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

Theorem atantan 26417
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 26029 . . . 4 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (cosβ€˜π΄) β‰  0)
2 atandmtan 26414 . . . 4 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) β‰  0) β†’ (tanβ€˜π΄) ∈ dom arctan)
31, 2syldan 591 . . 3 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (tanβ€˜π΄) ∈ dom arctan)
4 atanval 26378 . . 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 11164 . . . . . . 7 1 ∈ β„‚
7 ax-icn 11165 . . . . . . . 8 i ∈ β„‚
8 tancl 16068 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) β‰  0) β†’ (tanβ€˜π΄) ∈ β„‚)
91, 8syldan 591 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (tanβ€˜π΄) ∈ β„‚)
10 mulcl 11190 . . . . . . . 8 ((i ∈ β„‚ ∧ (tanβ€˜π΄) ∈ β„‚) β†’ (i Β· (tanβ€˜π΄)) ∈ β„‚)
117, 9, 10sylancr 587 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· (tanβ€˜π΄)) ∈ β„‚)
12 addcl 11188 . . . . . . 7 ((1 ∈ β„‚ ∧ (i Β· (tanβ€˜π΄)) ∈ β„‚) β†’ (1 + (i Β· (tanβ€˜π΄))) ∈ β„‚)
136, 11, 12sylancr 587 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (1 + (i Β· (tanβ€˜π΄))) ∈ β„‚)
14 atandm2 26371 . . . . . . . 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 1144 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (1 + (i Β· (tanβ€˜π΄))) β‰  0)
1713, 16logcld 26070 . . . . 5 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (logβ€˜(1 + (i Β· (tanβ€˜π΄)))) ∈ β„‚)
18 subcl 11455 . . . . . . 7 ((1 ∈ β„‚ ∧ (i Β· (tanβ€˜π΄)) ∈ β„‚) β†’ (1 βˆ’ (i Β· (tanβ€˜π΄))) ∈ β„‚)
196, 11, 18sylancr 587 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (1 βˆ’ (i Β· (tanβ€˜π΄))) ∈ β„‚)
2015simp2d 1143 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (1 βˆ’ (i Β· (tanβ€˜π΄))) β‰  0)
2119, 20logcld 26070 . . . . 5 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (logβ€˜(1 βˆ’ (i Β· (tanβ€˜π΄)))) ∈ β„‚)
2217, 21negsubdi2d 11583 . . . 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 16039 . . . . . . . . 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 584 . . . . . . . 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 16066 . . . . . . . . . . . . 13 (𝐴 ∈ β„‚ β†’ (cosβ€˜π΄) ∈ β„‚)
2625adantr 481 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (cosβ€˜π΄) ∈ β„‚)
27 sincl 16065 . . . . . . . . . . . . . 14 (𝐴 ∈ β„‚ β†’ (sinβ€˜π΄) ∈ β„‚)
2827adantr 481 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (sinβ€˜π΄) ∈ β„‚)
29 mulcl 11190 . . . . . . . . . . . . 13 ((i ∈ β„‚ ∧ (sinβ€˜π΄) ∈ β„‚) β†’ (i Β· (sinβ€˜π΄)) ∈ β„‚)
307, 28, 29sylancr 587 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· (sinβ€˜π΄)) ∈ β„‚)
3126, 30, 26, 1divdird 12024 . . . . . . . . . . 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 11984 . . . . . . . . . . . 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 12021 . . . . . . . . . . . . 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 16067 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) β‰  0) β†’ (tanβ€˜π΄) = ((sinβ€˜π΄) / (cosβ€˜π΄)))
361, 35syldan 591 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (tanβ€˜π΄) = ((sinβ€˜π΄) / (cosβ€˜π΄)))
3736oveq2d 7421 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· (tanβ€˜π΄)) = (i Β· ((sinβ€˜π΄) / (cosβ€˜π΄))))
3834, 37eqtr4d 2775 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i Β· (sinβ€˜π΄)) / (cosβ€˜π΄)) = (i Β· (tanβ€˜π΄)))
3932, 38oveq12d 7423 . . . . . . . . . . 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 2772 . . . . . . . . . 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 16091 . . . . . . . . . . . 12 (𝐴 ∈ β„‚ β†’ (expβ€˜(i Β· 𝐴)) = ((cosβ€˜π΄) + (i Β· (sinβ€˜π΄))))
4241adantr 481 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜(i Β· 𝐴)) = ((cosβ€˜π΄) + (i Β· (sinβ€˜π΄))))
4342oveq1d 7420 . . . . . . . . . 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 26076 . . . . . . . . . . 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 584 . . . . . . . . . 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 2782 . . . . . . . . 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 12025 . . . . . . . . . . 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 7423 . . . . . . . . . . 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 2772 . . . . . . . . . 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 11456 . . . . . . . . . . . . . . 15 (𝐴 ∈ β„‚ β†’ -𝐴 ∈ β„‚)
5150adantr 481 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -𝐴 ∈ β„‚)
52 efival 16091 . . . . . . . . . . . . . 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 16086 . . . . . . . . . . . . . . 15 (𝐴 ∈ β„‚ β†’ (cosβ€˜-𝐴) = (cosβ€˜π΄))
5554adantr 481 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (cosβ€˜-𝐴) = (cosβ€˜π΄))
56 sinneg 16085 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ β„‚ β†’ (sinβ€˜-𝐴) = -(sinβ€˜π΄))
5756adantr 481 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (sinβ€˜-𝐴) = -(sinβ€˜π΄))
5857oveq2d 7421 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· (sinβ€˜-𝐴)) = (i Β· -(sinβ€˜π΄)))
59 mulneg2 11647 . . . . . . . . . . . . . . . 16 ((i ∈ β„‚ ∧ (sinβ€˜π΄) ∈ β„‚) β†’ (i Β· -(sinβ€˜π΄)) = -(i Β· (sinβ€˜π΄)))
607, 28, 59sylancr 587 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· -(sinβ€˜π΄)) = -(i Β· (sinβ€˜π΄)))
6158, 60eqtrd 2772 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· (sinβ€˜-𝐴)) = -(i Β· (sinβ€˜π΄)))
6255, 61oveq12d 7423 . . . . . . . . . . . . 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 2772 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜(i Β· -𝐴)) = ((cosβ€˜π΄) + -(i Β· (sinβ€˜π΄))))
64 simpl 483 . . . . . . . . . . . . . 14 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 𝐴 ∈ β„‚)
65 mulneg2 11647 . . . . . . . . . . . . . 14 ((i ∈ β„‚ ∧ 𝐴 ∈ β„‚) β†’ (i Β· -𝐴) = -(i Β· 𝐴))
667, 64, 65sylancr 587 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· -𝐴) = -(i Β· 𝐴))
6766fveq2d 6892 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜(i Β· -𝐴)) = (expβ€˜-(i Β· 𝐴)))
6826, 30negsubd 11573 . . . . . . . . . . . 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 2780 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (expβ€˜-(i Β· 𝐴)) = ((cosβ€˜π΄) βˆ’ (i Β· (sinβ€˜π΄))))
7069oveq1d 7420 . . . . . . . . . 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 26076 . . . . . . . . . . 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 584 . . . . . . . . . 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 2782 . . . . . . . . 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 7423 . . . . . . . 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 11190 . . . . . . . . . . . 12 ((i ∈ β„‚ ∧ 𝐴 ∈ β„‚) β†’ (i Β· 𝐴) ∈ β„‚)
767, 64, 75sylancr 587 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· 𝐴) ∈ β„‚)
77 efcl 16022 . . . . . . . . . . 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 11554 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -(i Β· 𝐴) ∈ β„‚)
80 efcl 16022 . . . . . . . . . . 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 16036 . . . . . . . . . . 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 12014 . . . . . . . . 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 16039 . . . . . . . . . 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 584 . . . . . . . . 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 11574 . . . . . . . . . . 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 12451 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (2 Β· (i Β· 𝐴)) = ((i Β· 𝐴) + (i Β· 𝐴)))
8987, 88eqtr4d 2775 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i Β· 𝐴) βˆ’ -(i Β· 𝐴)) = (2 Β· (i Β· 𝐴)))
9089fveq2d 6892 . . . . . . . . 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 2778 . . . . . . . 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 2778 . . . . . . 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 6892 . . . . . 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 481 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ 𝐴 ∈ β„‚)
9594renegd 15152 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜-𝐴) = -(β„œβ€˜π΄))
9694recld 15137 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜π΄) ∈ ℝ)
9796renegcld 11637 . . . . . . . . . . . . . . 15 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ -(β„œβ€˜π΄) ∈ ℝ)
98 simpr 485 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜π΄) < 0)
9996lt0neg1d 11779 . . . . . . . . . . . . . . . 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 13379 . . . . . . . . . . . . . . . . . . 19 ((β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2)) β†’ (-(Ο€ / 2) < (β„œβ€˜π΄) ∧ (β„œβ€˜π΄) < (Ο€ / 2)))
102101adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (-(Ο€ / 2) < (β„œβ€˜π΄) ∧ (β„œβ€˜π΄) < (Ο€ / 2)))
103102simpld 495 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -(Ο€ / 2) < (β„œβ€˜π΄))
104103adantr 481 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ -(Ο€ / 2) < (β„œβ€˜π΄))
105 halfpire 25965 . . . . . . . . . . . . . . . . 17 (Ο€ / 2) ∈ ℝ
106 ltnegcon1 11711 . . . . . . . . . . . . . . . . 17 (((Ο€ / 2) ∈ ℝ ∧ (β„œβ€˜π΄) ∈ ℝ) β†’ (-(Ο€ / 2) < (β„œβ€˜π΄) ↔ -(β„œβ€˜π΄) < (Ο€ / 2)))
107105, 96, 106sylancr 587 . . . . . . . . . . . . . . . 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 11257 . . . . . . . . . . . . . . . 16 0 ∈ ℝ*
110105rexri 11268 . . . . . . . . . . . . . . . 16 (Ο€ / 2) ∈ ℝ*
111 elioo2 13361 . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . 15 (-(β„œβ€˜π΄) ∈ (0(,)(Ο€ / 2)) ↔ (-(β„œβ€˜π΄) ∈ ℝ ∧ 0 < -(β„œβ€˜π΄) ∧ -(β„œβ€˜π΄) < (Ο€ / 2)))
11397, 100, 108, 112syl3anbrc 1343 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ -(β„œβ€˜π΄) ∈ (0(,)(Ο€ / 2)))
11495, 113eqeltrd 2833 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜-𝐴) ∈ (0(,)(Ο€ / 2)))
115 tanregt0 26039 . . . . . . . . . . . . 13 ((-𝐴 ∈ β„‚ ∧ (β„œβ€˜-𝐴) ∈ (0(,)(Ο€ / 2))) β†’ 0 < (β„œβ€˜(tanβ€˜-𝐴)))
11651, 114, 115syl2an2r 683 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ 0 < (β„œβ€˜(tanβ€˜-𝐴)))
117 tanneg 16087 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ β„‚ ∧ (cosβ€˜π΄) β‰  0) β†’ (tanβ€˜-𝐴) = -(tanβ€˜π΄))
1181, 117syldan 591 . . . . . . . . . . . . . . 15 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (tanβ€˜-𝐴) = -(tanβ€˜π΄))
119118adantr 481 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (tanβ€˜-𝐴) = -(tanβ€˜π΄))
120119fveq2d 6892 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜(tanβ€˜-𝐴)) = (β„œβ€˜-(tanβ€˜π΄)))
1219adantr 481 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (tanβ€˜π΄) ∈ β„‚)
122121renegd 15152 . . . . . . . . . . . . 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 2772 . . . . . . . . . . . 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 5173 . . . . . . . . . . 11 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ 0 < -(β„œβ€˜(tanβ€˜π΄)))
1259recld 15137 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜(tanβ€˜π΄)) ∈ ℝ)
126125adantr 481 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜(tanβ€˜π΄)) ∈ ℝ)
127126lt0neg1d 11779 . . . . . . . . . . 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 256 . . . . . . . . . 10 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜(tanβ€˜π΄)) < 0)
129128lt0ne0d 11775 . . . . . . . . 9 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) < 0) β†’ (β„œβ€˜(tanβ€˜π΄)) β‰  0)
130 atanlogsub 26410 . . . . . . . . 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 683 . . . . . . . 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 11210 . . . . . . . . . . . . 13 1 ∈ ℝ
133 ioossre 13381 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (i Β· (tanβ€˜π΄)) ∈ β„‚)
136 ine0 11645 . . . . . . . . . . . . . . . . 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 11839 . . . . . . . . . . . . . . . . . . 19 (i Β· i) = -1
139138oveq1i 7415 . . . . . . . . . . . . . . . . . 18 ((i Β· i) Β· (tanβ€˜π΄)) = (-1 Β· (tanβ€˜π΄))
1409adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (tanβ€˜π΄) ∈ β„‚)
141140mulm1d 11662 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (-1 Β· (tanβ€˜π΄)) = -(tanβ€˜π΄))
142118adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (tanβ€˜-𝐴) = -(tanβ€˜π΄))
143141, 142eqtr4d 2775 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (-1 Β· (tanβ€˜π΄)) = (tanβ€˜-𝐴))
144139, 143eqtrid 2784 . . . . . . . . . . . . . . . . 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 11233 . . . . . . . . . . . . . . . . 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 7415 . . . . . . . . . . . . . . . . . . . 20 ((i Β· i) Β· 𝐴) = (-1 Β· 𝐴)
14764adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ 𝐴 ∈ β„‚)
148147mulm1d 11662 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (-1 Β· 𝐴) = -𝐴)
149146, 148eqtrid 2784 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ ((i Β· i) Β· 𝐴) = -𝐴)
150134, 134, 147mulassd 11233 . . . . . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ -𝐴 = (i Β· (i Β· 𝐴)))
152151fveq2d 6892 . . . . . . . . . . . . . . . . 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 2780 . . . . . . . . . . . . . . . 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 12042 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (i Β· 𝐴) ∈ β„‚)
156 reim 15052 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∈ β„‚ β†’ (β„œβ€˜π΄) = (β„‘β€˜(i Β· 𝐴)))
157156adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜π΄) = (β„‘β€˜(i Β· 𝐴)))
158157eqeq1d 2734 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((β„œβ€˜π΄) = 0 ↔ (β„‘β€˜(i Β· 𝐴)) = 0))
159158biimpa 477 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (β„‘β€˜(i Β· 𝐴)) = 0)
160155, 159reim0bd 15143 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (i Β· 𝐴) ∈ ℝ)
161 tanhbnd 16100 . . . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (i Β· (tanβ€˜π΄)) ∈ (-1(,)1))
164133, 163sselid 3979 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (i Β· (tanβ€˜π΄)) ∈ ℝ)
165 readdcl 11189 . . . . . . . . . . . . 13 ((1 ∈ ℝ ∧ (i Β· (tanβ€˜π΄)) ∈ ℝ) β†’ (1 + (i Β· (tanβ€˜π΄))) ∈ ℝ)
166132, 164, 165sylancr 587 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (1 + (i Β· (tanβ€˜π΄))) ∈ ℝ)
167 df-neg 11443 . . . . . . . . . . . . . 14 -1 = (0 βˆ’ 1)
168 eliooord 13379 . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . 14 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ -1 < (i Β· (tanβ€˜π΄)))
171167, 170eqbrtrrid 5183 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (0 βˆ’ 1) < (i Β· (tanβ€˜π΄)))
172 0red 11213 . . . . . . . . . . . . . 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 11808 . . . . . . . . . . . . 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 13009 . . . . . . . . . . 11 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (1 + (i Β· (tanβ€˜π΄))) ∈ ℝ+)
177176relogcld 26122 . . . . . . . . . 10 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (logβ€˜(1 + (i Β· (tanβ€˜π΄)))) ∈ ℝ)
178169simprd 496 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ (β„œβ€˜π΄) = 0) β†’ (i Β· (tanβ€˜π΄)) < 1)
179 difrp 13008 . . . . . . . . . . . . 13 (((i Β· (tanβ€˜π΄)) ∈ ℝ ∧ 1 ∈ ℝ) β†’ ((i Β· (tanβ€˜π΄)) < 1 ↔ (1 βˆ’ (i Β· (tanβ€˜π΄))) ∈ ℝ+))
180164, 132, 179sylancl 586 . . . . . . . . . . . 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 26122 . . . . . . . . . 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 11638 . . . . . . . . 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 26061 . . . . . . . . 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 481 . . . . . . . . . . . . 13 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ 0 < (β„œβ€˜π΄)) β†’ 𝐴 ∈ β„‚)
187186recld 15137 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜π΄) ∈ ℝ)
188 simpr 485 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ 0 < (β„œβ€˜π΄)) β†’ 0 < (β„œβ€˜π΄))
189102simprd 496 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜π΄) < (Ο€ / 2))
190189adantr 481 . . . . . . . . . . . 12 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜π΄) < (Ο€ / 2))
191 elioo2 13361 . . . . . . . . . . . . 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 690 . . . . . . . . . . . 12 ((β„œβ€˜π΄) ∈ (0(,)(Ο€ / 2)) ↔ ((β„œβ€˜π΄) ∈ ℝ ∧ 0 < (β„œβ€˜π΄) ∧ (β„œβ€˜π΄) < (Ο€ / 2)))
193187, 188, 190, 192syl3anbrc 1343 . . . . . . . . . . 11 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜π΄) ∈ (0(,)(Ο€ / 2)))
194 tanregt0 26039 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (0(,)(Ο€ / 2))) β†’ 0 < (β„œβ€˜(tanβ€˜π΄)))
19564, 193, 194syl2an2r 683 . . . . . . . . . 10 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ 0 < (β„œβ€˜π΄)) β†’ 0 < (β„œβ€˜(tanβ€˜π΄)))
196195gt0ne0d 11774 . . . . . . . . 9 (((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜(tanβ€˜π΄)) β‰  0)
1973, 196, 130syl2an2r 683 . . . . . . . 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 15053 . . . . . . . . . 10 (𝐴 ∈ β„‚ β†’ (β„œβ€˜π΄) ∈ ℝ)
199198adantr 481 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„œβ€˜π΄) ∈ ℝ)
200 0re 11212 . . . . . . . . 9 0 ∈ ℝ
201 lttri4 11294 . . . . . . . . 9 (((β„œβ€˜π΄) ∈ ℝ ∧ 0 ∈ ℝ) β†’ ((β„œβ€˜π΄) < 0 ∨ (β„œβ€˜π΄) = 0 ∨ 0 < (β„œβ€˜π΄)))
202199, 200, 201sylancl 586 . . . . . . . 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 1431 . . . . . . 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 26081 . . . . . . 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 12283 . . . . . . . . 9 2 ∈ β„‚
207 mulcl 11190 . . . . . . . . 9 ((2 ∈ β„‚ ∧ (i Β· 𝐴) ∈ β„‚) β†’ (2 Β· (i Β· 𝐴)) ∈ β„‚)
208206, 76, 207sylancr 587 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (2 Β· (i Β· 𝐴)) ∈ β„‚)
209 picn 25960 . . . . . . . . . . . 12 Ο€ ∈ β„‚
210 2ne0 12312 . . . . . . . . . . . 12 2 β‰  0
211 divneg 11902 . . . . . . . . . . . 12 ((Ο€ ∈ β„‚ ∧ 2 ∈ β„‚ ∧ 2 β‰  0) β†’ -(Ο€ / 2) = (-Ο€ / 2))
212209, 206, 210, 211mp3an 1461 . . . . . . . . . . 11 -(Ο€ / 2) = (-Ο€ / 2)
213212, 103eqbrtrrid 5183 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (-Ο€ / 2) < (β„œβ€˜π΄))
214 pire 25959 . . . . . . . . . . . . 13 Ο€ ∈ ℝ
215214renegcli 11517 . . . . . . . . . . . 12 -Ο€ ∈ ℝ
216215a1i 11 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -Ο€ ∈ ℝ)
217 2re 12282 . . . . . . . . . . . 12 2 ∈ ℝ
218217a1i 11 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 2 ∈ ℝ)
219 2pos 12311 . . . . . . . . . . . 12 0 < 2
220219a1i 11 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ 0 < 2)
221 ltdivmul 12085 . . . . . . . . . . 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 1374 . . . . . . . . . 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 15080 . . . . . . . . . . 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 587 . . . . . . . . . 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 7421 . . . . . . . . . 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 2775 . . . . . . . . 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 5175 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ -Ο€ < (β„‘β€˜(2 Β· (i Β· 𝐴))))
229 remulcl 11191 . . . . . . . . . . 11 ((2 ∈ ℝ ∧ (β„œβ€˜π΄) ∈ ℝ) β†’ (2 Β· (β„œβ€˜π΄)) ∈ ℝ)
230217, 199, 229sylancr 587 . . . . . . . . . 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 12084 . . . . . . . . . . . 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 1374 . . . . . . . . . . 11 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((2 Β· (β„œβ€˜π΄)) < Ο€ ↔ (β„œβ€˜π΄) < (Ο€ / 2)))
234189, 233mpbird 256 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (2 Β· (β„œβ€˜π΄)) < Ο€)
235230, 231, 234ltled 11358 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (2 Β· (β„œβ€˜π΄)) ≀ Ο€)
236227, 235eqbrtrd 5169 . . . . . . . 8 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (β„‘β€˜(2 Β· (i Β· 𝐴))) ≀ Ο€)
237 ellogrn 26059 . . . . . . . 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 1343 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (2 Β· (i Β· 𝐴)) ∈ ran log)
239 logef 26081 . . . . . . 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 2780 . . . . 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 11450 . . . 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 2774 . . 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 7421 . 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 12433 . . . . 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 11233 . . 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 11954 . . . . 5 ((i / 2) Β· 2) = i
250249oveq1i 7415 . . . 4 (((i / 2) Β· 2) Β· -(i Β· 𝐴)) = (i Β· -(i Β· 𝐴))
25133, 33, 51mulassd 11233 . . . . 5 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i Β· i) Β· -𝐴) = (i Β· (i Β· -𝐴)))
252138oveq1i 7415 . . . . . 6 ((i Β· i) Β· -𝐴) = (-1 Β· -𝐴)
253 mul2neg 11649 . . . . . . . 8 ((1 ∈ β„‚ ∧ 𝐴 ∈ β„‚) β†’ (-1 Β· -𝐴) = (1 Β· 𝐴))
2546, 64, 253sylancr 587 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (-1 Β· -𝐴) = (1 Β· 𝐴))
255 mullid 11209 . . . . . . . 8 (𝐴 ∈ β„‚ β†’ (1 Β· 𝐴) = 𝐴)
256255adantr 481 . . . . . . 7 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (1 Β· 𝐴) = 𝐴)
257254, 256eqtrd 2772 . . . . . 6 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (-1 Β· -𝐴) = 𝐴)
258252, 257eqtrid 2784 . . . . 5 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i Β· i) Β· -𝐴) = 𝐴)
25966oveq2d 7421 . . . . 5 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· (i Β· -𝐴)) = (i Β· -(i Β· 𝐴)))
260251, 258, 2593eqtr3rd 2781 . . . 4 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (i Β· -(i Β· 𝐴)) = 𝐴)
261250, 260eqtrid 2784 . . 3 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (((i / 2) Β· 2) Β· -(i Β· 𝐴)) = 𝐴)
262 mulneg2 11647 . . . . 5 ((2 ∈ β„‚ ∧ (i Β· 𝐴) ∈ β„‚) β†’ (2 Β· -(i Β· 𝐴)) = -(2 Β· (i Β· 𝐴)))
263206, 76, 262sylancr 587 . . . 4 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ (2 Β· -(i Β· 𝐴)) = -(2 Β· (i Β· 𝐴)))
264263oveq2d 7421 . . 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 2781 . 2 ((𝐴 ∈ β„‚ ∧ (β„œβ€˜π΄) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((i / 2) Β· -(2 Β· (i Β· 𝐴))) = 𝐴)
2665, 244, 2653eqtrd 2776 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 396   ∨ w3o 1086   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   class class class wbr 5147  dom cdm 5675  ran crn 5676  β€˜cfv 6540  (class class class)co 7405  β„‚cc 11104  β„cr 11105  0cc0 11106  1c1 11107  ici 11108   + caddc 11109   Β· cmul 11111  β„*cxr 11243   < clt 11244   ≀ cle 11245   βˆ’ cmin 11440  -cneg 11441   / cdiv 11867  2c2 12263  β„+crp 12970  (,)cioo 13320  β„œcre 15040  β„‘cim 15041  expce 16001  sincsin 16003  cosccos 16004  tanctan 16005  Ο€cpi 16006  logclog 26054  arctancatan 26358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184  ax-addf 11185  ax-mulf 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-fi 9402  df-sup 9433  df-inf 9434  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-ioo 13324  df-ioc 13325  df-ico 13326  df-icc 13327  df-fz 13481  df-fzo 13624  df-fl 13753  df-mod 13831  df-seq 13963  df-exp 14024  df-fac 14230  df-bc 14259  df-hash 14287  df-shft 15010  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-limsup 15411  df-clim 15428  df-rlim 15429  df-sum 15629  df-ef 16007  df-sin 16009  df-cos 16010  df-tan 16011  df-pi 16012  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-starv 17208  df-sca 17209  df-vsca 17210  df-ip 17211  df-tset 17212  df-ple 17213  df-ds 17215  df-unif 17216  df-hom 17217  df-cco 17218  df-rest 17364  df-topn 17365  df-0g 17383  df-gsum 17384  df-topgen 17385  df-pt 17386  df-prds 17389  df-xrs 17444  df-qtop 17449  df-imas 17450  df-xps 17452  df-mre 17526  df-mrc 17527  df-acs 17529  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-submnd 18668  df-mulg 18945  df-cntz 19175  df-cmn 19644  df-psmet 20928  df-xmet 20929  df-met 20930  df-bl 20931  df-mopn 20932  df-fbas 20933  df-fg 20934  df-cnfld 20937  df-top 22387  df-topon 22404  df-topsp 22426  df-bases 22440  df-cld 22514  df-ntr 22515  df-cls 22516  df-nei 22593  df-lp 22631  df-perf 22632  df-cn 22722  df-cnp 22723  df-haus 22810  df-tx 23057  df-hmeo 23250  df-fil 23341  df-fm 23433  df-flim 23434  df-flf 23435  df-xms 23817  df-ms 23818  df-tms 23819  df-cncf 24385  df-limc 25374  df-dv 25375  df-log 26056  df-atan 26361
This theorem is referenced by:  atantanb  26418  atan1  26422
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