3 ∈ ℂ
89		3ap0 8974	. . . . . . . . . . 11 ⊢ 3 # 0
90	88, 89	pm3.2i 270	. . . . . . . . . 10 ⊢ (3 ∈ ℂ ∧ 3 # 0)
91		div12ap 8611	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ (𝐴↑2) ∈ ℂ ∧ (3 ∈ ℂ ∧ 3 # 0)) → (2 · ((𝐴↑2) / 3)) = ((𝐴↑2) · (2 / 3)))
92	78, 90, 91	mp3an13 1323	. . . . . . . . 9 ⊢ ((𝐴↑2) ∈ ℂ → (2 · ((𝐴↑2) / 3)) = ((𝐴↑2) · (2 / 3)))
93	77, 92	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,]1) → (2 · ((𝐴↑2) / 3)) = ((𝐴↑2) · (2 / 3)))
94		divrecap 8605	. . . . . . . . . . 11 ⊢ (((𝐴↑2) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 # 0) → ((𝐴↑2) / 2) = ((𝐴↑2) · (1 / 2)))
95	78, 79, 94	mp3an23 1324	. . . . . . . . . 10 ⊢ ((𝐴↑2) ∈ ℂ → ((𝐴↑2) / 2) = ((𝐴↑2) · (1 / 2)))
96	77, 95	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴↑2) / 2) = ((𝐴↑2) · (1 / 2)))
97		divrecap 8605	. . . . . . . . . . 11 ⊢ (((𝐴↑2) ∈ ℂ ∧ 6 ∈ ℂ ∧ 6 # 0) → ((𝐴↑2) / 6) = ((𝐴↑2) · (1 / 6)))
98	81, 82, 97	mp3an23 1324	. . . . . . . . . 10 ⊢ ((𝐴↑2) ∈ ℂ → ((𝐴↑2) / 6) = ((𝐴↑2) · (1 / 6)))
99	77, 98	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴↑2) / 6) = ((𝐴↑2) · (1 / 6)))
100	96, 99	oveq12d 5871	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,]1) → (((𝐴↑2) / 2) + ((𝐴↑2) / 6)) = (((𝐴↑2) · (1 / 2)) + ((𝐴↑2) · (1 / 6))))
101	87, 93, 100	3eqtr4rd 2214	. . . . . . 7 ⊢ (𝐴 ∈ (0(,]1) → (((𝐴↑2) / 2) + ((𝐴↑2) / 6)) = (2 · ((𝐴↑2) / 3)))
102	101	oveq2d 5869	. . . . . 6 ⊢ (𝐴 ∈ (0(,]1) → (1 − (((𝐴↑2) / 2) + ((𝐴↑2) / 6))) = (1 − (2 · ((𝐴↑2) / 3))))
103	73, 102	eqtrd 2203	. . . . 5 ⊢ (𝐴 ∈ (0(,]1) → ((1 − ((𝐴↑2) / 2)) − ((𝐴↑2) / 6)) = (1 − (2 · ((𝐴↑2) / 3))))
104	103	breq1d 3999	. . . 4 ⊢ (𝐴 ∈ (0(,]1) → (((1 − ((𝐴↑2) / 2)) − ((𝐴↑2) / 6)) < (cos‘𝐴) ↔ (1 − (2 · ((𝐴↑2) / 3))) < (cos‘𝐴)))
105	70, 71, 72	subsubd 8258	. . . . . 6 ⊢ (𝐴 ∈ (0(,]1) → (1 − (((𝐴↑2) / 2) − ((𝐴↑2) / 6))) = ((1 − ((𝐴↑2) / 2)) + ((𝐴↑2) / 6)))
106	74	simpli 110	. . . . . . . . . 10 ⊢ ((1 / 2) − (1 / 6)) = (1 / 3)
107	106	oveq2i 5864	. . . . . . . . 9 ⊢ ((𝐴↑2) · ((1 / 2) − (1 / 6))) = ((𝐴↑2) · (1 / 3))
108		subdi 8304	. . . . . . . . . . 11 ⊢ (((𝐴↑2) ∈ ℂ ∧ (1 / 2) ∈ ℂ ∧ (1 / 6) ∈ ℂ) → ((𝐴↑2) · ((1 / 2) − (1 / 6))) = (((𝐴↑2) · (1 / 2)) − ((𝐴↑2) · (1 / 6))))
109	80, 83, 108	mp3an23 1324	. . . . . . . . . 10 ⊢ ((𝐴↑2) ∈ ℂ → ((𝐴↑2) · ((1 / 2) − (1 / 6))) = (((𝐴↑2) · (1 / 2)) − ((𝐴↑2) · (1 / 6))))
110	77, 109	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴↑2) · ((1 / 2) − (1 / 6))) = (((𝐴↑2) · (1 / 2)) − ((𝐴↑2) · (1 / 6))))
111	107, 110	eqtr3id 2217	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴↑2) · (1 / 3)) = (((𝐴↑2) · (1 / 2)) − ((𝐴↑2) · (1 / 6))))
112		divrecap 8605	. . . . . . . . . 10 ⊢ (((𝐴↑2) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 # 0) → ((𝐴↑2) / 3) = ((𝐴↑2) · (1 / 3)))
113	88, 89, 112	mp3an23 1324	. . . . . . . . 9 ⊢ ((𝐴↑2) ∈ ℂ → ((𝐴↑2) / 3) = ((𝐴↑2) · (1 / 3)))
114	77, 113	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,]1) → ((𝐴↑2) / 3) = ((𝐴↑2) · (1 / 3)))
115	96, 99	oveq12d 5871	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,]1) → (((𝐴↑2) / 2) − ((𝐴↑2) / 6)) = (((𝐴↑2) · (1 / 2)) − ((𝐴↑2) · (1 / 6))))
116	111, 114, 115	3eqtr4rd 2214	. . . . . . 7 ⊢ (𝐴 ∈ (0(,]1) → (((𝐴↑2) / 2) − ((𝐴↑2) / 6)) = ((𝐴↑2) / 3))
117	116	oveq2d 5869	. . . . . 6 ⊢ (𝐴 ∈ (0(,]1) → (1 − (((𝐴↑2) / 2) − ((𝐴↑2) / 6))) = (1 − ((𝐴↑2) / 3)))
118	105, 117	eqtr3d 2205	. . . . 5 ⊢ (𝐴 ∈ (0(,]1) → ((1 − ((𝐴↑2) / 2)) + ((𝐴↑2) / 6)) = (1 − ((𝐴↑2) / 3)))
119	118	breq2d 4001	. . . 4 ⊢ (𝐴 ∈ (0(,]1) → ((cos‘𝐴) < ((1 − ((𝐴↑2) / 2)) + ((𝐴↑2) / 6)) ↔ (cos‘𝐴) < (1 − ((𝐴↑2) / 3))))
120	104, 119	anbi12d 470	. . 3 ⊢ (𝐴 ∈ (0(,]1) → ((((1 − ((𝐴↑2) / 2)) − ((𝐴↑2) / 6)) < (cos‘𝐴) ∧ (cos‘𝐴) < ((1 − ((𝐴↑2) / 2)) + ((𝐴↑2) / 6))) ↔ ((1 − (2 · ((𝐴↑2) / 3))) < (cos‘𝐴) ∧ (cos‘𝐴) < (1 − ((𝐴↑2) / 3)))))
121	69, 120	bitrd 187	. 2 ⊢ (𝐴 ∈ (0(,]1) → ((abs‘((cos‘𝐴) − (1 − ((𝐴↑2) / 2)))) < ((𝐴↑2) / 6) ↔ ((1 − (2 · ((𝐴↑2) / 3))) < (cos‘𝐴) ∧ (cos‘𝐴) < (1 − ((𝐴↑2) / 3)))))
122	68, 121	mpbid 146	1 ⊢ (𝐴 ∈ (0(,]1) → ((1 − (2 · ((𝐴↑2) / 3))) < (cos‘𝐴) ∧ (cos‘𝐴) < (1 − ((𝐴↑2) / 3))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 class class class wbr 3989 ↦ cmpt 4050 ‘cfv 5198 (class class class)co 5853 ℂcc 7772 ℝcr 7773 0cc0 7774 1c1 7775 ici 7776 + caddc 7777 · cmul 7779 ℝ^*cxr 7953 < clt 7954 ≤ cle 7955 − cmin 8090 # cap 8500 / cdiv 8589 ℕcn 8878 2c2 8929 3c3 8930 4c4 8931 6c6 8933 ℕ₀cn0 9135 ℤcz 9212 ℤ_≥cuz 9487 (,]cioc 9846 ↑cexp 10475 !cfa 10659 ℜcre 10804 abscabs 10961 Σcsu 11316 cosccos 11608

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-5 8940 df-6 8941 df-7 8942 df-8 8943 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-ioc 9850 df-ico 9851 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-fac 10660 df-ihash 10710 df-shft 10779 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-sumdc 11317 df-ef 11611 df-cos 11614

This theorem is referenced by: cos1bnd 11722 cos01gt0 11725 tangtx 13553

W3C validator