fourierdlem56 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem56 46684

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 4087	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (-π[,]π))
3	2	sselda 3931	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (-π[,]π))
4		1ex 11166	. . . . . . . 8 ⊢ 1 ∈ V
5		ovex 7418	. . . . . . . 8 ⊢ (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈ V
6	4, 5	ifex 4525	. . . . . . 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 6976	. . . . . 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 592	. . . . 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 2956	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ¬ 𝑠 = 0)
13	12	iffalsed 4485	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
14		elioore 13369	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
15	14	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
16	15	recnd 11200	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
17	16	halfcld 12456	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℂ)
18	17	sincld 16138	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
19		2cnd 12286	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
20		fourierdlem44 46673	. . . . . . . 8 ⊢ ((𝑠 ∈ (-π[,]π) ∧ 𝑠 ≠ 0) → (sin‘(𝑠 / 2)) ≠ 0)
21	3, 11, 20	syl2anc 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ≠ 0)
22		2ne0 12314	. . . . . . . 8 ⊢ 2 ≠ 0
23	22	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
24	16, 18, 19, 21, 23	divdiv1d 11988	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝑠 / (sin‘(𝑠 / 2))) / 2) = (𝑠 / ((sin‘(𝑠 / 2)) · 2)))
25	18, 19	mulcomd 11193	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((sin‘(𝑠 / 2)) · 2) = (2 · (sin‘(𝑠 / 2))))
26	25	oveq2d 7401	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / ((sin‘(𝑠 / 2)) · 2)) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
27	24, 26	eqtr2d 2792	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) = ((𝑠 / (sin‘(𝑠 / 2))) / 2))
28	10, 13, 27	3eqtrd 2795	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐾‘𝑠) = ((𝑠 / (sin‘(𝑠 / 2))) / 2))
29	28	mpteq2dva 5187	. . 3 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝑠 / (sin‘(𝑠 / 2))) / 2)))
30	29	oveq2d 7401	. 2 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠))) = (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝑠 / (sin‘(𝑠 / 2))) / 2))))
31		reelprrecn 11155	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
32	31	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
33	16, 18, 21	divcld 11957	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / (sin‘(𝑠 / 2))) ∈ ℂ)
34		1red 11172	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
35	15	rehalfcld 12458	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℝ)
36	35	resincld 16151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℝ)
37	34, 36	remulcld 11202	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 · (sin‘(𝑠 / 2))) ∈ ℝ)
38	35	recoscld 16152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (cos‘(𝑠 / 2)) ∈ ℝ)
39	34	rehalfcld 12458	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 / 2) ∈ ℝ)
40	38, 39	remulcld 11202	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((cos‘(𝑠 / 2)) · (1 / 2)) ∈ ℝ)
41	40, 15	remulcld 11202	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠) ∈ ℝ)
42	37, 41	resubcld 11605	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((1 · (sin‘(𝑠 / 2))) − (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠)) ∈ ℝ)
43	36	resqcld 14128	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((sin‘(𝑠 / 2))↑2) ∈ ℝ)
44		2z 12593	. . . . . 6 ⊢ 2 ∈ ℤ
45	44	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℤ)
46	18, 21, 45	expne0d 14155	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((sin‘(𝑠 / 2))↑2) ≠ 0)
47	42, 43, 46	redivcld 12009	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((1 · (sin‘(𝑠 / 2))) − (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) ∈ ℝ)
48		1cnd 11165	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 1 ∈ ℂ)
49		recn 11153	. . . . . 6 ⊢ (𝑠 ∈ ℝ → 𝑠 ∈ ℂ)
50	49	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℂ)
51		1red 11172	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
52	32	dvmptid 25992	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑠)) = (𝑠 ∈ ℝ ↦ 1))
53		ioossre 13401	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℝ
54	53	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
55		tgioo4 24838	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
56		eqid 2756	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
57		iooretop 24798	. . . . . 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 25998	. . . 4 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 1))
60		elsni 4593	. . . . . . 7 ⊢ ((sin‘(𝑠 / 2)) ∈ {0} → (sin‘(𝑠 / 2)) = 0)
61	60	necon3ai 2976	. . . . . 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 3910	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ (ℂ ∖ {0}))
64	17	coscld 16139	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (cos‘(𝑠 / 2)) ∈ ℂ)
65	48	halfcld 12456	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 / 2) ∈ ℂ)
66	64, 65	mulcld 11192	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((cos‘(𝑠 / 2)) · (1 / 2)) ∈ ℂ)
67		cnelprrecn 11156	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
68	67	a1i 11	. . . . 5 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
69		sinf 16132	. . . . . . 7 ⊢ sin:ℂ⟶ℂ
70	69	a1i 11	. . . . . 6 ⊢ (𝜑 → sin:ℂ⟶ℂ)
71	70	ffvelcdmda 7054	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
72		cosf 16133	. . . . . . 7 ⊢ cos:ℂ⟶ℂ
73	72	a1i 11	. . . . . 6 ⊢ (𝜑 → cos:ℂ⟶ℂ)
74	73	ffvelcdmda 7054	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ ℂ)
75		2cnd 12286	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℂ)
76	22	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
77	32, 16, 34, 59, 75, 76	dvmptdivc 26000	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝑠 / 2))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (1 / 2)))
78		ffn 6680	. . . . . . . . . . 11 ⊢ (sin:ℂ⟶ℂ → sin Fn ℂ)
79	69, 78	ax-mp 5	. . . . . . . . . 10 ⊢ sin Fn ℂ
80		dffn5 6914	. . . . . . . . . 10 ⊢ (sin Fn ℂ ↔ sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
81	79, 80	mpbi 232	. . . . . . . . 9 ⊢ sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥))
82	81	eqcomi 2765	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ ↦ (sin‘𝑥)) = sin
83	82	oveq2i 7396	. . . . . . 7 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (ℂ D sin)
84		dvsin 26017	. . . . . . 7 ⊢ (ℂ D sin) = cos
85		ffn 6680	. . . . . . . . 9 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ)
86	72, 85	ax-mp 5	. . . . . . . 8 ⊢ cos Fn ℂ
87		dffn5 6914	. . . . . . . 8 ⊢ (cos Fn ℂ ↔ cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
88	86, 87	mpbi 232	. . . . . . 7 ⊢ cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥))
89	83, 84, 88	3eqtri 2783	. . . . . 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 6856	. . . . 5 ⊢ (𝑥 = (𝑠 / 2) → (sin‘𝑥) = (sin‘(𝑠 / 2)))
92		fveq2 6856	. . . . 5 ⊢ (𝑥 = (𝑠 / 2) → (cos‘𝑥) = (cos‘(𝑠 / 2)))
93	32, 68, 17, 39, 71, 74, 77, 90, 91, 92	dvmptco 26007	. . . 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 26009	. . 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 26000	. 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 11200	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℂ)
97	96	halfcld 12456	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (𝑠 / 2) ∈ ℂ)
98	97	sincld 16138	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (sin‘(𝑠 / 2)) ∈ ℂ)
99	98	mullidd 11190	. . . . . . 7 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (1 · (sin‘(𝑠 / 2))) = (sin‘(𝑠 / 2)))
100	97	coscld 16139	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (cos‘(𝑠 / 2)) ∈ ℂ)
101		2cnd 12286	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 2 ∈ ℂ)
102	22	a1i 11	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 2 ≠ 0)
103	100, 101, 102	divrecd 11960	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → ((cos‘(𝑠 / 2)) / 2) = ((cos‘(𝑠 / 2)) · (1 / 2)))
104	103	eqcomd 2762	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → ((cos‘(𝑠 / 2)) · (1 / 2)) = ((cos‘(𝑠 / 2)) / 2))
105	104	oveq1d 7400	. . . . . . 7 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠) = (((cos‘(𝑠 / 2)) / 2) · 𝑠))
106	99, 105	oveq12d 7403	. . . . . 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 7400	. . . . 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 7400	. . . 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 5189	. . 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 2795	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 398 = wceq 1554 ∈ wcel 2136 ≠ wne 2951 Vcvv 3448 ∖ cdif 3896 ⊆ wss 3899 ifcif 4474 {csn 4576 {cpr 4578 ↦ cmpt 5175 ran crn 5641 Fn wfn 6505 ⟶wf 6506 ‘cfv 6510 (class class class)co 7385 ℂcc 11061 ℝcr 11062 0cc0 11063 1c1 11064 · cmul 11068 − cmin 11404 -cneg 11405 / cdiv 11834 2c2 12262 ℤcz 12558 (,)cioo 13339 [,]cicc 13342 ↑cexp 14064 sincsin 16069 cosccos 16070 πcpi 16072 TopOpenctopn 17426 topGenctg 17442 ℂ_fldccnfld 21397 D cdv 25898

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-rep 5221 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707 ax-inf2 9586 ax-cnex 11119 ax-resscn 11120 ax-1cn 11121 ax-icn 11122 ax-addcl 11123 ax-addrcl 11124 ax-mulcl 11125 ax-mulrcl 11126 ax-mulcom 11127 ax-addass 11128 ax-mulass 11129 ax-distr 11130 ax-i2m1 11131 ax-1ne0 11132 ax-1rid 11133 ax-rnegex 11134 ax-rrecex 11135 ax-cnre 11136 ax-pre-lttri 11137 ax-pre-lttrn 11138 ax-pre-ltadd 11139 ax-pre-mulgt0 11140 ax-pre-sup 11141 ax-addf 11142

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-nel 3056 df-ral 3071 df-rex 3081 df-rmo 3361 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4900 df-iun 4945 df-iin 4946 df-br 5095 df-opab 5157 df-mpt 5176 df-tr 5202 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-isom 6519 df-riota 7342 df-ov 7388 df-oprab 7389 df-mpo 7390 df-of 7649 df-om 7836 df-1st 7959 df-2nd 7960 df-supp 8129 df-frecs 8250 df-wrecs 8281 df-recs 8330 df-rdg 8369 df-1o 8425 df-2o 8426 df-er 8666 df-map 8798 df-pm 8799 df-ixp 8869 df-en 8917 df-dom 8918 df-sdom 8919 df-fin 8920 df-fsupp 9298 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9448 df-card 9887 df-pnf 11208 df-mnf 11209 df-xr 11210 df-ltxr 11211 df-le 11212 df-sub 11406 df-neg 11407 df-div 11835 df-nn 12201 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12472 df-z 12559 df-dec 12679 df-uz 12830 df-q 12940 df-rp 12984 df-xneg 13104 df-xadd 13105 df-xmul 13106 df-ioo 13343 df-ioc 13344 df-ico 13345 df-icc 13346 df-fz 13503 df-fzo 13650 df-fl 13792 df-mod 13870 df-seq 14005 df-exp 14065 df-fac 14277 df-bc 14306 df-hash 14334 df-shft 15070 df-cj 15102 df-re 15103 df-im 15104 df-sqrt 15238 df-abs 15239 df-limsup 15474 df-clim 15491 df-rlim 15492 df-sum 15690 df-ef 16073 df-sin 16075 df-cos 16076 df-pi 16078 df-struct 17159 df-sets 17176 df-slot 17194 df-ndx 17206 df-base 17222 df-ress 17243 df-plusg 17275 df-mulr 17276 df-starv 17277 df-sca 17278 df-vsca 17279 df-ip 17280 df-tset 17281 df-ple 17282 df-ds 17284 df-unif 17285 df-hom 17286 df-cco 17287 df-rest 17427 df-topn 17428 df-0g 17446 df-gsum 17447 df-topgen 17448 df-pt 17449 df-prds 17452 df-xrs 17508 df-qtop 17513 df-imas 17514 df-xps 17516 df-mre 17590 df-mrc 17591 df-acs 17593 df-mgm 18650 df-sgrp 18729 df-mnd 18745 df-submnd 18794 df-mulg 19086 df-cntz 19333 df-cmn 19798 df-psmet 21389 df-xmet 21390 df-met 21391 df-bl 21392 df-mopn 21393 df-fbas 21394 df-fg 21395 df-cnfld 21398 df-top 22927 df-topon 22944 df-topsp 22966 df-bases 22979 df-cld 23052 df-ntr 23053 df-cls 23054 df-nei 23131 df-lp 23169 df-perf 23170 df-cn 23260 df-cnp 23261 df-t1 23347 df-haus 23348 df-tx 23595 df-hmeo 23788 df-fil 23879 df-fm 23971 df-flim 23972 df-flf 23973 df-xms 24353 df-ms 24354 df-tms 24355 df-cncf 24913 df-limc 25901 df-dv 25902

This theorem is referenced by: (None)

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