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Theorem asinsin 26397
Description: The arcsine function composed with sin is equal to the identity. This plus sinasin 26394 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 26401). (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 16069 . . . 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 26387 . . 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 11169 . . . . . . . 8 i ∈ β„‚
6 mulcl 11194 . . . . . . . 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 11194 . . . . . . . . 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 16026 . . . . . . . 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 11574 . . . . . 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 11571 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((expβ€˜(i Β· 𝐴)) βˆ’ (i Β· (sinβ€˜π΄))) ∈ β„‚)
15 ax-1cn 11168 . . . . . . . . 9 1 ∈ β„‚
162sqcld 14109 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((sinβ€˜π΄)↑2) ∈ β„‚)
17 subcl 11459 . . . . . . . . 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 14183 . . . . . . . . . 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 14108 . . . . . . . . . . . 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 12287 . . . . . . . . . . . . . 14 2 ∈ β„‚
2322a1i 11 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 2 ∈ β„‚)
2423, 12, 7mul12d 11423 . . . . . . . . . . . 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 7427 . . . . . . . . . . 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 16070 . . . . . . . . . . . . . 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 14174 . . . . . . . . . . . . 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 14084 . . . . . . . . . . . . . . . 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 14166 . . . . . . . . . . . . . . . . 17 (i↑2) = -1
3332oveq1i 7419 . . . . . . . . . . . . . . . 16 ((i↑2) Β· ((sinβ€˜π΄)↑2)) = (-1 Β· ((sinβ€˜π΄)↑2))
3416mulm1d 11666 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (-1 Β· ((sinβ€˜π΄)↑2)) = -((sinβ€˜π΄)↑2))
3533, 34eqtrid 2785 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i↑2) Β· ((sinβ€˜π΄)↑2)) = -((sinβ€˜π΄)↑2))
3631, 35eqtrd 2773 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i Β· (sinβ€˜π΄))↑2) = -((sinβ€˜π΄)↑2))
3736oveq2d 7425 . . . . . . . . . . . . 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 14109 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((cosβ€˜π΄)↑2) ∈ β„‚)
3938, 16subnegd 11578 . . . . . . . . . . . . 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 11416 . . . . . . . . . . . . 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 2777 . . . . . . . . . . . 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 16095 . . . . . . . . . . . . . . 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 12455 . . . . . . . . . . . . . . . 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 7427 . . . . . . . . . . . . . . 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 11609 . . . . . . . . . . . . . . 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 2773 . . . . . . . . . . . . . 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 7427 . . . . . . . . . . . . 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 11194 . . . . . . . . . . . . . . 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 11670 . . . . . . . . . . . . 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 2775 . . . . . . . . . . . 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 2781 . . . . . . . . . . 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 16119 . . . . . . . . . . . 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 2779 . . . . . . . . . 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 7427 . . . . . . . . 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 11508 . . . . . . . . . 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 2777 . . . . . . . 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 12426 . . . . . . . . . . . 12 (1 / 2) ∈ ℝ
6261a1i 11 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (1 / 2) ∈ ℝ)
63 negicn 11461 . . . . . . . . . . . . . . 15 -i ∈ β„‚
64 mulcl 11194 . . . . . . . . . . . . . . 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 16026 . . . . . . . . . . . . . 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 11233 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴))) ∈ β„‚)
6968recld 15141 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜((expβ€˜(i Β· 𝐴)) + (expβ€˜(-i Β· 𝐴)))) ∈ ℝ)
70 halfgt0 12428 . . . . . . . . . . . 12 0 < (1 / 2)
7170a1i 11 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 0 < (1 / 2))
7212recld 15141 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜(expβ€˜(i Β· 𝐴))) ∈ ℝ)
7367recld 15141 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜(expβ€˜(-i Β· 𝐴))) ∈ ℝ)
74 asinsinlem 26396 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 0 < (β„œβ€˜(expβ€˜(i Β· 𝐴))))
75 negcl 11460 . . . . . . . . . . . . . . . 16 (𝐴 ∈ β„‚ β†’ -𝐴 ∈ β„‚)
7675adantr 482 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -𝐴 ∈ β„‚)
77 reneg 15072 . . . . . . . . . . . . . . . . 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 25974 . . . . . . . . . . . . . . . . . . . 20 (Ο€ / 2) ∈ ℝ
8079renegcli 11521 . . . . . . . . . . . . . . . . . . 19 -(Ο€ / 2) ∈ ℝ
81 recl 15057 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ β„‚ β†’ (β„œβ€˜π΄) ∈ ℝ)
82 iooneg 13448 . . . . . . . . . . . . . . . . . . 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 11228 . . . . . . . . . . . . . . . . . . 19 (Ο€ / 2) ∈ β„‚
8685negnegi 11530 . . . . . . . . . . . . . . . . . 18 --(Ο€ / 2) = (Ο€ / 2)
8786oveq2i 7420 . . . . . . . . . . . . . . . . 17 (-(Ο€ / 2)(,)--(Ο€ / 2)) = (-(Ο€ / 2)(,)(Ο€ / 2))
8884, 87eleqtrdi 2844 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -(β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2)))
8978, 88eqeltrd 2834 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜-𝐴) ∈ (-(Ο€ / 2)(,)(Ο€ / 2)))
90 asinsinlem 26396 . . . . . . . . . . . . . . 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 11652 . . . . . . . . . . . . . . . . 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 6896 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜(-i Β· 𝐴)) = (expβ€˜(i Β· -𝐴)))
9594fveq2d 6896 . . . . . . . . . . . . . 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 5177 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 0 < (β„œβ€˜(expβ€˜(-i Β· 𝐴))))
9772, 73, 74, 96addgt0d 11789 . . . . . . . . . . . 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 15161 . . . . . . . . . . . 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 5177 . . . . . . . . . . 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 11369 . . . . . . . . . 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 16066 . . . . . . . . . . . . . 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 12316 . . . . . . . . . . . . . . 15 2 β‰  0
104103a1i 11 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 2 β‰  0)
10568, 23, 104divrec2d 11994 . . . . . . . . . . . . 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 2773 . . . . . . . . . . . 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 6896 . . . . . . . . . . 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 15077 . . . . . . . . . . . 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 2773 . . . . . . . . . 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 5177 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 0 < (β„œβ€˜(cosβ€˜π΄)))
11227, 7, 43mvrraddd 11626 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((expβ€˜(i Β· 𝐴)) βˆ’ (i Β· (sinβ€˜π΄))) = (cosβ€˜π΄))
113112fveq2d 6896 . . . . . . . . 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 5177 . . . . . . . 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 15315 . . . . . . 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 7425 . . . . . 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 2775 . . . . 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 6896 . . . 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 25968 . . . . . . . . . 10 Ο€ ∈ ℝ
120119renegcli 11521 . . . . . . . . 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 13354 . . . . . . . . 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 25971 . . . . . . . . . . 11 Ο€ ∈ ℝ+
126 rphalflt 13003 . . . . . . . . . . 11 (Ο€ ∈ ℝ+ β†’ (Ο€ / 2) < Ο€)
127125, 126ax-mp 5 . . . . . . . . . 10 (Ο€ / 2) < Ο€
12879, 119ltnegi 11758 . . . . . . . . . 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 13383 . . . . . . . . . 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 11375 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -Ο€ < (β„œβ€˜π΄))
135 imre 15055 . . . . . . . . 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 11660 . . . . . . . . . . . 12 (-i Β· i) = -(i Β· i)
138 ixi 11843 . . . . . . . . . . . . 13 (i Β· i) = -1
139138negeqi 11453 . . . . . . . . . . . 12 -(i Β· i) = --1
14015negnegi 11530 . . . . . . . . . . . 12 --1 = 1
141137, 139, 1403eqtri 2765 . . . . . . . . . . 11 (-i Β· i) = 1
142141oveq1i 7419 . . . . . . . . . 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 11237 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((-i Β· i) Β· 𝐴) = (-i Β· (i Β· 𝐴)))
146 mullid 11213 . . . . . . . . . . 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 2797 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (-i Β· (i Β· 𝐴)) = 𝐴)
149148fveq2d 6896 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜(-i Β· (i Β· 𝐴))) = (β„œβ€˜π΄))
150136, 149eqtrd 2773 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„‘β€˜(i Β· 𝐴)) = (β„œβ€˜π΄))
151134, 150breqtrrd 5177 . . . . . 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 11375 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜π΄) < Ο€)
157124, 152, 156ltled 11362 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜π΄) ≀ Ο€)
158150, 157eqbrtrd 5171 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„‘β€˜(i Β· 𝐴)) ≀ Ο€)
159 ellogrn 26068 . . . . . 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 26090 . . . . 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 2775 . . 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 7425 . 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 2777 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 2941   class class class wbr 5149  ran crn 5678  β€˜cfv 6544  (class class class)co 7409  β„‚cc 11108  β„cr 11109  0cc0 11110  1c1 11111  ici 11112   + caddc 11113   Β· cmul 11115   < clt 11248   ≀ cle 11249   βˆ’ cmin 11444  -cneg 11445   / cdiv 11871  2c2 12267  β„+crp 12974  (,)cioo 13324  β†‘cexp 14027  β„œcre 15044  β„‘cim 15045  βˆšcsqrt 15180  expce 16005  sincsin 16007  cosccos 16008  Ο€cpi 16010  logclog 26063  arcsincasin 26367
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188  ax-addf 11189  ax-mulf 11190
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9362  df-fi 9406  df-sup 9437  df-inf 9438  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ioo 13328  df-ioc 13329  df-ico 13330  df-icc 13331  df-fz 13485  df-fzo 13628  df-fl 13757  df-mod 13835  df-seq 13967  df-exp 14028  df-fac 14234  df-bc 14263  df-hash 14291  df-shft 15014  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-limsup 15415  df-clim 15432  df-rlim 15433  df-sum 15633  df-ef 16011  df-sin 16013  df-cos 16014  df-pi 16016  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-starv 17212  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-hom 17221  df-cco 17222  df-rest 17368  df-topn 17369  df-0g 17387  df-gsum 17388  df-topgen 17389  df-pt 17390  df-prds 17393  df-xrs 17448  df-qtop 17453  df-imas 17454  df-xps 17456  df-mre 17530  df-mrc 17531  df-acs 17533  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-submnd 18672  df-mulg 18951  df-cntz 19181  df-cmn 19650  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-fbas 20941  df-fg 20942  df-cnfld 20945  df-top 22396  df-topon 22413  df-topsp 22435  df-bases 22449  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602  df-lp 22640  df-perf 22641  df-cn 22731  df-cnp 22732  df-haus 22819  df-tx 23066  df-hmeo 23259  df-fil 23350  df-fm 23442  df-flim 23443  df-flf 23444  df-xms 23826  df-ms 23827  df-tms 23828  df-cncf 24394  df-limc 25383  df-dv 25384  df-log 26065  df-asin 26370
This theorem is referenced by:  acoscos  26398  reasinsin  26401  asinsinb  26402
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