fourierdlem56 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem56 46876

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 4093	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (-π[,]π))
3	2	sselda 3937	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ (-π[,]π))
4		1ex 11198	. . . . . . . 8 ⊢ 1 ∈ V
5		ovex 7443	. . . . . . . 8 ⊢ (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈ V
6	4, 5	ifex 4538	. . . . . . 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 7001	. . . . . 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 595	. . . . 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 2963	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ¬ 𝑠 = 0)
13	12	iffalsed 4498	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
14		elioore 13397	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
15	14	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
16	15	recnd 11232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
17	16	halfcld 12484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℂ)
18	17	sincld 16181	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
19		2cnd 12314	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
20		fourierdlem44 46865	. . . . . . . 8 ⊢ ((𝑠 ∈ (-π[,]π) ∧ 𝑠 ≠ 0) → (sin‘(𝑠 / 2)) ≠ 0)
21	3, 11, 20	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ≠ 0)
22		2ne0 12342	. . . . . . . 8 ⊢ 2 ≠ 0
23	22	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
24	16, 18, 19, 21, 23	divdiv1d 12017	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝑠 / (sin‘(𝑠 / 2))) / 2) = (𝑠 / ((sin‘(𝑠 / 2)) · 2)))
25	18, 19	mulcomd 11225	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((sin‘(𝑠 / 2)) · 2) = (2 · (sin‘(𝑠 / 2))))
26	25	oveq2d 7426	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / ((sin‘(𝑠 / 2)) · 2)) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
27	24, 26	eqtr2d 2799	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) = ((𝑠 / (sin‘(𝑠 / 2))) / 2))
28	10, 13, 27	3eqtrd 2802	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐾‘𝑠) = ((𝑠 / (sin‘(𝑠 / 2))) / 2))
29	28	mpteq2dva 5204	. . 3 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝑠 / (sin‘(𝑠 / 2))) / 2)))
30	29	oveq2d 7426	. 2 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠))) = (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝑠 / (sin‘(𝑠 / 2))) / 2))))
31		reelprrecn 11187	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
32	31	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
33	16, 18, 21	divcld 11986	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / (sin‘(𝑠 / 2))) ∈ ℂ)
34		1red 11204	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
35	15	rehalfcld 12486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℝ)
36	35	resincld 16194	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℝ)
37	34, 36	remulcld 11234	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 · (sin‘(𝑠 / 2))) ∈ ℝ)
38	35	recoscld 16195	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (cos‘(𝑠 / 2)) ∈ ℝ)
39	34	rehalfcld 12486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 / 2) ∈ ℝ)
40	38, 39	remulcld 11234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((cos‘(𝑠 / 2)) · (1 / 2)) ∈ ℝ)
41	40, 15	remulcld 11234	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠) ∈ ℝ)
42	37, 41	resubcld 11637	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((1 · (sin‘(𝑠 / 2))) − (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠)) ∈ ℝ)
43	36	resqcld 14157	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((sin‘(𝑠 / 2))↑2) ∈ ℝ)
44		2z 12621	. . . . . 6 ⊢ 2 ∈ ℤ
45	44	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℤ)
46	18, 21, 45	expne0d 14184	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((sin‘(𝑠 / 2))↑2) ≠ 0)
47	42, 43, 46	redivcld 12038	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((1 · (sin‘(𝑠 / 2))) − (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) ∈ ℝ)
48		1cnd 11197	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 1 ∈ ℂ)
49		recn 11185	. . . . . 6 ⊢ (𝑠 ∈ ℝ → 𝑠 ∈ ℂ)
50	49	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℂ)
51		1red 11204	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
52	32	dvmptid 26116	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑠)) = (𝑠 ∈ ℝ ↦ 1))
53		ioossre 13429	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℝ
54	53	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
55		tgioo4 24962	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
56		eqid 2763	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
57		iooretop 24922	. . . . . 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 26122	. . . 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 2983	. . . . . 6 ⊢ ((sin‘(𝑠 / 2)) ≠ 0 → ¬ (sin‘(𝑠 / 2)) ∈ {0})
62	21, 61	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ¬ (sin‘(𝑠 / 2)) ∈ {0})
63	18, 62	eldifd 3916	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ (ℂ ∖ {0}))
64	17	coscld 16182	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (cos‘(𝑠 / 2)) ∈ ℂ)
65	48	halfcld 12484	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 / 2) ∈ ℂ)
66	64, 65	mulcld 11224	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((cos‘(𝑠 / 2)) · (1 / 2)) ∈ ℂ)
67		cnelprrecn 11188	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
68	67	a1i 11	. . . . 5 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
69		sinf 16175	. . . . . . 7 ⊢ sin:ℂ⟶ℂ
70	69	a1i 11	. . . . . 6 ⊢ (𝜑 → sin:ℂ⟶ℂ)
71	70	ffvelcdmda 7079	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
72		cosf 16176	. . . . . . 7 ⊢ cos:ℂ⟶ℂ
73	72	a1i 11	. . . . . 6 ⊢ (𝜑 → cos:ℂ⟶ℂ)
74	73	ffvelcdmda 7079	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ ℂ)
75		2cnd 12314	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℂ)
76	22	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
77	32, 16, 34, 59, 75, 76	dvmptdivc 26124	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝑠 / 2))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (1 / 2)))
78		ffn 6705	. . . . . . . . . . 11 ⊢ (sin:ℂ⟶ℂ → sin Fn ℂ)
79	69, 78	ax-mp 5	. . . . . . . . . 10 ⊢ sin Fn ℂ
80		dffn5 6939	. . . . . . . . . 10 ⊢ (sin Fn ℂ ↔ sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
81	79, 80	mpbi 233	. . . . . . . . 9 ⊢ sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥))
82	81	eqcomi 2772	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ ↦ (sin‘𝑥)) = sin
83	82	oveq2i 7421	. . . . . . 7 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (ℂ D sin)
84		dvsin 26141	. . . . . . 7 ⊢ (ℂ D sin) = cos
85		ffn 6705	. . . . . . . . 9 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ)
86	72, 85	ax-mp 5	. . . . . . . 8 ⊢ cos Fn ℂ
87		dffn5 6939	. . . . . . . 8 ⊢ (cos Fn ℂ ↔ cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
88	86, 87	mpbi 233	. . . . . . 7 ⊢ cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥))
89	83, 84, 88	3eqtri 2790	. . . . . 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 6881	. . . . 5 ⊢ (𝑥 = (𝑠 / 2) → (sin‘𝑥) = (sin‘(𝑠 / 2)))
92		fveq2 6881	. . . . 5 ⊢ (𝑥 = (𝑠 / 2) → (cos‘𝑥) = (cos‘(𝑠 / 2)))
93	32, 68, 17, 39, 71, 74, 77, 90, 91, 92	dvmptco 26131	. . . 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 26133	. . 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 26124	. 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 11232	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℂ)
97	96	halfcld 12484	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (𝑠 / 2) ∈ ℂ)
98	97	sincld 16181	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (sin‘(𝑠 / 2)) ∈ ℂ)
99	98	mullidd 11222	. . . . . . 7 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (1 · (sin‘(𝑠 / 2))) = (sin‘(𝑠 / 2)))
100	97	coscld 16182	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (cos‘(𝑠 / 2)) ∈ ℂ)
101		2cnd 12314	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 2 ∈ ℂ)
102	22	a1i 11	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 2 ≠ 0)
103	100, 101, 102	divrecd 11989	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → ((cos‘(𝑠 / 2)) / 2) = ((cos‘(𝑠 / 2)) · (1 / 2)))
104	103	eqcomd 2769	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → ((cos‘(𝑠 / 2)) · (1 / 2)) = ((cos‘(𝑠 / 2)) / 2))
105	104	oveq1d 7425	. . . . . . 7 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠) = (((cos‘(𝑠 / 2)) / 2) · 𝑠))
106	99, 105	oveq12d 7428	. . . . . 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 7425	. . . . 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 7425	. . . 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 5206	. . 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 2802	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 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 Vcvv 3455 ∖ cdif 3902 ⊆ wss 3905 ifcif 4487 {csn 4589 {cpr 4591 ↦ cmpt 5192 ran crn 5662 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 ℝcr 11094 0cc0 11095 1c1 11096 · cmul 11100 − cmin 11436 -cneg 11437 / cdiv 11866 2c2 12290 ℤcz 12586 (,)cioo 13367 [,]cicc 13370 ↑cexp 14093 sincsin 16112 cosccos 16113 πcpi 16115 TopOpenctopn 17469 topGenctg 17485 ℂ_fldccnfld 21522 D cdv 26022

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 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174

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 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-fac 14306 df-bc 14335 df-hash 14363 df-shft 15100 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-limsup 15518 df-clim 15535 df-rlim 15536 df-sum 15734 df-ef 16116 df-sin 16118 df-cos 16119 df-pi 16121 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-fbas 21519 df-fg 21520 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cld 23176 df-ntr 23177 df-cls 23178 df-nei 23255 df-lp 23293 df-perf 23294 df-cn 23384 df-cnp 23385 df-t1 23471 df-haus 23472 df-tx 23719 df-hmeo 23912 df-fil 24003 df-fm 24095 df-flim 24096 df-flf 24097 df-xms 24477 df-ms 24478 df-tms 24479 df-cncf 25037 df-limc 26025 df-dv 26026

This theorem is referenced by: (None)

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