fourierdlem56 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem56		Structured version Visualization version GIF version

Theorem fourierdlem56 47266

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 4075	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ( -π[,]π))
3	2	sselda 3908	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ( -π[,]π))
4		1ex 11307	. . . . . . . 8 ⊢ 1 ∈ V
5		ovex 7454	. . . . . . . 8 ⊢ (𝑠 / (2 · (sin‘(𝑠 / 2)))) ∈ V
6	4, 5	ifex 4532	. . . . . . 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 7006	. . . . . 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 596	. . . . 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 2960	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ¬ 𝑠 = 0)
13	12	iffalsed 4492	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
14		elioore 13510	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
15	14	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
16	15	recnd 11341	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℂ)
17	16	halfcld 12595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℂ)
18	17	sincld 16304	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℂ)
19		2cnd 12425	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
20		fourierdlem44 47255	. . . . . . . 8 ⊢ ((𝑠 ∈ ( -π[,]π) ∧ 𝑠 ≠ 0) → (sin‘(𝑠 / 2)) ≠ 0)
21	3, 11, 20	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ≠ 0)
22		2ne0 12453	. . . . . . . 8 ⊢ 2 ≠ 0
23	22	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
24	16, 18, 19, 21, 23	divdiv1d 12128	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝑠 / (sin‘(𝑠 / 2))) / 2) = (𝑠 / ((sin‘(𝑠 / 2)) · 2)))
25	18, 19	mulcomd 11334	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((sin‘(𝑠 / 2)) · 2) = (2 · (sin‘(𝑠 / 2))))
26	25	oveq2d 7437	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / ((sin‘(𝑠 / 2)) · 2)) = (𝑠 / (2 · (sin‘(𝑠 / 2)))))
27	24, 26	eqtr2d 2796	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / (2 · (sin‘(𝑠 / 2)))) = ((𝑠 / (sin‘(𝑠 / 2))) / 2))
28	10, 13, 27	3eqtrd 2799	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐾‘𝑠) = ((𝑠 / (sin‘(𝑠 / 2))) / 2))
29	28	mpteq2dva 5197	. . 3 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝑠 / (sin‘(𝑠 / 2))) / 2)))
30	29	oveq2d 7437	. 2 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠))) = (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝑠 / (sin‘(𝑠 / 2))) / 2))))
31		reelprrecn 11296	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
32	31	a1i 11	. . 3 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
33	16, 18, 21	divcld 12097	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / (sin‘(𝑠 / 2))) ∈ ℂ)
34		1red 11313	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
35	15	rehalfcld 12597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝑠 / 2) ∈ ℝ)
36	35	resincld 16317	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ ℝ)
37	34, 36	remulcld 11343	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 · (sin‘(𝑠 / 2))) ∈ ℝ)
38	35	recoscld 16318	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (cos‘(𝑠 / 2)) ∈ ℝ)
39	34	rehalfcld 12597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 / 2) ∈ ℝ)
40	38, 39	remulcld 11343	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((cos‘(𝑠 / 2)) · (1 / 2)) ∈ ℝ)
41	40, 15	remulcld 11343	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠) ∈ ℝ)
42	37, 41	resubcld 11748	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((1 · (sin‘(𝑠 / 2))) − (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠)) ∈ ℝ)
43	36	resqcld 14272	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((sin‘(𝑠 / 2))↑2) ∈ ℝ)
44		2z 12732	. . . . . 6 ⊢ 2 ∈ ℤ
45	44	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 2 ∈ ℤ)
46	18, 21, 45	expne0d 14299	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((sin‘(𝑠 / 2))↑2) ≠ 0)
47	42, 43, 46	redivcld 12149	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((1 · (sin‘(𝑠 / 2))) − (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) ∈ ℝ)
48		1cnd 11306	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 1 ∈ ℂ)
49		recn 11294	. . . . . 6 ⊢ (𝑠 ∈ ℝ → 𝑠 ∈ ℂ)
50	49	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℂ)
51		1red 11313	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
52	32	dvmptid 26283	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑠)) = (𝑠 ∈ ℝ ↦ 1))
53		ioossre 13542	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℝ
54	53	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
55		tgioo4 25130	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
56		eqid 2760	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
57		iooretop 25090	. . . . . 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 26289	. . . 4 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ 𝑠)) = (𝑠 ∈ (𝐴(,)𝐵) ↦ 1))
60		elsni 4597	. . . . . . 7 ⊢ ((sin‘(𝑠 / 2)) ∈ {0} → (sin‘(𝑠 / 2)) = 0)
61	60	necon3ai 2980	. . . . . 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 4050	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (sin‘(𝑠 / 2)) ∈ (ℂ ∖ {0}))
64	17	coscld 16305	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (cos‘(𝑠 / 2)) ∈ ℂ)
65	48	halfcld 12595	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (1 / 2) ∈ ℂ)
66	64, 65	mulcld 11333	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((cos‘(𝑠 / 2)) · (1 / 2)) ∈ ℂ)
67		cnelprrecn 11297	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
68	67	a1i 11	. . . . 5 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
69		sinf 16298	. . . . . . 7 ⊢ sin:ℂ⟶ℂ
70	69	a1i 11	. . . . . 6 ⊢ (𝜑 → sin:ℂ⟶ℂ)
71	70	ffvelcdmda 7085	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
72		cosf 16299	. . . . . . 7 ⊢ cos:ℂ⟶ℂ
73	72	a1i 11	. . . . . 6 ⊢ (𝜑 → cos:ℂ⟶ℂ)
74	73	ffvelcdmda 7085	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ ℂ)
75		2cnd 12425	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℂ)
76	22	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
77	32, 16, 34, 59, 75, 76	dvmptdivc 26291	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝑠 / 2))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (1 / 2)))
78		ffn 6710	. . . . . . . . . . 11 ⊢ (sin:ℂ⟶ℂ → sin Fn ℂ)
79	69, 78	ax-mp 5	. . . . . . . . . 10 ⊢ sin Fn ℂ
80		dffn5 6944	. . . . . . . . . 10 ⊢ (sin Fn ℂ ↔ sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
81	79, 80	mpbi 233	. . . . . . . . 9 ⊢ sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥))
82	81	eqcomi 2769	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ ↦ (sin‘𝑥)) = sin
83	82	oveq2i 7432	. . . . . . 7 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (ℂ D sin)
84		dvsin 26308	. . . . . . 7 ⊢ (ℂ D sin) = cos
85		ffn 6710	. . . . . . . . 9 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ)
86	72, 85	ax-mp 5	. . . . . . . 8 ⊢ cos Fn ℂ
87		dffn5 6944	. . . . . . . 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 2787	. . . . . 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 6886	. . . . 5 ⊢ (𝑥 = (𝑠 / 2) → (sin‘𝑥) = (sin‘(𝑠 / 2)))
92		fveq2 6886	. . . . 5 ⊢ (𝑥 = (𝑠 / 2) → (cos‘𝑥) = (cos‘(𝑠 / 2)))
93	32, 68, 17, 39, 71, 74, 77, 90, 91, 92	dvmptco 26298	. . . 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 26300	. . 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 26291	. 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 11341	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℂ)
97	96	halfcld 12595	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (𝑠 / 2) ∈ ℂ)
98	97	sincld 16304	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (sin‘(𝑠 / 2)) ∈ ℂ)
99	98	mullidd 11331	. . . . . . 7 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (1 · (sin‘(𝑠 / 2))) = (sin‘(𝑠 / 2)))
100	97	coscld 16305	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (cos‘(𝑠 / 2)) ∈ ℂ)
101		2cnd 12425	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 2 ∈ ℂ)
102	22	a1i 11	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 2 ≠ 0)
103	100, 101, 102	divrecd 12100	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → ((cos‘(𝑠 / 2)) / 2) = ((cos‘(𝑠 / 2)) · (1 / 2)))
104	103	eqcomd 2766	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → ((cos‘(𝑠 / 2)) · (1 / 2)) = ((cos‘(𝑠 / 2)) / 2))
105	104	oveq1d 7436	. . . . . . 7 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → (((cos‘(𝑠 / 2)) · (1 / 2)) · 𝑠) = (((cos‘(𝑠 / 2)) / 2) · 𝑠))
106	99, 105	oveq12d 7439	. . . . . 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 7436	. . . . 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 7436	. . . 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 5199	. . 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 2799	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 Vcvv 3450 ⊆ wss 3895 ∖ cdif 4045 ifcif 4481 {csn 4584 {cpr 4586 ↦ cmpt 5185 ran crn 5651 Fn wfn 6534 ⟶wf 6535 ‘cfv 6539 (class class class)co 7421 ℂcc 11202 ℝcr 11203 0cc0 11204 1c1 11205 · cmul 11209 − cmin 11545 -cneg 11546 / cdiv 11977 2c2 12401 ℤcz 12697 (,)cioo 13480 [,]cicc 13483 ↑cexp 14208 sincsin 16235 cosccos 16236 πcpi 16238 TopOpenctopn 17598 topGenctg 17614 ℂ_fldccnfld 21684 D cdv 26189

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-inf2 9646 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-pre-sup 11282 ax-addf 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-supp 8181 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-er 8721 df-map 8853 df-pm 8854 df-ixp 8930 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-fsupp 9358 df-fi 9407 df-sup 9438 df-inf 9439 df-oi 9508 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-z 12698 df-dec 12819 df-uz 12970 df-q 13080 df-rp 13125 df-xneg 13245 df-xadd 13246 df-xmul 13247 df-ioo 13484 df-ioc 13485 df-ico 13486 df-icc 13487 df-fz 13644 df-fzo 13793 df-fl 13936 df-mod 14014 df-seq 14149 df-exp 14209 df-fac 14422 df-bc 14451 df-hash 14479 df-shft 15226 df-cj 15272 df-re 15273 df-im 15274 df-sqrt 15408 df-abs 15409 df-limsup 15644 df-clim 15661 df-rlim 15662 df-sum 15860 df-ef 16239 df-sin 16241 df-cos 16242 df-pi 16244 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-starv 17449 df-sca 17450 df-vsca 17451 df-ip 17452 df-tset 17453 df-ple 17454 df-ds 17456 df-unif 17457 df-hom 17458 df-cco 17459 df-rest 17599 df-topn 17600 df-0g 17618 df-gsum 17619 df-topgen 17620 df-pt 17621 df-prds 17624 df-xrs 17680 df-qtop 17685 df-imas 17686 df-xps 17688 df-mre 17762 df-mrc 17763 df-acs 17765 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-submnd 18985 df-mulg 19284 df-cntz 19537 df-cmn 20002 df-psmet 21676 df-xmet 21677 df-met 21678 df-bl 21679 df-mopn 21680 df-fbas 21681 df-fg 21682 df-cnfld 21685 df-top 23218 df-topon 23235 df-topsp 23257 df-bases 23270 df-cld 23343 df-ntr 23344 df-cls 23345 df-nei 23422 df-lp 23460 df-perf 23461 df-cn 23551 df-cnp 23552 df-t1 23638 df-haus 23639 df-tx 23887 df-hmeo 24080 df-fil 24171 df-fm 24263 df-flim 24264 df-flf 24265 df-xms 24645 df-ms 24646 df-tms 24647 df-cncf 25205 df-limc 26192 df-dv 26193

This theorem is used by: (None)

W3C validator