fourierdlem56 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem56 46160

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 4102	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (-π[,]π))
3	2	sselda 3946	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (-π[,]π))
4		1ex 11170	. . . . . . . 8 ⊢ 1 ∈ V
5		ovex 7420	. . . . . . . 8 ⊢ (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈ V
6	4, 5	ifex 4539	. . . . . . 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 6979	. . . . . 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 2930	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ¬ 𝑠 = 0)
13	12	iffalsed 4499	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
14		elioore 13336	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
15	14	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
16	15	recnd 11202	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
17	16	halfcld 12427	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℂ)
18	17	sincld 16098	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
19		2cnd 12264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
20		fourierdlem44 46149	. . . . . . . 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 12290	. . . . . . . 8 ⊢ 2 ≠ 0
23	22	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
24	16, 18, 19, 21, 23	divdiv1d 11989	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝑠 / (sin‘(𝑠 / 2))) / 2) = (𝑠 / ((sin‘(𝑠 / 2)) · 2)))
25	18, 19	mulcomd 11195	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((sin‘(𝑠 / 2)) · 2) = (2 · (sin‘(𝑠 / 2))))
26	25	oveq2d 7403	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / ((sin‘(𝑠 / 2)) · 2)) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
27	24, 26	eqtr2d 2765	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) = ((𝑠 / (sin‘(𝑠 / 2))) / 2))
28	10, 13, 27	3eqtrd 2768	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐾‘𝑠) = ((𝑠 / (sin‘(𝑠 / 2))) / 2))
29	28	mpteq2dva 5200	. . 3 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝑠 / (sin‘(𝑠 / 2))) / 2)))
30	29	oveq2d 7403	. 2 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠))) = (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝑠 / (sin‘(𝑠 / 2))) / 2))))
31		reelprrecn 11160	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
32	31	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
33	16, 18, 21	divcld 11958	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / (sin‘(𝑠 / 2))) ∈ ℂ)
34		1red 11175	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
35	15	rehalfcld 12429	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℝ)
36	35	resincld 16111	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℝ)
37	34, 36	remulcld 11204	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 · (sin‘(𝑠 / 2))) ∈ ℝ)
38	35	recoscld 16112	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (cos‘(𝑠 / 2)) ∈ ℝ)
39	34	rehalfcld 12429	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 / 2) ∈ ℝ)
40	38, 39	remulcld 11204	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((cos‘(𝑠 / 2)) · (1 / 2)) ∈ ℝ)
41	40, 15	remulcld 11204	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠) ∈ ℝ)
42	37, 41	resubcld 11606	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((1 · (sin‘(𝑠 / 2))) − (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠)) ∈ ℝ)
43	36	resqcld 14090	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((sin‘(𝑠 / 2))↑2) ∈ ℝ)
44		2z 12565	. . . . . 6 ⊢ 2 ∈ ℤ
45	44	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℤ)
46	18, 21, 45	expne0d 14117	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((sin‘(𝑠 / 2))↑2) ≠ 0)
47	42, 43, 46	redivcld 12010	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((1 · (sin‘(𝑠 / 2))) − (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) ∈ ℝ)
48		1cnd 11169	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 1 ∈ ℂ)
49		recn 11158	. . . . . 6 ⊢ (𝑠 ∈ ℝ → 𝑠 ∈ ℂ)
50	49	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℂ)
51		1red 11175	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
52	32	dvmptid 25861	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑠)) = (𝑠 ∈ ℝ ↦ 1))
53		ioossre 13368	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℝ
54	53	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
55		tgioo4 24693	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
56		eqid 2729	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
57		iooretop 24653	. . . . . 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 25867	. . . 4 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 1))
60		elsni 4606	. . . . . . 7 ⊢ ((sin‘(𝑠 / 2)) ∈ {0} → (sin‘(𝑠 / 2)) = 0)
61	60	necon3ai 2950	. . . . . 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 3925	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ (ℂ ∖ {0}))
64	17	coscld 16099	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (cos‘(𝑠 / 2)) ∈ ℂ)
65	48	halfcld 12427	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 / 2) ∈ ℂ)
66	64, 65	mulcld 11194	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((cos‘(𝑠 / 2)) · (1 / 2)) ∈ ℂ)
67		cnelprrecn 11161	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
68	67	a1i 11	. . . . 5 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
69		sinf 16092	. . . . . . 7 ⊢ sin:ℂ⟶ℂ
70	69	a1i 11	. . . . . 6 ⊢ (𝜑 → sin:ℂ⟶ℂ)
71	70	ffvelcdmda 7056	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
72		cosf 16093	. . . . . . 7 ⊢ cos:ℂ⟶ℂ
73	72	a1i 11	. . . . . 6 ⊢ (𝜑 → cos:ℂ⟶ℂ)
74	73	ffvelcdmda 7056	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ ℂ)
75		2cnd 12264	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℂ)
76	22	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
77	32, 16, 34, 59, 75, 76	dvmptdivc 25869	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝑠 / 2))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (1 / 2)))
78		ffn 6688	. . . . . . . . . . 11 ⊢ (sin:ℂ⟶ℂ → sin Fn ℂ)
79	69, 78	ax-mp 5	. . . . . . . . . 10 ⊢ sin Fn ℂ
80		dffn5 6919	. . . . . . . . . 10 ⊢ (sin Fn ℂ ↔ sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
81	79, 80	mpbi 230	. . . . . . . . 9 ⊢ sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥))
82	81	eqcomi 2738	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ ↦ (sin‘𝑥)) = sin
83	82	oveq2i 7398	. . . . . . 7 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (ℂ D sin)
84		dvsin 25886	. . . . . . 7 ⊢ (ℂ D sin) = cos
85		ffn 6688	. . . . . . . . 9 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ)
86	72, 85	ax-mp 5	. . . . . . . 8 ⊢ cos Fn ℂ
87		dffn5 6919	. . . . . . . 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 2756	. . . . . 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 6858	. . . . 5 ⊢ (𝑥 = (𝑠 / 2) → (sin‘𝑥) = (sin‘(𝑠 / 2)))
92		fveq2 6858	. . . . 5 ⊢ (𝑥 = (𝑠 / 2) → (cos‘𝑥) = (cos‘(𝑠 / 2)))
93	32, 68, 17, 39, 71, 74, 77, 90, 91, 92	dvmptco 25876	. . . 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 25878	. . 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 25869	. 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 11202	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℂ)
97	96	halfcld 12427	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (𝑠 / 2) ∈ ℂ)
98	97	sincld 16098	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (sin‘(𝑠 / 2)) ∈ ℂ)
99	98	mullidd 11192	. . . . . . 7 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (1 · (sin‘(𝑠 / 2))) = (sin‘(𝑠 / 2)))
100	97	coscld 16099	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (cos‘(𝑠 / 2)) ∈ ℂ)
101		2cnd 12264	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 2 ∈ ℂ)
102	22	a1i 11	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 2 ≠ 0)
103	100, 101, 102	divrecd 11961	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → ((cos‘(𝑠 / 2)) / 2) = ((cos‘(𝑠 / 2)) · (1 / 2)))
104	103	eqcomd 2735	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → ((cos‘(𝑠 / 2)) · (1 / 2)) = ((cos‘(𝑠 / 2)) / 2))
105	104	oveq1d 7402	. . . . . . 7 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠) = (((cos‘(𝑠 / 2)) / 2) · 𝑠))
106	99, 105	oveq12d 7405	. . . . . 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 7402	. . . . 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 7402	. . . 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 5202	. . 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 2768	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 1540 ∈ wcel 2109 ≠ wne 2925 Vcvv 3447 ∖ cdif 3911 ⊆ wss 3914 ifcif 4488 {csn 4589 {cpr 4591 ↦ cmpt 5188 ran crn 5639 Fn wfn 6506 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 · cmul 11073 − cmin 11405 -cneg 11406 / cdiv 11835 2c2 12241 ℤcz 12529 (,)cioo 13306 [,]cicc 13309 ↑cexp 14026 sincsin 16029 cosccos 16030 πcpi 16032 TopOpenctopn 17384 topGenctg 17400 ℂ_fldccnfld 21264 D cdv 25764

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-sin 16035 df-cos 16036 df-pi 16038 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-t1 23201 df-haus 23202 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768

This theorem is referenced by: (None)

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