fourierdlem56 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem56 46182

Description: Derivative of the 𝐾 function on an interval not containing ' 0 '. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem56.k	⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
fourierdlem56.a	⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ((-π[,]π) ∖ {0}))
fourierdlem56.r4	⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≠ 0)

**Assertion**
Ref	Expression
fourierdlem56	⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((((sin‘(𝑠 / 2)) − (((cos‘(𝑠 / 2)) / 2) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) / 2)))

Distinct variable groups: 𝐴,𝑠 𝐵,𝑠 𝜑,𝑠 Allowed substitution hint: 𝐾(𝑠)

Proof of Theorem fourierdlem56 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem56.a	. . . . . . . 8 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ((-π[,]π) ∖ {0}))
2	1	difss2d 4138	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (-π[,]π))
3	2	sselda 3982	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (-π[,]π))
4		1ex 11258	. . . . . . . 8 ⊢ 1 ∈ V
5		ovex 7465	. . . . . . . 8 ⊢ (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈ V
6	4, 5	ifex 4575	. . . . . . 7 ⊢ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈ V
7	6	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈ V)
8		fourierdlem56.k	. . . . . . 7 ⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
9	8	fvmpt2 7026	. . . . . 6 ⊢ ((𝑠 ∈ (-π[,]π) ∧ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) ∈ V) → (𝐾‘𝑠) = if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
10	3, 7, 9	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐾‘𝑠) = if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
11		fourierdlem56.r4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≠ 0)
12	11	neneqd 2944	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ¬ 𝑠 = 0)
13	12	iffalsed 4535	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
14		elioore 13418	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
15	14	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
16	15	recnd 11290	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
17	16	halfcld 12513	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℂ)
18	17	sincld 16167	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
19		2cnd 12345	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
20		fourierdlem44 46171	. . . . . . . 8 ⊢ ((𝑠 ∈ (-π[,]π) ∧ 𝑠 ≠ 0) → (sin‘(𝑠 / 2)) ≠ 0)
21	3, 11, 20	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ≠ 0)
22		2ne0 12371	. . . . . . . 8 ⊢ 2 ≠ 0
23	22	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
24	16, 18, 19, 21, 23	divdiv1d 12075	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝑠 / (sin‘(𝑠 / 2))) / 2) = (𝑠 / ((sin‘(𝑠 / 2)) · 2)))
25	18, 19	mulcomd 11283	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((sin‘(𝑠 / 2)) · 2) = (2 · (sin‘(𝑠 / 2))))
26	25	oveq2d 7448	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / ((sin‘(𝑠 / 2)) · 2)) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
27	24, 26	eqtr2d 2777	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) = ((𝑠 / (sin‘(𝑠 / 2))) / 2))
28	10, 13, 27	3eqtrd 2780	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐾‘𝑠) = ((𝑠 / (sin‘(𝑠 / 2))) / 2))
29	28	mpteq2dva 5241	. . 3 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝑠 / (sin‘(𝑠 / 2))) / 2)))
30	29	oveq2d 7448	. 2 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠))) = (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝑠 / (sin‘(𝑠 / 2))) / 2))))
31		reelprrecn 11248	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
32	31	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
33	16, 18, 21	divcld 12044	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / (sin‘(𝑠 / 2))) ∈ ℂ)
34		1red 11263	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
35	15	rehalfcld 12515	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℝ)
36	35	resincld 16180	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℝ)
37	34, 36	remulcld 11292	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 · (sin‘(𝑠 / 2))) ∈ ℝ)
38	35	recoscld 16181	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (cos‘(𝑠 / 2)) ∈ ℝ)
39	34	rehalfcld 12515	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 / 2) ∈ ℝ)
40	38, 39	remulcld 11292	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((cos‘(𝑠 / 2)) · (1 / 2)) ∈ ℝ)
41	40, 15	remulcld 11292	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠) ∈ ℝ)
42	37, 41	resubcld 11692	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((1 · (sin‘(𝑠 / 2))) − (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠)) ∈ ℝ)
43	36	resqcld 14166	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((sin‘(𝑠 / 2))↑2) ∈ ℝ)
44		2z 12651	. . . . . 6 ⊢ 2 ∈ ℤ
45	44	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℤ)
46	18, 21, 45	expne0d 14193	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((sin‘(𝑠 / 2))↑2) ≠ 0)
47	42, 43, 46	redivcld 12096	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((1 · (sin‘(𝑠 / 2))) − (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) ∈ ℝ)
48		1cnd 11257	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 1 ∈ ℂ)
49		recn 11246	. . . . . 6 ⊢ (𝑠 ∈ ℝ → 𝑠 ∈ ℂ)
50	49	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℂ)
51		1red 11263	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
52	32	dvmptid 25996	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑠)) = (𝑠 ∈ ℝ ↦ 1))
53		ioossre 13449	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℝ
54	53	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
55		tgioo4 24827	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
56		eqid 2736	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
57		iooretop 24787	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
58	57	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
59	32, 50, 51, 52, 54, 55, 56, 58	dvmptres 26002	. . . 4 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 1))
60		elsni 4642	. . . . . . 7 ⊢ ((sin‘(𝑠 / 2)) ∈ {0} → (sin‘(𝑠 / 2)) = 0)
61	60	necon3ai 2964	. . . . . 6 ⊢ ((sin‘(𝑠 / 2)) ≠ 0 → ¬ (sin‘(𝑠 / 2)) ∈ {0})
62	21, 61	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ¬ (sin‘(𝑠 / 2)) ∈ {0})
63	18, 62	eldifd 3961	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ (ℂ ∖ {0}))
64	17	coscld 16168	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (cos‘(𝑠 / 2)) ∈ ℂ)
65	48	halfcld 12513	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 / 2) ∈ ℂ)
66	64, 65	mulcld 11282	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((cos‘(𝑠 / 2)) · (1 / 2)) ∈ ℂ)
67		cnelprrecn 11249	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
68	67	a1i 11	. . . . 5 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
69		sinf 16161	. . . . . . 7 ⊢ sin:ℂ⟶ℂ
70	69	a1i 11	. . . . . 6 ⊢ (𝜑 → sin:ℂ⟶ℂ)
71	70	ffvelcdmda 7103	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
72		cosf 16162	. . . . . . 7 ⊢ cos:ℂ⟶ℂ
73	72	a1i 11	. . . . . 6 ⊢ (𝜑 → cos:ℂ⟶ℂ)
74	73	ffvelcdmda 7103	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ ℂ)
75		2cnd 12345	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℂ)
76	22	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
77	32, 16, 34, 59, 75, 76	dvmptdivc 26004	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝑠 / 2))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (1 / 2)))
78		ffn 6735	. . . . . . . . . . 11 ⊢ (sin:ℂ⟶ℂ → sin Fn ℂ)
79	69, 78	ax-mp 5	. . . . . . . . . 10 ⊢ sin Fn ℂ
80		dffn5 6966	. . . . . . . . . 10 ⊢ (sin Fn ℂ ↔ sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
81	79, 80	mpbi 230	. . . . . . . . 9 ⊢ sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥))
82	81	eqcomi 2745	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ ↦ (sin‘𝑥)) = sin
83	82	oveq2i 7443	. . . . . . 7 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (ℂ D sin)
84		dvsin 26021	. . . . . . 7 ⊢ (ℂ D sin) = cos
85		ffn 6735	. . . . . . . . 9 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ)
86	72, 85	ax-mp 5	. . . . . . . 8 ⊢ cos Fn ℂ
87		dffn5 6966	. . . . . . . 8 ⊢ (cos Fn ℂ ↔ cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
88	86, 87	mpbi 230	. . . . . . 7 ⊢ cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥))
89	83, 84, 88	3eqtri 2768	. . . . . 6 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (cos‘𝑥))
90	89	a1i 11	. . . . 5 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
91		fveq2 6905	. . . . 5 ⊢ (𝑥 = (𝑠 / 2) → (sin‘𝑥) = (sin‘(𝑠 / 2)))
92		fveq2 6905	. . . . 5 ⊢ (𝑥 = (𝑠 / 2) → (cos‘𝑥) = (cos‘(𝑠 / 2)))
93	32, 68, 17, 39, 71, 74, 77, 90, 91, 92	dvmptco 26011	. . . 4 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (sin‘(𝑠 / 2)))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((cos‘(𝑠 / 2)) · (1 / 2))))
94	32, 16, 48, 59, 63, 66, 93	dvmptdiv 26013	. . 3 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝑠 / (sin‘(𝑠 / 2))))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((1 · (sin‘(𝑠 / 2))) − (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠)) / ((sin‘(𝑠 / 2))↑2))))
95	32, 33, 47, 94, 75, 76	dvmptdivc 26004	. 2 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝑠 / (sin‘(𝑠 / 2))) / 2))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((((1 · (sin‘(𝑠 / 2))) − (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) / 2)))
96	14	recnd 11290	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℂ)
97	96	halfcld 12513	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (𝑠 / 2) ∈ ℂ)
98	97	sincld 16167	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (sin‘(𝑠 / 2)) ∈ ℂ)
99	98	mullidd 11280	. . . . . . 7 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (1 · (sin‘(𝑠 / 2))) = (sin‘(𝑠 / 2)))
100	97	coscld 16168	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (cos‘(𝑠 / 2)) ∈ ℂ)
101		2cnd 12345	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 2 ∈ ℂ)
102	22	a1i 11	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 2 ≠ 0)
103	100, 101, 102	divrecd 12047	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → ((cos‘(𝑠 / 2)) / 2) = ((cos‘(𝑠 / 2)) · (1 / 2)))
104	103	eqcomd 2742	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → ((cos‘(𝑠 / 2)) · (1 / 2)) = ((cos‘(𝑠 / 2)) / 2))
105	104	oveq1d 7447	. . . . . . 7 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠) = (((cos‘(𝑠 / 2)) / 2) · 𝑠))
106	99, 105	oveq12d 7450	. . . . . 6 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → ((1 · (sin‘(𝑠 / 2))) − (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠)) = ((sin‘(𝑠 / 2)) − (((cos‘(𝑠 / 2)) / 2) · 𝑠)))
107	106	oveq1d 7447	. . . . 5 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (((1 · (sin‘(𝑠 / 2))) − (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) = (((sin‘(𝑠 / 2)) − (((cos‘(𝑠 / 2)) / 2) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)))
108	107	oveq1d 7447	. . . 4 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → ((((1 · (sin‘(𝑠 / 2))) − (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) / 2) = ((((sin‘(𝑠 / 2)) − (((cos‘(𝑠 / 2)) / 2) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) / 2))
109	108	mpteq2ia 5244	. . 3 ⊢ (𝑠 ∈ (𝐴(,)𝐵) ↦ ((((1 · (sin‘(𝑠 / 2))) − (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) / 2)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((((sin‘(𝑠 / 2)) − (((cos‘(𝑠 / 2)) / 2) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) / 2))
110	109	a1i 11	. 2 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((((1 · (sin‘(𝑠 / 2))) − (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) / 2)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((((sin‘(𝑠 / 2)) − (((cos‘(𝑠 / 2)) / 2) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) / 2)))
111	30, 95, 110	3eqtrd 2780	1 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((((sin‘(𝑠 / 2)) − (((cos‘(𝑠 / 2)) / 2) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) / 2)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 Vcvv 3479 ∖ cdif 3947 ⊆ wss 3950 ifcif 4524 {csn 4625 {cpr 4627 ↦ cmpt 5224 ran crn 5685 Fn wfn 6555 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 ℝcr 11155 0cc0 11156 1c1 11157 · cmul 11161 − cmin 11493 -cneg 11494 / cdiv 11921 2c2 12322 ℤcz 12615 (,)cioo 13388 [,]cicc 13391 ↑cexp 14103 sincsin 16100 cosccos 16101 πcpi 16103 TopOpenctopn 17467 topGenctg 17483 ℂ_fldccnfld 21365 D cdv 25899

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-ef 16104 df-sin 16106 df-cos 16107 df-pi 16109 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-t1 23323 df-haus 23324 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-limc 25902 df-dv 25903

This theorem is referenced by: (None)

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