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Theorem asinsin 26633

Description: The arcsine function composed with sin is equal to the identity. This plus sinasin 26630 allow to view sin and arcsin as inverse operations to each other. For ease of use, we have not defined precisely the correct domain of correctness of this identity; in addition to the main region described here it is also true for some points on the branch cuts, namely when 𝐴 = (π / 2) − i𝑦 for nonnegative real 𝑦 and also symmetrically at 𝐴 = i𝑦 − (π / 2). In particular, when restricted to reals this identity extends to the closed interval [-(π / 2), (π / 2)], not just the open interval (see reasinsin 26637). (Contributed by Mario Carneiro, 2-Apr-2015.)

**Assertion**
Ref	Expression
asinsin	⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (arcsin‘(sin‘𝐴)) = 𝐴)

**Proof of Theorem asinsin**
Step	Hyp	Ref	Expression
1		sincl 16073	. . . 4 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ)
2	1	adantr 479	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (sin‘𝐴) ∈ ℂ)
3		asinval 26623	. . 3 ⊢ ((sin‘𝐴) ∈ ℂ → (arcsin‘(sin‘𝐴)) = (-i · (log‘((i · (sin‘𝐴)) + (√‘(1 − ((sin‘𝐴)↑2)))))))
4	2, 3	syl 17	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (arcsin‘(sin‘𝐴)) = (-i · (log‘((i · (sin‘𝐴)) + (√‘(1 − ((sin‘𝐴)↑2)))))))
5		ax-icn 11171	. . . . . . . 8 ⊢ i ∈ ℂ
6		mulcl 11196	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ (sin‘𝐴) ∈ ℂ) → (i · (sin‘𝐴)) ∈ ℂ)
7	5, 2, 6	sylancr 585	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (i · (sin‘𝐴)) ∈ ℂ)
8		simpl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 𝐴 ∈ ℂ)
9		mulcl 11196	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
10	5, 8, 9	sylancr 585	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (i · 𝐴) ∈ ℂ)
11		efcl 16030	. . . . . . . 8 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) ∈ ℂ)
12	10, 11	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (exp‘(i · 𝐴)) ∈ ℂ)
13	7, 12	pncan3d 11578	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((i · (sin‘𝐴)) + ((exp‘(i · 𝐴)) − (i · (sin‘𝐴)))) = (exp‘(i · 𝐴)))
14	12, 7	subcld 11575	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((exp‘(i · 𝐴)) − (i · (sin‘𝐴))) ∈ ℂ)
15		ax-1cn 11170	. . . . . . . . 9 ⊢ 1 ∈ ℂ
16	2	sqcld 14113	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((sin‘𝐴)↑2) ∈ ℂ)
17		subcl 11463	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ ((sin‘𝐴)↑2) ∈ ℂ) → (1 − ((sin‘𝐴)↑2)) ∈ ℂ)
18	15, 16, 17	sylancr 585	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (1 − ((sin‘𝐴)↑2)) ∈ ℂ)
19		binom2sub 14187	. . . . . . . . . 10 ⊢ (((exp‘(i · 𝐴)) ∈ ℂ ∧ (i · (sin‘𝐴)) ∈ ℂ) → (((exp‘(i · 𝐴)) − (i · (sin‘𝐴)))↑2) = ((((exp‘(i · 𝐴))↑2) − (2 · ((exp‘(i · 𝐴)) · (i · (sin‘𝐴))))) + ((i · (sin‘𝐴))↑2)))
20	12, 7, 19	syl2anc 582	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (((exp‘(i · 𝐴)) − (i · (sin‘𝐴)))↑2) = ((((exp‘(i · 𝐴))↑2) − (2 · ((exp‘(i · 𝐴)) · (i · (sin‘𝐴))))) + ((i · (sin‘𝐴))↑2)))
21	12	sqvald 14112	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((exp‘(i · 𝐴))↑2) = ((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))))
22		2cn 12291	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
23	22	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 2 ∈ ℂ)
24	23, 12, 7	mul12d 11427	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (2 · ((exp‘(i · 𝐴)) · (i · (sin‘𝐴)))) = ((exp‘(i · 𝐴)) · (2 · (i · (sin‘𝐴)))))
25	21, 24	oveq12d 7429	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (((exp‘(i · 𝐴))↑2) − (2 · ((exp‘(i · 𝐴)) · (i · (sin‘𝐴))))) = (((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))) − ((exp‘(i · 𝐴)) · (2 · (i · (sin‘𝐴))))))
26		coscl 16074	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ)
27	26	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝐴) ∈ ℂ)
28		subsq 14178	. . . . . . . . . . . . 13 ⊢ (((cos‘𝐴) ∈ ℂ ∧ (i · (sin‘𝐴)) ∈ ℂ) → (((cos‘𝐴)↑2) − ((i · (sin‘𝐴))↑2)) = (((cos‘𝐴) + (i · (sin‘𝐴))) · ((cos‘𝐴) − (i · (sin‘𝐴)))))
29	27, 7, 28	syl2anc 582	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (((cos‘𝐴)↑2) − ((i · (sin‘𝐴))↑2)) = (((cos‘𝐴) + (i · (sin‘𝐴))) · ((cos‘𝐴) − (i · (sin‘𝐴)))))
30		sqmul 14088	. . . . . . . . . . . . . . . 16 ⊢ ((i ∈ ℂ ∧ (sin‘𝐴) ∈ ℂ) → ((i · (sin‘𝐴))↑2) = ((i↑2) · ((sin‘𝐴)↑2)))
31	5, 2, 30	sylancr 585	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((i · (sin‘𝐴))↑2) = ((i↑2) · ((sin‘𝐴)↑2)))
32		i2 14170	. . . . . . . . . . . . . . . . 17 ⊢ (i↑2) = -1
33	32	oveq1i 7421	. . . . . . . . . . . . . . . 16 ⊢ ((i↑2) · ((sin‘𝐴)↑2)) = (-1 · ((sin‘𝐴)↑2))
34	16	mulm1d 11670	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (-1 · ((sin‘𝐴)↑2)) = -((sin‘𝐴)↑2))
35	33, 34	eqtrid 2782	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((i↑2) · ((sin‘𝐴)↑2)) = -((sin‘𝐴)↑2))
36	31, 35	eqtrd 2770	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((i · (sin‘𝐴))↑2) = -((sin‘𝐴)↑2))
37	36	oveq2d 7427	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (((cos‘𝐴)↑2) − ((i · (sin‘𝐴))↑2)) = (((cos‘𝐴)↑2) − -((sin‘𝐴)↑2)))
38	27	sqcld 14113	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((cos‘𝐴)↑2) ∈ ℂ)
39	38, 16	subnegd 11582	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (((cos‘𝐴)↑2) − -((sin‘𝐴)↑2)) = (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2)))
40	38, 16	addcomd 11420	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (((cos‘𝐴)↑2) + ((sin‘𝐴)↑2)) = (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)))
41	37, 39, 40	3eqtrd 2774	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (((cos‘𝐴)↑2) − ((i · (sin‘𝐴))↑2)) = (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)))
42		efival 16099	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) = ((cos‘𝐴) + (i · (sin‘𝐴))))
43	42	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (exp‘(i · 𝐴)) = ((cos‘𝐴) + (i · (sin‘𝐴))))
44	7	2timesd 12459	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (2 · (i · (sin‘𝐴))) = ((i · (sin‘𝐴)) + (i · (sin‘𝐴))))
45	43, 44	oveq12d 7429	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((exp‘(i · 𝐴)) − (2 · (i · (sin‘𝐴)))) = (((cos‘𝐴) + (i · (sin‘𝐴))) − ((i · (sin‘𝐴)) + (i · (sin‘𝐴)))))
46	27, 7, 7	pnpcan2d 11613	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (((cos‘𝐴) + (i · (sin‘𝐴))) − ((i · (sin‘𝐴)) + (i · (sin‘𝐴)))) = ((cos‘𝐴) − (i · (sin‘𝐴))))
47	45, 46	eqtrd 2770	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((exp‘(i · 𝐴)) − (2 · (i · (sin‘𝐴)))) = ((cos‘𝐴) − (i · (sin‘𝐴))))
48	43, 47	oveq12d 7429	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((exp‘(i · 𝐴)) · ((exp‘(i · 𝐴)) − (2 · (i · (sin‘𝐴))))) = (((cos‘𝐴) + (i · (sin‘𝐴))) · ((cos‘𝐴) − (i · (sin‘𝐴)))))
49		mulcl 11196	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℂ ∧ (i · (sin‘𝐴)) ∈ ℂ) → (2 · (i · (sin‘𝐴))) ∈ ℂ)
50	22, 7, 49	sylancr 585	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (2 · (i · (sin‘𝐴))) ∈ ℂ)
51	12, 12, 50	subdid 11674	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((exp‘(i · 𝐴)) · ((exp‘(i · 𝐴)) − (2 · (i · (sin‘𝐴))))) = (((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))) − ((exp‘(i · 𝐴)) · (2 · (i · (sin‘𝐴))))))
52	48, 51	eqtr3d 2772	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (((cos‘𝐴) + (i · (sin‘𝐴))) · ((cos‘𝐴) − (i · (sin‘𝐴)))) = (((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))) − ((exp‘(i · 𝐴)) · (2 · (i · (sin‘𝐴))))))
53	29, 41, 52	3eqtr3d 2778	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = (((exp‘(i · 𝐴)) · (exp‘(i · 𝐴))) − ((exp‘(i · 𝐴)) · (2 · (i · (sin‘𝐴))))))
54		sincossq 16123	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = 1)
55	54	adantr 479	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = 1)
56	25, 53, 55	3eqtr2d 2776	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (((exp‘(i · 𝐴))↑2) − (2 · ((exp‘(i · 𝐴)) · (i · (sin‘𝐴))))) = 1)
57	56, 36	oveq12d 7429	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((((exp‘(i · 𝐴))↑2) − (2 · ((exp‘(i · 𝐴)) · (i · (sin‘𝐴))))) + ((i · (sin‘𝐴))↑2)) = (1 + -((sin‘𝐴)↑2)))
58		negsub 11512	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ ((sin‘𝐴)↑2) ∈ ℂ) → (1 + -((sin‘𝐴)↑2)) = (1 − ((sin‘𝐴)↑2)))
59	15, 16, 58	sylancr 585	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (1 + -((sin‘𝐴)↑2)) = (1 − ((sin‘𝐴)↑2)))
60	20, 57, 59	3eqtrd 2774	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (((exp‘(i · 𝐴)) − (i · (sin‘𝐴)))↑2) = (1 − ((sin‘𝐴)↑2)))
61		halfre 12430	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℝ
62	61	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (1 / 2) ∈ ℝ)
63		negicn 11465	. . . . . . . . . . . . . . 15 ⊢ -i ∈ ℂ
64		mulcl 11196	. . . . . . . . . . . . . . 15 ⊢ ((-i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) ∈ ℂ)
65	63, 8, 64	sylancr 585	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (-i · 𝐴) ∈ ℂ)
66		efcl 16030	. . . . . . . . . . . . . 14 ⊢ ((-i · 𝐴) ∈ ℂ → (exp‘(-i · 𝐴)) ∈ ℂ)
67	65, 66	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (exp‘(-i · 𝐴)) ∈ ℂ)
68	12, 67	addcld 11237	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) ∈ ℂ)
69	68	recld 15145	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℜ‘((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴)))) ∈ ℝ)
70		halfgt0 12432	. . . . . . . . . . . 12 ⊢ 0 < (1 / 2)
71	70	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 0 < (1 / 2))
72	12	recld 15145	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℜ‘(exp‘(i · 𝐴))) ∈ ℝ)
73	67	recld 15145	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℜ‘(exp‘(-i · 𝐴))) ∈ ℝ)
74		asinsinlem 26632	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 0 < (ℜ‘(exp‘(i · 𝐴))))
75		negcl 11464	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ)
76	75	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → -𝐴 ∈ ℂ)
77		reneg 15076	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) = -(ℜ‘𝐴))
78	77	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℜ‘-𝐴) = -(ℜ‘𝐴))
79		halfpire 26210	. . . . . . . . . . . . . . . . . . . 20 ⊢ (π / 2) ∈ ℝ
80	79	renegcli 11525	. . . . . . . . . . . . . . . . . . 19 ⊢ -(π / 2) ∈ ℝ
81		recl 15061	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ)
82		iooneg 13452	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-(π / 2) ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ) → ((ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2)) ↔ -(ℜ‘𝐴) ∈ (-(π / 2)(,)--(π / 2))))
83	80, 79, 81, 82	mp3an12i 1463	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℂ → ((ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2)) ↔ -(ℜ‘𝐴) ∈ (-(π / 2)(,)--(π / 2))))
84	83	biimpa 475	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → -(ℜ‘𝐴) ∈ (-(π / 2)(,)--(π / 2)))
85	79	recni 11232	. . . . . . . . . . . . . . . . . . 19 ⊢ (π / 2) ∈ ℂ
86	85	negnegi 11534	. . . . . . . . . . . . . . . . . 18 ⊢ --(π / 2) = (π / 2)
87	86	oveq2i 7422	. . . . . . . . . . . . . . . . 17 ⊢ (-(π / 2)(,)--(π / 2)) = (-(π / 2)(,)(π / 2))
88	84, 87	eleqtrdi 2841	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → -(ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2)))
89	78, 88	eqeltrd 2831	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℜ‘-𝐴) ∈ (-(π / 2)(,)(π / 2)))
90		asinsinlem 26632	. . . . . . . . . . . . . . 15 ⊢ ((-𝐴 ∈ ℂ ∧ (ℜ‘-𝐴) ∈ (-(π / 2)(,)(π / 2))) → 0 < (ℜ‘(exp‘(i · -𝐴))))
91	76, 89, 90	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 0 < (ℜ‘(exp‘(i · -𝐴))))
92		mulneg12 11656	. . . . . . . . . . . . . . . . 17 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (-i · 𝐴) = (i · -𝐴))
93	5, 8, 92	sylancr 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (-i · 𝐴) = (i · -𝐴))
94	93	fveq2d 6894	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (exp‘(-i · 𝐴)) = (exp‘(i · -𝐴)))
95	94	fveq2d 6894	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℜ‘(exp‘(-i · 𝐴))) = (ℜ‘(exp‘(i · -𝐴))))
96	91, 95	breqtrrd 5175	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 0 < (ℜ‘(exp‘(-i · 𝐴))))
97	72, 73, 74, 96	addgt0d 11793	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 0 < ((ℜ‘(exp‘(i · 𝐴))) + (ℜ‘(exp‘(-i · 𝐴)))))
98	12, 67	readdd 15165	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℜ‘((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴)))) = ((ℜ‘(exp‘(i · 𝐴))) + (ℜ‘(exp‘(-i · 𝐴)))))
99	97, 98	breqtrrd 5175	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 0 < (ℜ‘((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴)))))
100	62, 69, 71, 99	mulgt0d 11373	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 0 < ((1 / 2) · (ℜ‘((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))))))
101		cosval 16070	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2))
102	101	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2))
103		2ne0 12320	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
104	103	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 2 ≠ 0)
105	68, 23, 104	divrec2d 11998	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2) = ((1 / 2) · ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴)))))
106	102, 105	eqtrd 2770	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝐴) = ((1 / 2) · ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴)))))
107	106	fveq2d 6894	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℜ‘(cos‘𝐴)) = (ℜ‘((1 / 2) · ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))))))
108		remul2 15081	. . . . . . . . . . . 12 ⊢ (((1 / 2) ∈ ℝ ∧ ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) ∈ ℂ) → (ℜ‘((1 / 2) · ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))))) = ((1 / 2) · (ℜ‘((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))))))
109	61, 68, 108	sylancr 585	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℜ‘((1 / 2) · ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))))) = ((1 / 2) · (ℜ‘((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))))))
110	107, 109	eqtrd 2770	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℜ‘(cos‘𝐴)) = ((1 / 2) · (ℜ‘((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))))))
111	100, 110	breqtrrd 5175	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 0 < (ℜ‘(cos‘𝐴)))
112	27, 7, 43	mvrraddd 11630	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((exp‘(i · 𝐴)) − (i · (sin‘𝐴))) = (cos‘𝐴))
113	112	fveq2d 6894	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℜ‘((exp‘(i · 𝐴)) − (i · (sin‘𝐴)))) = (ℜ‘(cos‘𝐴)))
114	111, 113	breqtrrd 5175	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → 0 < (ℜ‘((exp‘(i · 𝐴)) − (i · (sin‘𝐴)))))
115	14, 18, 60, 114	eqsqrt2d 15319	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((exp‘(i · 𝐴)) − (i · (sin‘𝐴))) = (√‘(1 − ((sin‘𝐴)↑2))))
116	115	oveq2d 7427	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((i · (sin‘𝐴)) + ((exp‘(i · 𝐴)) − (i · (sin‘𝐴)))) = ((i · (sin‘𝐴)) + (√‘(1 − ((sin‘𝐴)↑2)))))
117	13, 116	eqtr3d 2772	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (exp‘(i · 𝐴)) = ((i · (sin‘𝐴)) + (√‘(1 − ((sin‘𝐴)↑2)))))
118	117	fveq2d 6894	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (log‘(exp‘(i · 𝐴))) = (log‘((i · (sin‘𝐴)) + (√‘(1 − ((sin‘𝐴)↑2))))))
119		pire 26204	. . . . . . . . . 10 ⊢ π ∈ ℝ
120	119	renegcli 11525	. . . . . . . . 9 ⊢ -π ∈ ℝ
121	120	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → -π ∈ ℝ)
122	80	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → -(π / 2) ∈ ℝ)
123		elioore 13358	. . . . . . . . 9 ⊢ ((ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝐴) ∈ ℝ)
124	123	adantl 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℜ‘𝐴) ∈ ℝ)
125		pirp 26207	. . . . . . . . . . 11 ⊢ π ∈ ℝ⁺
126		rphalflt 13007	. . . . . . . . . . 11 ⊢ (π ∈ ℝ⁺ → (π / 2) < π)
127	125, 126	ax-mp 5	. . . . . . . . . 10 ⊢ (π / 2) < π
128	79, 119	ltnegi 11762	. . . . . . . . . 10 ⊢ ((π / 2) < π ↔ -π < -(π / 2))
129	127, 128	mpbi 229	. . . . . . . . 9 ⊢ -π < -(π / 2)
130	129	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → -π < -(π / 2))
131		eliooord 13387	. . . . . . . . . 10 ⊢ ((ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2)) → (-(π / 2) < (ℜ‘𝐴) ∧ (ℜ‘𝐴) < (π / 2)))
132	131	adantl 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (-(π / 2) < (ℜ‘𝐴) ∧ (ℜ‘𝐴) < (π / 2)))
133	132	simpld 493	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → -(π / 2) < (ℜ‘𝐴))
134	121, 122, 124, 130, 133	lttrd 11379	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → -π < (ℜ‘𝐴))
135		imre 15059	. . . . . . . . 9 ⊢ ((i · 𝐴) ∈ ℂ → (ℑ‘(i · 𝐴)) = (ℜ‘(-i · (i · 𝐴))))
136	10, 135	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℑ‘(i · 𝐴)) = (ℜ‘(-i · (i · 𝐴))))
137	5, 5	mulneg1i 11664	. . . . . . . . . . . 12 ⊢ (-i · i) = -(i · i)
138		ixi 11847	. . . . . . . . . . . . 13 ⊢ (i · i) = -1
139	138	negeqi 11457	. . . . . . . . . . . 12 ⊢ -(i · i) = --1
140	15	negnegi 11534	. . . . . . . . . . . 12 ⊢ --1 = 1
141	137, 139, 140	3eqtri 2762	. . . . . . . . . . 11 ⊢ (-i · i) = 1
142	141	oveq1i 7421	. . . . . . . . . 10 ⊢ ((-i · i) · 𝐴) = (1 · 𝐴)
143	63	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → -i ∈ ℂ)
144	5	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → i ∈ ℂ)
145	143, 144, 8	mulassd 11241	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → ((-i · i) · 𝐴) = (-i · (i · 𝐴)))
146		mullid 11217	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
147	146	adantr 479	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (1 · 𝐴) = 𝐴)
148	142, 145, 147	3eqtr3a 2794	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (-i · (i · 𝐴)) = 𝐴)
149	148	fveq2d 6894	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℜ‘(-i · (i · 𝐴))) = (ℜ‘𝐴))
150	136, 149	eqtrd 2770	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℑ‘(i · 𝐴)) = (ℜ‘𝐴))
151	134, 150	breqtrrd 5175	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → -π < (ℑ‘(i · 𝐴)))
152	119	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → π ∈ ℝ)
153	79	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (π / 2) ∈ ℝ)
154	132	simprd 494	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℜ‘𝐴) < (π / 2))
155	127	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (π / 2) < π)
156	124, 153, 152, 154, 155	lttrd 11379	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℜ‘𝐴) < π)
157	124, 152, 156	ltled 11366	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℜ‘𝐴) ≤ π)
158	150, 157	eqbrtrd 5169	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (ℑ‘(i · 𝐴)) ≤ π)
159		ellogrn 26304	. . . . . 6 ⊢ ((i · 𝐴) ∈ ran log ↔ ((i · 𝐴) ∈ ℂ ∧ -π < (ℑ‘(i · 𝐴)) ∧ (ℑ‘(i · 𝐴)) ≤ π))
160	10, 151, 158, 159	syl3anbrc 1341	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (i · 𝐴) ∈ ran log)
161		logef 26326	. . . . 5 ⊢ ((i · 𝐴) ∈ ran log → (log‘(exp‘(i · 𝐴))) = (i · 𝐴))
162	160, 161	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (log‘(exp‘(i · 𝐴))) = (i · 𝐴))
163	118, 162	eqtr3d 2772	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (log‘((i · (sin‘𝐴)) + (√‘(1 − ((sin‘𝐴)↑2))))) = (i · 𝐴))
164	163	oveq2d 7427	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (-i · (log‘((i · (sin‘𝐴)) + (√‘(1 − ((sin‘𝐴)↑2)))))) = (-i · (i · 𝐴)))
165	4, 164, 148	3eqtrd 2774	1 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ∈ (-(π / 2)(,)(π / 2))) → (arcsin‘(sin‘𝐴)) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 class class class wbr 5147 ran crn 5676 ‘cfv 6542 (class class class)co 7411 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 ici 11114 + caddc 11115 · cmul 11117 < clt 11252 ≤ cle 11253 − cmin 11448 -cneg 11449 / cdiv 11875 2c2 12271 ℝ⁺crp 12978 (,)cioo 13328 ↑cexp 14031 ℜcre 15048 ℑcim 15049 √csqrt 15184 expce 16009 sincsin 16011 cosccos 16012 πcpi 16014 logclog 26299 arcsincasin 26603

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ioc 13333 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-fac 14238 df-bc 14267 df-hash 14295 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-pi 16020 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 df-perf 22861 df-cn 22951 df-cnp 22952 df-haus 23039 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-xms 24046 df-ms 24047 df-tms 24048 df-cncf 24618 df-limc 25615 df-dv 25616 df-log 26301 df-asin 26606

This theorem is referenced by: acoscos 26634 reasinsin 26637 asinsinb 26638

W3C validator