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Theorem asinsin 26258
Description: The arcsine function composed with sin is equal to the identity. This plus sinasin 26255 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 = (Ο€ / 2) βˆ’ i𝑦 for nonnegative real 𝑦 and also symmetrically at 𝐴 = i𝑦 βˆ’ (Ο€ / 2). In particular, when restricted to reals this identity extends to the closed interval [-(Ο€ / 2), (Ο€ / 2)], not just the open interval (see reasinsin 26262). (Contributed by Mario Carneiro, 2-Apr-2015.)
Assertion
Ref Expression
asinsin ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (arcsinβ€˜(sinβ€˜π΄)) = 𝐴)

Proof of Theorem asinsin
StepHypRef Expression
1 sincl 16015 . . . 4 (𝐴 ∈ β„‚ β†’ (sinβ€˜π΄) ∈ β„‚)
21adantr 482 . . 3 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (sinβ€˜π΄) ∈ β„‚)
3 asinval 26248 . . 3 ((sinβ€˜π΄) ∈ β„‚ β†’ (arcsinβ€˜(sinβ€˜π΄)) = (-i Β· (logβ€˜((i Β· (sinβ€˜π΄)) + (βˆšβ€˜(1 βˆ’ ((sinβ€˜π΄)↑2)))))))
42, 3syl 17 . 2 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (arcsinβ€˜(sinβ€˜π΄)) = (-i Β· (logβ€˜((i Β· (sinβ€˜π΄)) + (βˆšβ€˜(1 βˆ’ ((sinβ€˜π΄)↑2)))))))
5 ax-icn 11117 . . . . . . . 8 i ∈ β„‚
6 mulcl 11142 . . . . . . . 8 ((i ∈ β„‚ ∧ (sinβ€˜π΄) ∈ β„‚) β†’ (i Β· (sinβ€˜π΄)) ∈ β„‚)
75, 2, 6sylancr 588 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· (sinβ€˜π΄)) ∈ β„‚)
8 simpl 484 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 𝐴 ∈ β„‚)
9 mulcl 11142 . . . . . . . . 9 ((i ∈ β„‚ ∧ 𝐴 ∈ β„‚) β†’ (i Β· 𝐴) ∈ β„‚)
105, 8, 9sylancr 588 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· 𝐴) ∈ β„‚)
11 efcl 15972 . . . . . . . 8 ((i Β· 𝐴) ∈ β„‚ β†’ (expβ€˜(i Β· 𝐴)) ∈ β„‚)
1210, 11syl 17 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜(i Β· 𝐴)) ∈ β„‚)
137, 12pncan3d 11522 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i Β· (sinβ€˜π΄)) + ((expβ€˜(i Β· 𝐴)) βˆ’ (i Β· (sinβ€˜π΄)))) = (expβ€˜(i Β· 𝐴)))
1412, 7subcld 11519 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((expβ€˜(i Β· 𝐴)) βˆ’ (i Β· (sinβ€˜π΄))) ∈ β„‚)
15 ax-1cn 11116 . . . . . . . . 9 1 ∈ β„‚
162sqcld 14056 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((sinβ€˜π΄)↑2) ∈ β„‚)
17 subcl 11407 . . . . . . . . 9 ((1 ∈ β„‚ ∧ ((sinβ€˜π΄)↑2) ∈ β„‚) β†’ (1 βˆ’ ((sinβ€˜π΄)↑2)) ∈ β„‚)
1815, 16, 17sylancr 588 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (1 βˆ’ ((sinβ€˜π΄)↑2)) ∈ β„‚)
19 binom2sub 14130 . . . . . . . . . 10 (((expβ€˜(i Β· 𝐴)) ∈ β„‚ ∧ (i Β· (sinβ€˜π΄)) ∈ β„‚) β†’ (((expβ€˜(i Β· 𝐴)) βˆ’ (i Β· (sinβ€˜π΄)))↑2) = ((((expβ€˜(i Β· 𝐴))↑2) βˆ’ (2 Β· ((expβ€˜(i Β· 𝐴)) Β· (i Β· (sinβ€˜π΄))))) + ((i Β· (sinβ€˜π΄))↑2)))
2012, 7, 19syl2anc 585 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((expβ€˜(i Β· 𝐴)) βˆ’ (i Β· (sinβ€˜π΄)))↑2) = ((((expβ€˜(i Β· 𝐴))↑2) βˆ’ (2 Β· ((expβ€˜(i Β· 𝐴)) Β· (i Β· (sinβ€˜π΄))))) + ((i Β· (sinβ€˜π΄))↑2)))
2112sqvald 14055 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((expβ€˜(i Β· 𝐴))↑2) = ((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(i Β· 𝐴))))
22 2cn 12235 . . . . . . . . . . . . . 14 2 ∈ β„‚
2322a1i 11 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 2 ∈ β„‚)
2423, 12, 7mul12d 11371 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (2 Β· ((expβ€˜(i Β· 𝐴)) Β· (i Β· (sinβ€˜π΄)))) = ((expβ€˜(i Β· 𝐴)) Β· (2 Β· (i Β· (sinβ€˜π΄)))))
2521, 24oveq12d 7380 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((expβ€˜(i Β· 𝐴))↑2) βˆ’ (2 Β· ((expβ€˜(i Β· 𝐴)) Β· (i Β· (sinβ€˜π΄))))) = (((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(i Β· 𝐴))) βˆ’ ((expβ€˜(i Β· 𝐴)) Β· (2 Β· (i Β· (sinβ€˜π΄))))))
26 coscl 16016 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ (cosβ€˜π΄) ∈ β„‚)
2726adantr 482 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (cosβ€˜π΄) ∈ β„‚)
28 subsq 14121 . . . . . . . . . . . . 13 (((cosβ€˜π΄) ∈ β„‚ ∧ (i Β· (sinβ€˜π΄)) ∈ β„‚) β†’ (((cosβ€˜π΄)↑2) βˆ’ ((i Β· (sinβ€˜π΄))↑2)) = (((cosβ€˜π΄) + (i Β· (sinβ€˜π΄))) Β· ((cosβ€˜π΄) βˆ’ (i Β· (sinβ€˜π΄)))))
2927, 7, 28syl2anc 585 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((cosβ€˜π΄)↑2) βˆ’ ((i Β· (sinβ€˜π΄))↑2)) = (((cosβ€˜π΄) + (i Β· (sinβ€˜π΄))) Β· ((cosβ€˜π΄) βˆ’ (i Β· (sinβ€˜π΄)))))
30 sqmul 14031 . . . . . . . . . . . . . . . 16 ((i ∈ β„‚ ∧ (sinβ€˜π΄) ∈ β„‚) β†’ ((i Β· (sinβ€˜π΄))↑2) = ((i↑2) Β· ((sinβ€˜π΄)↑2)))
315, 2, 30sylancr 588 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i Β· (sinβ€˜π΄))↑2) = ((i↑2) Β· ((sinβ€˜π΄)↑2)))
32 i2 14113 . . . . . . . . . . . . . . . . 17 (i↑2) = -1
3332oveq1i 7372 . . . . . . . . . . . . . . . 16 ((i↑2) Β· ((sinβ€˜π΄)↑2)) = (-1 Β· ((sinβ€˜π΄)↑2))
3416mulm1d 11614 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (-1 Β· ((sinβ€˜π΄)↑2)) = -((sinβ€˜π΄)↑2))
3533, 34eqtrid 2789 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i↑2) Β· ((sinβ€˜π΄)↑2)) = -((sinβ€˜π΄)↑2))
3631, 35eqtrd 2777 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i Β· (sinβ€˜π΄))↑2) = -((sinβ€˜π΄)↑2))
3736oveq2d 7378 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((cosβ€˜π΄)↑2) βˆ’ ((i Β· (sinβ€˜π΄))↑2)) = (((cosβ€˜π΄)↑2) βˆ’ -((sinβ€˜π΄)↑2)))
3827sqcld 14056 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((cosβ€˜π΄)↑2) ∈ β„‚)
3938, 16subnegd 11526 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((cosβ€˜π΄)↑2) βˆ’ -((sinβ€˜π΄)↑2)) = (((cosβ€˜π΄)↑2) + ((sinβ€˜π΄)↑2)))
4038, 16addcomd 11364 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((cosβ€˜π΄)↑2) + ((sinβ€˜π΄)↑2)) = (((sinβ€˜π΄)↑2) + ((cosβ€˜π΄)↑2)))
4137, 39, 403eqtrd 2781 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((cosβ€˜π΄)↑2) βˆ’ ((i Β· (sinβ€˜π΄))↑2)) = (((sinβ€˜π΄)↑2) + ((cosβ€˜π΄)↑2)))
42 efival 16041 . . . . . . . . . . . . . . 15 (𝐴 ∈ β„‚ β†’ (expβ€˜(i Β· 𝐴)) = ((cosβ€˜π΄) + (i Β· (sinβ€˜π΄))))
4342adantr 482 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜(i Β· 𝐴)) = ((cosβ€˜π΄) + (i Β· (sinβ€˜π΄))))
4472timesd 12403 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (2 Β· (i Β· (sinβ€˜π΄))) = ((i Β· (sinβ€˜π΄)) + (i Β· (sinβ€˜π΄))))
4543, 44oveq12d 7380 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((expβ€˜(i Β· 𝐴)) βˆ’ (2 Β· (i Β· (sinβ€˜π΄)))) = (((cosβ€˜π΄) + (i Β· (sinβ€˜π΄))) βˆ’ ((i Β· (sinβ€˜π΄)) + (i Β· (sinβ€˜π΄)))))
4627, 7, 7pnpcan2d 11557 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((cosβ€˜π΄) + (i Β· (sinβ€˜π΄))) βˆ’ ((i Β· (sinβ€˜π΄)) + (i Β· (sinβ€˜π΄)))) = ((cosβ€˜π΄) βˆ’ (i Β· (sinβ€˜π΄))))
4745, 46eqtrd 2777 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((expβ€˜(i Β· 𝐴)) βˆ’ (2 Β· (i Β· (sinβ€˜π΄)))) = ((cosβ€˜π΄) βˆ’ (i Β· (sinβ€˜π΄))))
4843, 47oveq12d 7380 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((expβ€˜(i Β· 𝐴)) Β· ((expβ€˜(i Β· 𝐴)) βˆ’ (2 Β· (i Β· (sinβ€˜π΄))))) = (((cosβ€˜π΄) + (i Β· (sinβ€˜π΄))) Β· ((cosβ€˜π΄) βˆ’ (i Β· (sinβ€˜π΄)))))
49 mulcl 11142 . . . . . . . . . . . . . . 15 ((2 ∈ β„‚ ∧ (i Β· (sinβ€˜π΄)) ∈ β„‚) β†’ (2 Β· (i Β· (sinβ€˜π΄))) ∈ β„‚)
5022, 7, 49sylancr 588 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (2 Β· (i Β· (sinβ€˜π΄))) ∈ β„‚)
5112, 12, 50subdid 11618 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((expβ€˜(i Β· 𝐴)) Β· ((expβ€˜(i Β· 𝐴)) βˆ’ (2 Β· (i Β· (sinβ€˜π΄))))) = (((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(i Β· 𝐴))) βˆ’ ((expβ€˜(i Β· 𝐴)) Β· (2 Β· (i Β· (sinβ€˜π΄))))))
5248, 51eqtr3d 2779 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((cosβ€˜π΄) + (i Β· (sinβ€˜π΄))) Β· ((cosβ€˜π΄) βˆ’ (i Β· (sinβ€˜π΄)))) = (((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(i Β· 𝐴))) βˆ’ ((expβ€˜(i Β· 𝐴)) Β· (2 Β· (i Β· (sinβ€˜π΄))))))
5329, 41, 523eqtr3d 2785 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((sinβ€˜π΄)↑2) + ((cosβ€˜π΄)↑2)) = (((expβ€˜(i Β· 𝐴)) Β· (expβ€˜(i Β· 𝐴))) βˆ’ ((expβ€˜(i Β· 𝐴)) Β· (2 Β· (i Β· (sinβ€˜π΄))))))
54 sincossq 16065 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ (((sinβ€˜π΄)↑2) + ((cosβ€˜π΄)↑2)) = 1)
5554adantr 482 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((sinβ€˜π΄)↑2) + ((cosβ€˜π΄)↑2)) = 1)
5625, 53, 553eqtr2d 2783 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((expβ€˜(i Β· 𝐴))↑2) βˆ’ (2 Β· ((expβ€˜(i Β· 𝐴)) Β· (i Β· (sinβ€˜π΄))))) = 1)
5756, 36oveq12d 7380 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((((expβ€˜(i Β· 𝐴))↑2) βˆ’ (2 Β· ((expβ€˜(i Β· 𝐴)) Β· (i Β· (sinβ€˜π΄))))) + ((i Β· (sinβ€˜π΄))↑2)) = (1 + -((sinβ€˜π΄)↑2)))
58 negsub 11456 . . . . . . . . . 10 ((1 ∈ β„‚ ∧ ((sinβ€˜π΄)↑2) ∈ β„‚) β†’ (1 + -((sinβ€˜π΄)↑2)) = (1 βˆ’ ((sinβ€˜π΄)↑2)))
5915, 16, 58sylancr 588 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (1 + -((sinβ€˜π΄)↑2)) = (1 βˆ’ ((sinβ€˜π΄)↑2)))
6020, 57, 593eqtrd 2781 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((expβ€˜(i Β· 𝐴)) βˆ’ (i Β· (sinβ€˜π΄)))↑2) = (1 βˆ’ ((sinβ€˜π΄)↑2)))
61 halfre 12374 . . . . . . . . . . . 12 (1 / 2) ∈ ℝ
6261a1i 11 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (1 / 2) ∈ ℝ)
63 negicn 11409 . . . . . . . . . . . . . . 15 -i ∈ β„‚
64 mulcl 11142 . . . . . . . . . . . . . . 15 ((-i ∈ β„‚ ∧ 𝐴 ∈ β„‚) β†’ (-i Β· 𝐴) ∈ β„‚)
6563, 8, 64sylancr 588 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (-i Β· 𝐴) ∈ β„‚)
66 efcl 15972 . . . . . . . . . . . . . 14 ((-i Β· 𝐴) ∈ β„‚ β†’ (expβ€˜(-i Β· 𝐴)) ∈ β„‚)
6765, 66syl 17 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜(-i Β· 𝐴)) ∈ β„‚)
6812, 67addcld 11181 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))) ∈ β„‚)
6968recld 15086 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴)))) ∈ ℝ)
70 halfgt0 12376 . . . . . . . . . . . 12 0 < (1 / 2)
7170a1i 11 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 0 < (1 / 2))
7212recld 15086 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜(expβ€˜(i Β· 𝐴))) ∈ ℝ)
7367recld 15086 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜(expβ€˜(-i Β· 𝐴))) ∈ ℝ)
74 asinsinlem 26257 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 0 < (β„œβ€˜(expβ€˜(i Β· 𝐴))))
75 negcl 11408 . . . . . . . . . . . . . . . 16 (𝐴 ∈ β„‚ β†’ -𝐴 ∈ β„‚)
7675adantr 482 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -𝐴 ∈ β„‚)
77 reneg 15017 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ β„‚ β†’ (β„œβ€˜-𝐴) = -(β„œβ€˜π΄))
7877adantr 482 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜-𝐴) = -(β„œβ€˜π΄))
79 halfpire 25837 . . . . . . . . . . . . . . . . . . . 20 (Ο€ / 2) ∈ ℝ
8079renegcli 11469 . . . . . . . . . . . . . . . . . . 19 -(Ο€ / 2) ∈ ℝ
81 recl 15002 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ β„‚ β†’ (β„œβ€˜π΄) ∈ ℝ)
82 iooneg 13395 . . . . . . . . . . . . . . . . . . 19 ((-(Ο€ / 2) ∈ ℝ ∧ (Ο€ / 2) ∈ ℝ ∧ (β„œβ€˜π΄) ∈ ℝ) β†’ ((β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2)) ↔ -(β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)--(Ο€ / 2))))
8380, 79, 81, 82mp3an12i 1466 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ β„‚ β†’ ((β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2)) ↔ -(β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)--(Ο€ / 2))))
8483biimpa 478 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -(β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)--(Ο€ / 2)))
8579recni 11176 . . . . . . . . . . . . . . . . . . 19 (Ο€ / 2) ∈ β„‚
8685negnegi 11478 . . . . . . . . . . . . . . . . . 18 --(Ο€ / 2) = (Ο€ / 2)
8786oveq2i 7373 . . . . . . . . . . . . . . . . 17 (-(Ο€ / 2)(,)--(Ο€ / 2)) = (-(Ο€ / 2)(,)(Ο€ / 2))
8884, 87eleqtrdi 2848 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -(β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2)))
8978, 88eqeltrd 2838 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜-𝐴) ∈ (-(Ο€ / 2)(,)(Ο€ / 2)))
90 asinsinlem 26257 . . . . . . . . . . . . . . 15 ((-𝐴 ∈ β„‚ ∧ (β„œβ€˜-𝐴) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 0 < (β„œβ€˜(expβ€˜(i Β· -𝐴))))
9176, 89, 90syl2anc 585 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 0 < (β„œβ€˜(expβ€˜(i Β· -𝐴))))
92 mulneg12 11600 . . . . . . . . . . . . . . . . 17 ((i ∈ β„‚ ∧ 𝐴 ∈ β„‚) β†’ (-i Β· 𝐴) = (i Β· -𝐴))
935, 8, 92sylancr 588 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (-i Β· 𝐴) = (i Β· -𝐴))
9493fveq2d 6851 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜(-i Β· 𝐴)) = (expβ€˜(i Β· -𝐴)))
9594fveq2d 6851 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜(expβ€˜(-i Β· 𝐴))) = (β„œβ€˜(expβ€˜(i Β· -𝐴))))
9691, 95breqtrrd 5138 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 0 < (β„œβ€˜(expβ€˜(-i Β· 𝐴))))
9772, 73, 74, 96addgt0d 11737 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 0 < ((β„œβ€˜(expβ€˜(i Β· 𝐴))) + (β„œβ€˜(expβ€˜(-i Β· 𝐴)))))
9812, 67readdd 15106 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴)))) = ((β„œβ€˜(expβ€˜(i Β· 𝐴))) + (β„œβ€˜(expβ€˜(-i Β· 𝐴)))))
9997, 98breqtrrd 5138 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 0 < (β„œβ€˜((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴)))))
10062, 69, 71, 99mulgt0d 11317 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 0 < ((1 / 2) Β· (β„œβ€˜((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))))))
101 cosval 16012 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ (cosβ€˜π΄) = (((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))) / 2))
102101adantr 482 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (cosβ€˜π΄) = (((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))) / 2))
103 2ne0 12264 . . . . . . . . . . . . . . 15 2 β‰  0
104103a1i 11 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 2 β‰  0)
10568, 23, 104divrec2d 11942 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))) / 2) = ((1 / 2) Β· ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴)))))
106102, 105eqtrd 2777 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (cosβ€˜π΄) = ((1 / 2) Β· ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴)))))
107106fveq2d 6851 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜(cosβ€˜π΄)) = (β„œβ€˜((1 / 2) Β· ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))))))
108 remul2 15022 . . . . . . . . . . . 12 (((1 / 2) ∈ ℝ ∧ ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))) ∈ β„‚) β†’ (β„œβ€˜((1 / 2) Β· ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))))) = ((1 / 2) Β· (β„œβ€˜((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))))))
10961, 68, 108sylancr 588 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜((1 / 2) Β· ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))))) = ((1 / 2) Β· (β„œβ€˜((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))))))
110107, 109eqtrd 2777 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜(cosβ€˜π΄)) = ((1 / 2) Β· (β„œβ€˜((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))))))
111100, 110breqtrrd 5138 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 0 < (β„œβ€˜(cosβ€˜π΄)))
11227, 7, 43mvrraddd 11574 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((expβ€˜(i Β· 𝐴)) βˆ’ (i Β· (sinβ€˜π΄))) = (cosβ€˜π΄))
113112fveq2d 6851 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜((expβ€˜(i Β· 𝐴)) βˆ’ (i Β· (sinβ€˜π΄)))) = (β„œβ€˜(cosβ€˜π΄)))
114111, 113breqtrrd 5138 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 0 < (β„œβ€˜((expβ€˜(i Β· 𝐴)) βˆ’ (i Β· (sinβ€˜π΄)))))
11514, 18, 60, 114eqsqrt2d 15260 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((expβ€˜(i Β· 𝐴)) βˆ’ (i Β· (sinβ€˜π΄))) = (βˆšβ€˜(1 βˆ’ ((sinβ€˜π΄)↑2))))
116115oveq2d 7378 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i Β· (sinβ€˜π΄)) + ((expβ€˜(i Β· 𝐴)) βˆ’ (i Β· (sinβ€˜π΄)))) = ((i Β· (sinβ€˜π΄)) + (βˆšβ€˜(1 βˆ’ ((sinβ€˜π΄)↑2)))))
11713, 116eqtr3d 2779 . . . . 5 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜(i Β· 𝐴)) = ((i Β· (sinβ€˜π΄)) + (βˆšβ€˜(1 βˆ’ ((sinβ€˜π΄)↑2)))))
118117fveq2d 6851 . . . 4 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (logβ€˜(expβ€˜(i Β· 𝐴))) = (logβ€˜((i Β· (sinβ€˜π΄)) + (βˆšβ€˜(1 βˆ’ ((sinβ€˜π΄)↑2))))))
119 pire 25831 . . . . . . . . . 10 Ο€ ∈ ℝ
120119renegcli 11469 . . . . . . . . 9 -Ο€ ∈ ℝ
121120a1i 11 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -Ο€ ∈ ℝ)
12280a1i 11 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -(Ο€ / 2) ∈ ℝ)
123 elioore 13301 . . . . . . . . 9 ((β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2)) β†’ (β„œβ€˜π΄) ∈ ℝ)
124123adantl 483 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜π΄) ∈ ℝ)
125 pirp 25834 . . . . . . . . . . 11 Ο€ ∈ ℝ+
126 rphalflt 12951 . . . . . . . . . . 11 (Ο€ ∈ ℝ+ β†’ (Ο€ / 2) < Ο€)
127125, 126ax-mp 5 . . . . . . . . . 10 (Ο€ / 2) < Ο€
12879, 119ltnegi 11706 . . . . . . . . . 10 ((Ο€ / 2) < Ο€ ↔ -Ο€ < -(Ο€ / 2))
129127, 128mpbi 229 . . . . . . . . 9 -Ο€ < -(Ο€ / 2)
130129a1i 11 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -Ο€ < -(Ο€ / 2))
131 eliooord 13330 . . . . . . . . . 10 ((β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2)) β†’ (-(Ο€ / 2) < (β„œβ€˜π΄) ∧ (β„œβ€˜π΄) < (Ο€ / 2)))
132131adantl 483 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (-(Ο€ / 2) < (β„œβ€˜π΄) ∧ (β„œβ€˜π΄) < (Ο€ / 2)))
133132simpld 496 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -(Ο€ / 2) < (β„œβ€˜π΄))
134121, 122, 124, 130, 133lttrd 11323 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -Ο€ < (β„œβ€˜π΄))
135 imre 15000 . . . . . . . . 9 ((i Β· 𝐴) ∈ β„‚ β†’ (β„‘β€˜(i Β· 𝐴)) = (β„œβ€˜(-i Β· (i Β· 𝐴))))
13610, 135syl 17 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„‘β€˜(i Β· 𝐴)) = (β„œβ€˜(-i Β· (i Β· 𝐴))))
1375, 5mulneg1i 11608 . . . . . . . . . . . 12 (-i Β· i) = -(i Β· i)
138 ixi 11791 . . . . . . . . . . . . 13 (i Β· i) = -1
139138negeqi 11401 . . . . . . . . . . . 12 -(i Β· i) = --1
14015negnegi 11478 . . . . . . . . . . . 12 --1 = 1
141137, 139, 1403eqtri 2769 . . . . . . . . . . 11 (-i Β· i) = 1
142141oveq1i 7372 . . . . . . . . . 10 ((-i Β· i) Β· 𝐴) = (1 Β· 𝐴)
14363a1i 11 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -i ∈ β„‚)
1445a1i 11 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ i ∈ β„‚)
145143, 144, 8mulassd 11185 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((-i Β· i) Β· 𝐴) = (-i Β· (i Β· 𝐴)))
146 mulid2 11161 . . . . . . . . . . 11 (𝐴 ∈ β„‚ β†’ (1 Β· 𝐴) = 𝐴)
147146adantr 482 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (1 Β· 𝐴) = 𝐴)
148142, 145, 1473eqtr3a 2801 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (-i Β· (i Β· 𝐴)) = 𝐴)
149148fveq2d 6851 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜(-i Β· (i Β· 𝐴))) = (β„œβ€˜π΄))
150136, 149eqtrd 2777 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„‘β€˜(i Β· 𝐴)) = (β„œβ€˜π΄))
151134, 150breqtrrd 5138 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -Ο€ < (β„‘β€˜(i Β· 𝐴)))
152119a1i 11 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ Ο€ ∈ ℝ)
15379a1i 11 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (Ο€ / 2) ∈ ℝ)
154132simprd 497 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜π΄) < (Ο€ / 2))
155127a1i 11 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (Ο€ / 2) < Ο€)
156124, 153, 152, 154, 155lttrd 11323 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜π΄) < Ο€)
157124, 152, 156ltled 11310 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜π΄) ≀ Ο€)
158150, 157eqbrtrd 5132 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„‘β€˜(i Β· 𝐴)) ≀ Ο€)
159 ellogrn 25931 . . . . . 6 ((i Β· 𝐴) ∈ ran log ↔ ((i Β· 𝐴) ∈ β„‚ ∧ -Ο€ < (β„‘β€˜(i Β· 𝐴)) ∧ (β„‘β€˜(i Β· 𝐴)) ≀ Ο€))
16010, 151, 158, 159syl3anbrc 1344 . . . . 5 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· 𝐴) ∈ ran log)
161 logef 25953 . . . . 5 ((i Β· 𝐴) ∈ ran log β†’ (logβ€˜(expβ€˜(i Β· 𝐴))) = (i Β· 𝐴))
162160, 161syl 17 . . . 4 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (logβ€˜(expβ€˜(i Β· 𝐴))) = (i Β· 𝐴))
163118, 162eqtr3d 2779 . . 3 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (logβ€˜((i Β· (sinβ€˜π΄)) + (βˆšβ€˜(1 βˆ’ ((sinβ€˜π΄)↑2))))) = (i Β· 𝐴))
164163oveq2d 7378 . 2 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (-i Β· (logβ€˜((i Β· (sinβ€˜π΄)) + (βˆšβ€˜(1 βˆ’ ((sinβ€˜π΄)↑2)))))) = (-i Β· (i Β· 𝐴)))
1654, 164, 1483eqtrd 2781 1 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (arcsinβ€˜(sinβ€˜π΄)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944   class class class wbr 5110  ran crn 5639  β€˜cfv 6501  (class class class)co 7362  β„‚cc 11056  β„cr 11057  0cc0 11058  1c1 11059  ici 11060   + caddc 11061   Β· cmul 11063   < clt 11196   ≀ cle 11197   βˆ’ cmin 11392  -cneg 11393   / cdiv 11819  2c2 12215  β„+crp 12922  (,)cioo 13271  β†‘cexp 13974  β„œcre 14989  β„‘cim 14990  βˆšcsqrt 15125  expce 15951  sincsin 15953  cosccos 15954  Ο€cpi 15956  logclog 25926  arcsincasin 26228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136  ax-addf 11137  ax-mulf 11138
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  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 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622  df-om 7808  df-1st 7926  df-2nd 7927  df-supp 8098  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-er 8655  df-map 8774  df-pm 8775  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9313  df-fi 9354  df-sup 9385  df-inf 9386  df-oi 9453  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-z 12507  df-dec 12626  df-uz 12771  df-q 12881  df-rp 12923  df-xneg 13040  df-xadd 13041  df-xmul 13042  df-ioo 13275  df-ioc 13276  df-ico 13277  df-icc 13278  df-fz 13432  df-fzo 13575  df-fl 13704  df-mod 13782  df-seq 13914  df-exp 13975  df-fac 14181  df-bc 14210  df-hash 14238  df-shft 14959  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-limsup 15360  df-clim 15377  df-rlim 15378  df-sum 15578  df-ef 15957  df-sin 15959  df-cos 15960  df-pi 15962  df-struct 17026  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-mulr 17154  df-starv 17155  df-sca 17156  df-vsca 17157  df-ip 17158  df-tset 17159  df-ple 17160  df-ds 17162  df-unif 17163  df-hom 17164  df-cco 17165  df-rest 17311  df-topn 17312  df-0g 17330  df-gsum 17331  df-topgen 17332  df-pt 17333  df-prds 17336  df-xrs 17391  df-qtop 17396  df-imas 17397  df-xps 17399  df-mre 17473  df-mrc 17474  df-acs 17476  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-submnd 18609  df-mulg 18880  df-cntz 19104  df-cmn 19571  df-psmet 20804  df-xmet 20805  df-met 20806  df-bl 20807  df-mopn 20808  df-fbas 20809  df-fg 20810  df-cnfld 20813  df-top 22259  df-topon 22276  df-topsp 22298  df-bases 22312  df-cld 22386  df-ntr 22387  df-cls 22388  df-nei 22465  df-lp 22503  df-perf 22504  df-cn 22594  df-cnp 22595  df-haus 22682  df-tx 22929  df-hmeo 23122  df-fil 23213  df-fm 23305  df-flim 23306  df-flf 23307  df-xms 23689  df-ms 23690  df-tms 23691  df-cncf 24257  df-limc 25246  df-dv 25247  df-log 25928  df-asin 26231
This theorem is referenced by:  acoscos  26259  reasinsin  26262  asinsinb  26263
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