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Mirrors  >  Home  >  MPE Home  >  Th. List  >  atantan Structured version   Visualization version   GIF version

Theorem atantan 26289
Description: The arctangent function is an inverse to tan. (Contributed by Mario Carneiro, 5-Apr-2015.)
Assertion
Ref Expression
atantan ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (arctanβ€˜(tanβ€˜π΄)) = 𝐴)

Proof of Theorem atantan
StepHypRef Expression
1 cosne0 25901 . . . 4 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (cosβ€˜π΄) β‰  0)
2 atandmtan 26286 . . . 4 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) β‰  0) β†’ (tanβ€˜π΄) ∈ dom arctan)
31, 2syldan 592 . . 3 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (tanβ€˜π΄) ∈ dom arctan)
4 atanval 26250 . . 3 ((tanβ€˜π΄) ∈ dom arctan β†’ (arctanβ€˜(tanβ€˜π΄)) = ((i / 2) Β· ((logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 + (i Β· (tanβ€˜π΄)))))))
53, 4syl 17 . 2 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (arctanβ€˜(tanβ€˜π΄)) = ((i / 2) Β· ((logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 + (i Β· (tanβ€˜π΄)))))))
6 ax-1cn 11116 . . . . . . 7 1 ∈ β„‚
7 ax-icn 11117 . . . . . . . 8 i ∈ β„‚
8 tancl 16018 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) β‰  0) β†’ (tanβ€˜π΄) ∈ β„‚)
91, 8syldan 592 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (tanβ€˜π΄) ∈ β„‚)
10 mulcl 11142 . . . . . . . 8 ((i ∈ β„‚ ∧ (tanβ€˜π΄) ∈ β„‚) β†’ (i Β· (tanβ€˜π΄)) ∈ β„‚)
117, 9, 10sylancr 588 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· (tanβ€˜π΄)) ∈ β„‚)
12 addcl 11140 . . . . . . 7 ((1 ∈ β„‚ ∧ (i Β· (tanβ€˜π΄)) ∈ β„‚) β†’ (1 + (i Β· (tanβ€˜π΄))) ∈ β„‚)
136, 11, 12sylancr 588 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (1 + (i Β· (tanβ€˜π΄))) ∈ β„‚)
14 atandm2 26243 . . . . . . . 8 ((tanβ€˜π΄) ∈ dom arctan ↔ ((tanβ€˜π΄) ∈ β„‚ ∧ (1 βˆ’ (i Β· (tanβ€˜π΄))) β‰  0 ∧ (1 + (i Β· (tanβ€˜π΄))) β‰  0))
153, 14sylib 217 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((tanβ€˜π΄) ∈ β„‚ ∧ (1 βˆ’ (i Β· (tanβ€˜π΄))) β‰  0 ∧ (1 + (i Β· (tanβ€˜π΄))) β‰  0))
1615simp3d 1145 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (1 + (i Β· (tanβ€˜π΄))) β‰  0)
1713, 16logcld 25942 . . . . 5 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (logβ€˜(1 + (i Β· (tanβ€˜π΄)))) ∈ β„‚)
18 subcl 11407 . . . . . . 7 ((1 ∈ β„‚ ∧ (i Β· (tanβ€˜π΄)) ∈ β„‚) β†’ (1 βˆ’ (i Β· (tanβ€˜π΄))) ∈ β„‚)
196, 11, 18sylancr 588 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (1 βˆ’ (i Β· (tanβ€˜π΄))) ∈ β„‚)
2015simp2d 1144 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (1 βˆ’ (i Β· (tanβ€˜π΄))) β‰  0)
2119, 20logcld 25942 . . . . 5 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄)))) ∈ β„‚)
2217, 21negsubdi2d 11535 . . . 4 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))) = ((logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 + (i Β· (tanβ€˜π΄))))))
23 efsub 15989 . . . . . . . . 9 (((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) ∈ β„‚ ∧ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄)))) ∈ β„‚) β†’ (expβ€˜((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄)))))) = ((expβ€˜(logβ€˜(1 + (i Β· (tanβ€˜π΄))))) / (expβ€˜(logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄)))))))
2417, 21, 23syl2anc 585 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄)))))) = ((expβ€˜(logβ€˜(1 + (i Β· (tanβ€˜π΄))))) / (expβ€˜(logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄)))))))
25 coscl 16016 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ (cosβ€˜π΄) ∈ β„‚)
2625adantr 482 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (cosβ€˜π΄) ∈ β„‚)
27 sincl 16015 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ (sinβ€˜π΄) ∈ β„‚)
2827adantr 482 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (sinβ€˜π΄) ∈ β„‚)
29 mulcl 11142 . . . . . . . . . . . . 13 ((i ∈ β„‚ ∧ (sinβ€˜π΄) ∈ β„‚) β†’ (i Β· (sinβ€˜π΄)) ∈ β„‚)
307, 28, 29sylancr 588 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· (sinβ€˜π΄)) ∈ β„‚)
3126, 30, 26, 1divdird 11976 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((cosβ€˜π΄) + (i Β· (sinβ€˜π΄))) / (cosβ€˜π΄)) = (((cosβ€˜π΄) / (cosβ€˜π΄)) + ((i Β· (sinβ€˜π΄)) / (cosβ€˜π΄))))
3226, 1dividd 11936 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((cosβ€˜π΄) / (cosβ€˜π΄)) = 1)
337a1i 11 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ i ∈ β„‚)
3433, 28, 26, 1divassd 11973 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i Β· (sinβ€˜π΄)) / (cosβ€˜π΄)) = (i Β· ((sinβ€˜π΄) / (cosβ€˜π΄))))
35 tanval 16017 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) β‰  0) β†’ (tanβ€˜π΄) = ((sinβ€˜π΄) / (cosβ€˜π΄)))
361, 35syldan 592 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (tanβ€˜π΄) = ((sinβ€˜π΄) / (cosβ€˜π΄)))
3736oveq2d 7378 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· (tanβ€˜π΄)) = (i Β· ((sinβ€˜π΄) / (cosβ€˜π΄))))
3834, 37eqtr4d 2780 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i Β· (sinβ€˜π΄)) / (cosβ€˜π΄)) = (i Β· (tanβ€˜π΄)))
3932, 38oveq12d 7380 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((cosβ€˜π΄) / (cosβ€˜π΄)) + ((i Β· (sinβ€˜π΄)) / (cosβ€˜π΄))) = (1 + (i Β· (tanβ€˜π΄))))
4031, 39eqtrd 2777 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((cosβ€˜π΄) + (i Β· (sinβ€˜π΄))) / (cosβ€˜π΄)) = (1 + (i Β· (tanβ€˜π΄))))
41 efival 16041 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ (expβ€˜(i Β· 𝐴)) = ((cosβ€˜π΄) + (i Β· (sinβ€˜π΄))))
4241adantr 482 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜(i Β· 𝐴)) = ((cosβ€˜π΄) + (i Β· (sinβ€˜π΄))))
4342oveq1d 7377 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((expβ€˜(i Β· 𝐴)) / (cosβ€˜π΄)) = (((cosβ€˜π΄) + (i Β· (sinβ€˜π΄))) / (cosβ€˜π΄)))
44 eflog 25948 . . . . . . . . . . 11 (((1 + (i Β· (tanβ€˜π΄))) ∈ β„‚ ∧ (1 + (i Β· (tanβ€˜π΄))) β‰  0) β†’ (expβ€˜(logβ€˜(1 + (i Β· (tanβ€˜π΄))))) = (1 + (i Β· (tanβ€˜π΄))))
4513, 16, 44syl2anc 585 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜(logβ€˜(1 + (i Β· (tanβ€˜π΄))))) = (1 + (i Β· (tanβ€˜π΄))))
4640, 43, 453eqtr4d 2787 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((expβ€˜(i Β· 𝐴)) / (cosβ€˜π΄)) = (expβ€˜(logβ€˜(1 + (i Β· (tanβ€˜π΄))))))
4726, 30, 26, 1divsubdird 11977 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((cosβ€˜π΄) βˆ’ (i Β· (sinβ€˜π΄))) / (cosβ€˜π΄)) = (((cosβ€˜π΄) / (cosβ€˜π΄)) βˆ’ ((i Β· (sinβ€˜π΄)) / (cosβ€˜π΄))))
4832, 38oveq12d 7380 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((cosβ€˜π΄) / (cosβ€˜π΄)) βˆ’ ((i Β· (sinβ€˜π΄)) / (cosβ€˜π΄))) = (1 βˆ’ (i Β· (tanβ€˜π΄))))
4947, 48eqtrd 2777 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((cosβ€˜π΄) βˆ’ (i Β· (sinβ€˜π΄))) / (cosβ€˜π΄)) = (1 βˆ’ (i Β· (tanβ€˜π΄))))
50 negcl 11408 . . . . . . . . . . . . . . 15 (𝐴 ∈ β„‚ β†’ -𝐴 ∈ β„‚)
5150adantr 482 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -𝐴 ∈ β„‚)
52 efival 16041 . . . . . . . . . . . . . 14 (-𝐴 ∈ β„‚ β†’ (expβ€˜(i Β· -𝐴)) = ((cosβ€˜-𝐴) + (i Β· (sinβ€˜-𝐴))))
5351, 52syl 17 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜(i Β· -𝐴)) = ((cosβ€˜-𝐴) + (i Β· (sinβ€˜-𝐴))))
54 cosneg 16036 . . . . . . . . . . . . . . 15 (𝐴 ∈ β„‚ β†’ (cosβ€˜-𝐴) = (cosβ€˜π΄))
5554adantr 482 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (cosβ€˜-𝐴) = (cosβ€˜π΄))
56 sinneg 16035 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ β„‚ β†’ (sinβ€˜-𝐴) = -(sinβ€˜π΄))
5756adantr 482 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (sinβ€˜-𝐴) = -(sinβ€˜π΄))
5857oveq2d 7378 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· (sinβ€˜-𝐴)) = (i Β· -(sinβ€˜π΄)))
59 mulneg2 11599 . . . . . . . . . . . . . . . 16 ((i ∈ β„‚ ∧ (sinβ€˜π΄) ∈ β„‚) β†’ (i Β· -(sinβ€˜π΄)) = -(i Β· (sinβ€˜π΄)))
607, 28, 59sylancr 588 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· -(sinβ€˜π΄)) = -(i Β· (sinβ€˜π΄)))
6158, 60eqtrd 2777 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· (sinβ€˜-𝐴)) = -(i Β· (sinβ€˜π΄)))
6255, 61oveq12d 7380 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((cosβ€˜-𝐴) + (i Β· (sinβ€˜-𝐴))) = ((cosβ€˜π΄) + -(i Β· (sinβ€˜π΄))))
6353, 62eqtrd 2777 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜(i Β· -𝐴)) = ((cosβ€˜π΄) + -(i Β· (sinβ€˜π΄))))
64 simpl 484 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 𝐴 ∈ β„‚)
65 mulneg2 11599 . . . . . . . . . . . . . 14 ((i ∈ β„‚ ∧ 𝐴 ∈ β„‚) β†’ (i Β· -𝐴) = -(i Β· 𝐴))
667, 64, 65sylancr 588 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· -𝐴) = -(i Β· 𝐴))
6766fveq2d 6851 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜(i Β· -𝐴)) = (expβ€˜-(i Β· 𝐴)))
6826, 30negsubd 11525 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((cosβ€˜π΄) + -(i Β· (sinβ€˜π΄))) = ((cosβ€˜π΄) βˆ’ (i Β· (sinβ€˜π΄))))
6963, 67, 683eqtr3d 2785 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜-(i Β· 𝐴)) = ((cosβ€˜π΄) βˆ’ (i Β· (sinβ€˜π΄))))
7069oveq1d 7377 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((expβ€˜-(i Β· 𝐴)) / (cosβ€˜π΄)) = (((cosβ€˜π΄) βˆ’ (i Β· (sinβ€˜π΄))) / (cosβ€˜π΄)))
71 eflog 25948 . . . . . . . . . . 11 (((1 βˆ’ (i Β· (tanβ€˜π΄))) ∈ β„‚ ∧ (1 βˆ’ (i Β· (tanβ€˜π΄))) β‰  0) β†’ (expβ€˜(logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))) = (1 βˆ’ (i Β· (tanβ€˜π΄))))
7219, 20, 71syl2anc 585 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜(logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))) = (1 βˆ’ (i Β· (tanβ€˜π΄))))
7349, 70, 723eqtr4d 2787 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((expβ€˜-(i Β· 𝐴)) / (cosβ€˜π΄)) = (expβ€˜(logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))))
7446, 73oveq12d 7380 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((expβ€˜(i Β· 𝐴)) / (cosβ€˜π΄)) / ((expβ€˜-(i Β· 𝐴)) / (cosβ€˜π΄))) = ((expβ€˜(logβ€˜(1 + (i Β· (tanβ€˜π΄))))) / (expβ€˜(logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄)))))))
75 mulcl 11142 . . . . . . . . . . . 12 ((i ∈ β„‚ ∧ 𝐴 ∈ β„‚) β†’ (i Β· 𝐴) ∈ β„‚)
767, 64, 75sylancr 588 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· 𝐴) ∈ β„‚)
77 efcl 15972 . . . . . . . . . . 11 ((i Β· 𝐴) ∈ β„‚ β†’ (expβ€˜(i Β· 𝐴)) ∈ β„‚)
7876, 77syl 17 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜(i Β· 𝐴)) ∈ β„‚)
7976negcld 11506 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -(i Β· 𝐴) ∈ β„‚)
80 efcl 15972 . . . . . . . . . . 11 (-(i Β· 𝐴) ∈ β„‚ β†’ (expβ€˜-(i Β· 𝐴)) ∈ β„‚)
8179, 80syl 17 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜-(i Β· 𝐴)) ∈ β„‚)
82 efne0 15986 . . . . . . . . . . 11 (-(i Β· 𝐴) ∈ β„‚ β†’ (expβ€˜-(i Β· 𝐴)) β‰  0)
8379, 82syl 17 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜-(i Β· 𝐴)) β‰  0)
8478, 81, 26, 83, 1divcan7d 11966 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((expβ€˜(i Β· 𝐴)) / (cosβ€˜π΄)) / ((expβ€˜-(i Β· 𝐴)) / (cosβ€˜π΄))) = ((expβ€˜(i Β· 𝐴)) / (expβ€˜-(i Β· 𝐴))))
85 efsub 15989 . . . . . . . . . 10 (((i Β· 𝐴) ∈ β„‚ ∧ -(i Β· 𝐴) ∈ β„‚) β†’ (expβ€˜((i Β· 𝐴) βˆ’ -(i Β· 𝐴))) = ((expβ€˜(i Β· 𝐴)) / (expβ€˜-(i Β· 𝐴))))
8676, 79, 85syl2anc 585 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜((i Β· 𝐴) βˆ’ -(i Β· 𝐴))) = ((expβ€˜(i Β· 𝐴)) / (expβ€˜-(i Β· 𝐴))))
8776, 76subnegd 11526 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i Β· 𝐴) βˆ’ -(i Β· 𝐴)) = ((i Β· 𝐴) + (i Β· 𝐴)))
88762timesd 12403 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (2 Β· (i Β· 𝐴)) = ((i Β· 𝐴) + (i Β· 𝐴)))
8987, 88eqtr4d 2780 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i Β· 𝐴) βˆ’ -(i Β· 𝐴)) = (2 Β· (i Β· 𝐴)))
9089fveq2d 6851 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜((i Β· 𝐴) βˆ’ -(i Β· 𝐴))) = (expβ€˜(2 Β· (i Β· 𝐴))))
9184, 86, 903eqtr2d 2783 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((expβ€˜(i Β· 𝐴)) / (cosβ€˜π΄)) / ((expβ€˜-(i Β· 𝐴)) / (cosβ€˜π΄))) = (expβ€˜(2 Β· (i Β· 𝐴))))
9224, 74, 913eqtr2d 2783 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄)))))) = (expβ€˜(2 Β· (i Β· 𝐴))))
9392fveq2d 6851 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (logβ€˜(expβ€˜((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))))) = (logβ€˜(expβ€˜(2 Β· (i Β· 𝐴)))))
9464adantr 482 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ 𝐴 ∈ β„‚)
9594renegd 15101 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜-𝐴) = -(β„œβ€˜π΄))
9694recld 15086 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜π΄) ∈ ℝ)
9796renegcld 11589 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ -(β„œβ€˜π΄) ∈ ℝ)
98 simpr 486 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜π΄) < 0)
9996lt0neg1d 11731 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ ((β„œβ€˜π΄) < 0 ↔ 0 < -(β„œβ€˜π΄)))
10098, 99mpbid 231 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ 0 < -(β„œβ€˜π΄))
101 eliooord 13330 . . . . . . . . . . . . . . . . . . 19 ((β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2)) β†’ (-(Ο€ / 2) < (β„œβ€˜π΄) ∧ (β„œβ€˜π΄) < (Ο€ / 2)))
102101adantl 483 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (-(Ο€ / 2) < (β„œβ€˜π΄) ∧ (β„œβ€˜π΄) < (Ο€ / 2)))
103102simpld 496 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -(Ο€ / 2) < (β„œβ€˜π΄))
104103adantr 482 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ -(Ο€ / 2) < (β„œβ€˜π΄))
105 halfpire 25837 . . . . . . . . . . . . . . . . 17 (Ο€ / 2) ∈ ℝ
106 ltnegcon1 11663 . . . . . . . . . . . . . . . . 17 (((Ο€ / 2) ∈ ℝ ∧ (β„œβ€˜π΄) ∈ ℝ) β†’ (-(Ο€ / 2) < (β„œβ€˜π΄) ↔ -(β„œβ€˜π΄) < (Ο€ / 2)))
107105, 96, 106sylancr 588 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (-(Ο€ / 2) < (β„œβ€˜π΄) ↔ -(β„œβ€˜π΄) < (Ο€ / 2)))
108104, 107mpbid 231 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ -(β„œβ€˜π΄) < (Ο€ / 2))
109 0xr 11209 . . . . . . . . . . . . . . . 16 0 ∈ ℝ*
110105rexri 11220 . . . . . . . . . . . . . . . 16 (Ο€ / 2) ∈ ℝ*
111 elioo2 13312 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ* ∧ (Ο€ / 2) ∈ ℝ*) β†’ (-(β„œβ€˜π΄) ∈ (0(,)(Ο€ / 2)) ↔ (-(β„œβ€˜π΄) ∈ ℝ ∧ 0 < -(β„œβ€˜π΄) ∧ -(β„œβ€˜π΄) < (Ο€ / 2))))
112109, 110, 111mp2an 691 . . . . . . . . . . . . . . 15 (-(β„œβ€˜π΄) ∈ (0(,)(Ο€ / 2)) ↔ (-(β„œβ€˜π΄) ∈ ℝ ∧ 0 < -(β„œβ€˜π΄) ∧ -(β„œβ€˜π΄) < (Ο€ / 2)))
11397, 100, 108, 112syl3anbrc 1344 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ -(β„œβ€˜π΄) ∈ (0(,)(Ο€ / 2)))
11495, 113eqeltrd 2838 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜-𝐴) ∈ (0(,)(Ο€ / 2)))
115 tanregt0 25911 . . . . . . . . . . . . 13 ((-𝐴 ∈ β„‚ ∧ (β„œβ€˜-𝐴) ∈ (0(,)(Ο€ / 2))) β†’ 0 < (β„œβ€˜(tanβ€˜-𝐴)))
11651, 114, 115syl2an2r 684 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ 0 < (β„œβ€˜(tanβ€˜-𝐴)))
117 tanneg 16037 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) β‰  0) β†’ (tanβ€˜-𝐴) = -(tanβ€˜π΄))
1181, 117syldan 592 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (tanβ€˜-𝐴) = -(tanβ€˜π΄))
119118adantr 482 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (tanβ€˜-𝐴) = -(tanβ€˜π΄))
120119fveq2d 6851 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜(tanβ€˜-𝐴)) = (β„œβ€˜-(tanβ€˜π΄)))
1219adantr 482 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (tanβ€˜π΄) ∈ β„‚)
122121renegd 15101 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜-(tanβ€˜π΄)) = -(β„œβ€˜(tanβ€˜π΄)))
123120, 122eqtrd 2777 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜(tanβ€˜-𝐴)) = -(β„œβ€˜(tanβ€˜π΄)))
124116, 123breqtrd 5136 . . . . . . . . . . 11 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ 0 < -(β„œβ€˜(tanβ€˜π΄)))
1259recld 15086 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜(tanβ€˜π΄)) ∈ ℝ)
126125adantr 482 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜(tanβ€˜π΄)) ∈ ℝ)
127126lt0neg1d 11731 . . . . . . . . . . 11 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ ((β„œβ€˜(tanβ€˜π΄)) < 0 ↔ 0 < -(β„œβ€˜(tanβ€˜π΄))))
128124, 127mpbird 257 . . . . . . . . . 10 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜(tanβ€˜π΄)) < 0)
129128lt0ne0d 11727 . . . . . . . . 9 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜(tanβ€˜π΄)) β‰  0)
130 atanlogsub 26282 . . . . . . . . 9 (((tanβ€˜π΄) ∈ dom arctan ∧ (β„œβ€˜(tanβ€˜π΄)) β‰  0) β†’ ((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))) ∈ ran log)
1313, 129, 130syl2an2r 684 . . . . . . . 8 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ ((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))) ∈ ran log)
132 1re 11162 . . . . . . . . . . . . 13 1 ∈ ℝ
133 ioossre 13332 . . . . . . . . . . . . . 14 (-1(,)1) βŠ† ℝ
1347a1i 11 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ i ∈ β„‚)
13511adantr 482 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (i Β· (tanβ€˜π΄)) ∈ β„‚)
136 ine0 11597 . . . . . . . . . . . . . . . . 17 i β‰  0
137136a1i 11 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ i β‰  0)
138 ixi 11791 . . . . . . . . . . . . . . . . . . 19 (i Β· i) = -1
139138oveq1i 7372 . . . . . . . . . . . . . . . . . 18 ((i Β· i) Β· (tanβ€˜π΄)) = (-1 Β· (tanβ€˜π΄))
1409adantr 482 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (tanβ€˜π΄) ∈ β„‚)
141140mulm1d 11614 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (-1 Β· (tanβ€˜π΄)) = -(tanβ€˜π΄))
142118adantr 482 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (tanβ€˜-𝐴) = -(tanβ€˜π΄))
143141, 142eqtr4d 2780 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (-1 Β· (tanβ€˜π΄)) = (tanβ€˜-𝐴))
144139, 143eqtrid 2789 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ ((i Β· i) Β· (tanβ€˜π΄)) = (tanβ€˜-𝐴))
145134, 134, 140mulassd 11185 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ ((i Β· i) Β· (tanβ€˜π΄)) = (i Β· (i Β· (tanβ€˜π΄))))
146138oveq1i 7372 . . . . . . . . . . . . . . . . . . . 20 ((i Β· i) Β· 𝐴) = (-1 Β· 𝐴)
14764adantr 482 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ 𝐴 ∈ β„‚)
148147mulm1d 11614 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (-1 Β· 𝐴) = -𝐴)
149146, 148eqtrid 2789 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ ((i Β· i) Β· 𝐴) = -𝐴)
150134, 134, 147mulassd 11185 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ ((i Β· i) Β· 𝐴) = (i Β· (i Β· 𝐴)))
151149, 150eqtr3d 2779 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ -𝐴 = (i Β· (i Β· 𝐴)))
152151fveq2d 6851 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (tanβ€˜-𝐴) = (tanβ€˜(i Β· (i Β· 𝐴))))
153144, 145, 1523eqtr3d 2785 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (i Β· (i Β· (tanβ€˜π΄))) = (tanβ€˜(i Β· (i Β· 𝐴))))
154134, 135, 137, 153mvllmuld 11994 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (i Β· (tanβ€˜π΄)) = ((tanβ€˜(i Β· (i Β· 𝐴))) / i))
15576adantr 482 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (i Β· 𝐴) ∈ β„‚)
156 reim 15001 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∈ β„‚ β†’ (β„œβ€˜π΄) = (β„‘β€˜(i Β· 𝐴)))
157156adantr 482 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜π΄) = (β„‘β€˜(i Β· 𝐴)))
158157eqeq1d 2739 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((β„œβ€˜π΄) = 0 ↔ (β„‘β€˜(i Β· 𝐴)) = 0))
159158biimpa 478 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (β„‘β€˜(i Β· 𝐴)) = 0)
160155, 159reim0bd 15092 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (i Β· 𝐴) ∈ ℝ)
161 tanhbnd 16050 . . . . . . . . . . . . . . . 16 ((i Β· 𝐴) ∈ ℝ β†’ ((tanβ€˜(i Β· (i Β· 𝐴))) / i) ∈ (-1(,)1))
162160, 161syl 17 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ ((tanβ€˜(i Β· (i Β· 𝐴))) / i) ∈ (-1(,)1))
163154, 162eqeltrd 2838 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (i Β· (tanβ€˜π΄)) ∈ (-1(,)1))
164133, 163sselid 3947 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (i Β· (tanβ€˜π΄)) ∈ ℝ)
165 readdcl 11141 . . . . . . . . . . . . 13 ((1 ∈ ℝ ∧ (i Β· (tanβ€˜π΄)) ∈ ℝ) β†’ (1 + (i Β· (tanβ€˜π΄))) ∈ ℝ)
166132, 164, 165sylancr 588 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (1 + (i Β· (tanβ€˜π΄))) ∈ ℝ)
167 df-neg 11395 . . . . . . . . . . . . . 14 -1 = (0 βˆ’ 1)
168 eliooord 13330 . . . . . . . . . . . . . . . 16 ((i Β· (tanβ€˜π΄)) ∈ (-1(,)1) β†’ (-1 < (i Β· (tanβ€˜π΄)) ∧ (i Β· (tanβ€˜π΄)) < 1))
169163, 168syl 17 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (-1 < (i Β· (tanβ€˜π΄)) ∧ (i Β· (tanβ€˜π΄)) < 1))
170169simpld 496 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ -1 < (i Β· (tanβ€˜π΄)))
171167, 170eqbrtrrid 5146 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (0 βˆ’ 1) < (i Β· (tanβ€˜π΄)))
172 0red 11165 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ 0 ∈ ℝ)
173132a1i 11 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ 1 ∈ ℝ)
174172, 173, 164ltsubadd2d 11760 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ ((0 βˆ’ 1) < (i Β· (tanβ€˜π΄)) ↔ 0 < (1 + (i Β· (tanβ€˜π΄)))))
175171, 174mpbid 231 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ 0 < (1 + (i Β· (tanβ€˜π΄))))
176166, 175elrpd 12961 . . . . . . . . . . 11 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (1 + (i Β· (tanβ€˜π΄))) ∈ ℝ+)
177176relogcld 25994 . . . . . . . . . 10 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (logβ€˜(1 + (i Β· (tanβ€˜π΄)))) ∈ ℝ)
178169simprd 497 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (i Β· (tanβ€˜π΄)) < 1)
179 difrp 12960 . . . . . . . . . . . . 13 (((i Β· (tanβ€˜π΄)) ∈ ℝ ∧ 1 ∈ ℝ) β†’ ((i Β· (tanβ€˜π΄)) < 1 ↔ (1 βˆ’ (i Β· (tanβ€˜π΄))) ∈ ℝ+))
180164, 132, 179sylancl 587 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ ((i Β· (tanβ€˜π΄)) < 1 ↔ (1 βˆ’ (i Β· (tanβ€˜π΄))) ∈ ℝ+))
181178, 180mpbid 231 . . . . . . . . . . 11 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (1 βˆ’ (i Β· (tanβ€˜π΄))) ∈ ℝ+)
182181relogcld 25994 . . . . . . . . . 10 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄)))) ∈ ℝ)
183177, 182resubcld 11590 . . . . . . . . 9 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ ((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))) ∈ ℝ)
184 relogrn 25933 . . . . . . . . 9 (((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))) ∈ ℝ β†’ ((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))) ∈ ran log)
185183, 184syl 17 . . . . . . . 8 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ ((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))) ∈ ran log)
18664adantr 482 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ 0 < (β„œβ€˜π΄)) β†’ 𝐴 ∈ β„‚)
187186recld 15086 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜π΄) ∈ ℝ)
188 simpr 486 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ 0 < (β„œβ€˜π΄)) β†’ 0 < (β„œβ€˜π΄))
189102simprd 497 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜π΄) < (Ο€ / 2))
190189adantr 482 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜π΄) < (Ο€ / 2))
191 elioo2 13312 . . . . . . . . . . . . 13 ((0 ∈ ℝ* ∧ (Ο€ / 2) ∈ ℝ*) β†’ ((β„œβ€˜π΄) ∈ (0(,)(Ο€ / 2)) ↔ ((β„œβ€˜π΄) ∈ ℝ ∧ 0 < (β„œβ€˜π΄) ∧ (β„œβ€˜π΄) < (Ο€ / 2))))
192109, 110, 191mp2an 691 . . . . . . . . . . . 12 ((β„œβ€˜π΄) ∈ (0(,)(Ο€ / 2)) ↔ ((β„œβ€˜π΄) ∈ ℝ ∧ 0 < (β„œβ€˜π΄) ∧ (β„œβ€˜π΄) < (Ο€ / 2)))
193187, 188, 190, 192syl3anbrc 1344 . . . . . . . . . . 11 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜π΄) ∈ (0(,)(Ο€ / 2)))
194 tanregt0 25911 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (0(,)(Ο€ / 2))) β†’ 0 < (β„œβ€˜(tanβ€˜π΄)))
19564, 193, 194syl2an2r 684 . . . . . . . . . 10 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ 0 < (β„œβ€˜π΄)) β†’ 0 < (β„œβ€˜(tanβ€˜π΄)))
196195gt0ne0d 11726 . . . . . . . . 9 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜(tanβ€˜π΄)) β‰  0)
1973, 196, 130syl2an2r 684 . . . . . . . 8 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ 0 < (β„œβ€˜π΄)) β†’ ((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))) ∈ ran log)
198 recl 15002 . . . . . . . . . 10 (𝐴 ∈ β„‚ β†’ (β„œβ€˜π΄) ∈ ℝ)
199198adantr 482 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜π΄) ∈ ℝ)
200 0re 11164 . . . . . . . . 9 0 ∈ ℝ
201 lttri4 11246 . . . . . . . . 9 (((β„œβ€˜π΄) ∈ ℝ ∧ 0 ∈ ℝ) β†’ ((β„œβ€˜π΄) < 0 ∨ (β„œβ€˜π΄) = 0 ∨ 0 < (β„œβ€˜π΄)))
202199, 200, 201sylancl 587 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((β„œβ€˜π΄) < 0 ∨ (β„œβ€˜π΄) = 0 ∨ 0 < (β„œβ€˜π΄)))
203131, 185, 197, 202mpjao3dan 1432 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))) ∈ ran log)
204 logef 25953 . . . . . . 7 (((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))) ∈ ran log β†’ (logβ€˜(expβ€˜((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))))) = ((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))))
205203, 204syl 17 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (logβ€˜(expβ€˜((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))))) = ((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))))
206 2cn 12235 . . . . . . . . 9 2 ∈ β„‚
207 mulcl 11142 . . . . . . . . 9 ((2 ∈ β„‚ ∧ (i Β· 𝐴) ∈ β„‚) β†’ (2 Β· (i Β· 𝐴)) ∈ β„‚)
208206, 76, 207sylancr 588 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (2 Β· (i Β· 𝐴)) ∈ β„‚)
209 picn 25832 . . . . . . . . . . . 12 Ο€ ∈ β„‚
210 2ne0 12264 . . . . . . . . . . . 12 2 β‰  0
211 divneg 11854 . . . . . . . . . . . 12 ((Ο€ ∈ β„‚ ∧ 2 ∈ β„‚ ∧ 2 β‰  0) β†’ -(Ο€ / 2) = (-Ο€ / 2))
212209, 206, 210, 211mp3an 1462 . . . . . . . . . . 11 -(Ο€ / 2) = (-Ο€ / 2)
213212, 103eqbrtrrid 5146 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (-Ο€ / 2) < (β„œβ€˜π΄))
214 pire 25831 . . . . . . . . . . . . 13 Ο€ ∈ ℝ
215214renegcli 11469 . . . . . . . . . . . 12 -Ο€ ∈ ℝ
216215a1i 11 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -Ο€ ∈ ℝ)
217 2re 12234 . . . . . . . . . . . 12 2 ∈ ℝ
218217a1i 11 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 2 ∈ ℝ)
219 2pos 12263 . . . . . . . . . . . 12 0 < 2
220219a1i 11 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 0 < 2)
221 ltdivmul 12037 . . . . . . . . . . 11 ((-Ο€ ∈ ℝ ∧ (β„œβ€˜π΄) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) β†’ ((-Ο€ / 2) < (β„œβ€˜π΄) ↔ -Ο€ < (2 Β· (β„œβ€˜π΄))))
222216, 199, 218, 220, 221syl112anc 1375 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((-Ο€ / 2) < (β„œβ€˜π΄) ↔ -Ο€ < (2 Β· (β„œβ€˜π΄))))
223213, 222mpbid 231 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -Ο€ < (2 Β· (β„œβ€˜π΄)))
224 immul2 15029 . . . . . . . . . . 11 ((2 ∈ ℝ ∧ (i Β· 𝐴) ∈ β„‚) β†’ (β„‘β€˜(2 Β· (i Β· 𝐴))) = (2 Β· (β„‘β€˜(i Β· 𝐴))))
225217, 76, 224sylancr 588 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„‘β€˜(2 Β· (i Β· 𝐴))) = (2 Β· (β„‘β€˜(i Β· 𝐴))))
226157oveq2d 7378 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (2 Β· (β„œβ€˜π΄)) = (2 Β· (β„‘β€˜(i Β· 𝐴))))
227225, 226eqtr4d 2780 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„‘β€˜(2 Β· (i Β· 𝐴))) = (2 Β· (β„œβ€˜π΄)))
228223, 227breqtrrd 5138 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -Ο€ < (β„‘β€˜(2 Β· (i Β· 𝐴))))
229 remulcl 11143 . . . . . . . . . . 11 ((2 ∈ ℝ ∧ (β„œβ€˜π΄) ∈ ℝ) β†’ (2 Β· (β„œβ€˜π΄)) ∈ ℝ)
230217, 199, 229sylancr 588 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (2 Β· (β„œβ€˜π΄)) ∈ ℝ)
231214a1i 11 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ Ο€ ∈ ℝ)
232 ltmuldiv2 12036 . . . . . . . . . . . 12 (((β„œβ€˜π΄) ∈ ℝ ∧ Ο€ ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) β†’ ((2 Β· (β„œβ€˜π΄)) < Ο€ ↔ (β„œβ€˜π΄) < (Ο€ / 2)))
233199, 231, 218, 220, 232syl112anc 1375 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((2 Β· (β„œβ€˜π΄)) < Ο€ ↔ (β„œβ€˜π΄) < (Ο€ / 2)))
234189, 233mpbird 257 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (2 Β· (β„œβ€˜π΄)) < Ο€)
235230, 231, 234ltled 11310 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (2 Β· (β„œβ€˜π΄)) ≀ Ο€)
236227, 235eqbrtrd 5132 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„‘β€˜(2 Β· (i Β· 𝐴))) ≀ Ο€)
237 ellogrn 25931 . . . . . . . 8 ((2 Β· (i Β· 𝐴)) ∈ ran log ↔ ((2 Β· (i Β· 𝐴)) ∈ β„‚ ∧ -Ο€ < (β„‘β€˜(2 Β· (i Β· 𝐴))) ∧ (β„‘β€˜(2 Β· (i Β· 𝐴))) ≀ Ο€))
238208, 228, 236, 237syl3anbrc 1344 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (2 Β· (i Β· 𝐴)) ∈ ran log)
239 logef 25953 . . . . . . 7 ((2 Β· (i Β· 𝐴)) ∈ ran log β†’ (logβ€˜(expβ€˜(2 Β· (i Β· 𝐴)))) = (2 Β· (i Β· 𝐴)))
240238, 239syl 17 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (logβ€˜(expβ€˜(2 Β· (i Β· 𝐴)))) = (2 Β· (i Β· 𝐴)))
24193, 205, 2403eqtr3d 2785 . . . . 5 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))) = (2 Β· (i Β· 𝐴)))
242241negeqd 11402 . . . 4 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -((logβ€˜(1 + (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄))))) = -(2 Β· (i Β· 𝐴)))
24322, 242eqtr3d 2779 . . 3 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 + (i Β· (tanβ€˜π΄))))) = -(2 Β· (i Β· 𝐴)))
244243oveq2d 7378 . 2 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i / 2) Β· ((logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄)))) βˆ’ (logβ€˜(1 + (i Β· (tanβ€˜π΄)))))) = ((i / 2) Β· -(2 Β· (i Β· 𝐴))))
245 halfcl 12385 . . . . 5 (i ∈ β„‚ β†’ (i / 2) ∈ β„‚)
2467, 245mp1i 13 . . . 4 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i / 2) ∈ β„‚)
247206a1i 11 . . . 4 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 2 ∈ β„‚)
248246, 247, 79mulassd 11185 . . 3 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((i / 2) Β· 2) Β· -(i Β· 𝐴)) = ((i / 2) Β· (2 Β· -(i Β· 𝐴))))
2497, 206, 210divcan1i 11906 . . . . 5 ((i / 2) Β· 2) = i
250249oveq1i 7372 . . . 4 (((i / 2) Β· 2) Β· -(i Β· 𝐴)) = (i Β· -(i Β· 𝐴))
25133, 33, 51mulassd 11185 . . . . 5 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i Β· i) Β· -𝐴) = (i Β· (i Β· -𝐴)))
252138oveq1i 7372 . . . . . 6 ((i Β· i) Β· -𝐴) = (-1 Β· -𝐴)
253 mul2neg 11601 . . . . . . . 8 ((1 ∈ β„‚ ∧ 𝐴 ∈ β„‚) β†’ (-1 Β· -𝐴) = (1 Β· 𝐴))
2546, 64, 253sylancr 588 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (-1 Β· -𝐴) = (1 Β· 𝐴))
255 mulid2 11161 . . . . . . . 8 (𝐴 ∈ β„‚ β†’ (1 Β· 𝐴) = 𝐴)
256255adantr 482 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (1 Β· 𝐴) = 𝐴)
257254, 256eqtrd 2777 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (-1 Β· -𝐴) = 𝐴)
258252, 257eqtrid 2789 . . . . 5 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i Β· i) Β· -𝐴) = 𝐴)
25966oveq2d 7378 . . . . 5 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· (i Β· -𝐴)) = (i Β· -(i Β· 𝐴)))
260251, 258, 2593eqtr3rd 2786 . . . 4 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· -(i Β· 𝐴)) = 𝐴)
261250, 260eqtrid 2789 . . 3 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((i / 2) Β· 2) Β· -(i Β· 𝐴)) = 𝐴)
262 mulneg2 11599 . . . . 5 ((2 ∈ β„‚ ∧ (i Β· 𝐴) ∈ β„‚) β†’ (2 Β· -(i Β· 𝐴)) = -(2 Β· (i Β· 𝐴)))
263206, 76, 262sylancr 588 . . . 4 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (2 Β· -(i Β· 𝐴)) = -(2 Β· (i Β· 𝐴)))
264263oveq2d 7378 . . 3 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i / 2) Β· (2 Β· -(i Β· 𝐴))) = ((i / 2) Β· -(2 Β· (i Β· 𝐴))))
265248, 261, 2643eqtr3rd 2786 . 2 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i / 2) Β· -(2 Β· (i Β· 𝐴))) = 𝐴)
2665, 244, 2653eqtrd 2781 1 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (arctanβ€˜(tanβ€˜π΄)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ w3o 1087   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944   class class class wbr 5110  dom cdm 5638  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  β„*cxr 11195   < clt 11196   ≀ cle 11197   βˆ’ cmin 11392  -cneg 11393   / cdiv 11819  2c2 12215  β„+crp 12922  (,)cioo 13271  β„œcre 14989  β„‘cim 14990  expce 15951  sincsin 15953  cosccos 15954  tanctan 15955  Ο€cpi 15956  logclog 25926  arctancatan 26230
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-tan 15961  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-atan 26233
This theorem is referenced by:  atantanb  26290  atan1  26294
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