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Theorem atanlogsublem 26420
Description: Lemma for atanlogsub 26421. (Contributed by Mario Carneiro, 4-Apr-2015.)
Assertion
Ref Expression
atanlogsublem ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜((logβ€˜(1 + (i Β· 𝐴))) βˆ’ (logβ€˜(1 βˆ’ (i Β· 𝐴))))) ∈ (-Ο€(,)Ο€))
Proof of Theorem atanlogsublem
StepHypRef Expression
1 ax-1cn 11168 . . . . . 6 1 ∈ β„‚
2 ax-icn 11169 . . . . . . 7 i ∈ β„‚
3 simpl 484 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ 𝐴 ∈ dom arctan)
4 atandm2 26382 . . . . . . . . 9 (𝐴 ∈ dom arctan ↔ (𝐴 ∈ β„‚ ∧ (1 βˆ’ (i Β· 𝐴)) β‰  0 ∧ (1 + (i Β· 𝐴)) β‰  0))
53, 4sylib 217 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (𝐴 ∈ β„‚ ∧ (1 βˆ’ (i Β· 𝐴)) β‰  0 ∧ (1 + (i Β· 𝐴)) β‰  0))
65simp1d 1143 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ 𝐴 ∈ β„‚)
7 mulcl 11194 . . . . . . 7 ((i ∈ β„‚ ∧ 𝐴 ∈ β„‚) β†’ (i Β· 𝐴) ∈ β„‚)
82, 6, 7sylancr 588 . . . . . 6 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (i Β· 𝐴) ∈ β„‚)
9 addcl 11192 . . . . . 6 ((1 ∈ β„‚ ∧ (i Β· 𝐴) ∈ β„‚) β†’ (1 + (i Β· 𝐴)) ∈ β„‚)
101, 8, 9sylancr 588 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (1 + (i Β· 𝐴)) ∈ β„‚)
115simp3d 1145 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (1 + (i Β· 𝐴)) β‰  0)
1210, 11logcld 26079 . . . 4 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (logβ€˜(1 + (i Β· 𝐴))) ∈ β„‚)
13 subcl 11459 . . . . . 6 ((1 ∈ β„‚ ∧ (i Β· 𝐴) ∈ β„‚) β†’ (1 βˆ’ (i Β· 𝐴)) ∈ β„‚)
141, 8, 13sylancr 588 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (1 βˆ’ (i Β· 𝐴)) ∈ β„‚)
155simp2d 1144 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (1 βˆ’ (i Β· 𝐴)) β‰  0)
1614, 15logcld 26079 . . . 4 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (logβ€˜(1 βˆ’ (i Β· 𝐴))) ∈ β„‚)
1712, 16imsubd 15164 . . 3 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜((logβ€˜(1 + (i Β· 𝐴))) βˆ’ (logβ€˜(1 βˆ’ (i Β· 𝐴))))) = ((β„‘β€˜(logβ€˜(1 + (i Β· 𝐴)))) βˆ’ (β„‘β€˜(logβ€˜(1 βˆ’ (i Β· 𝐴))))))
182a1i 11 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ i ∈ β„‚)
1918, 6, 18subdid 11670 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (i Β· (𝐴 βˆ’ i)) = ((i Β· 𝐴) βˆ’ (i Β· i)))
20 ixi 11843 . . . . . . . . . . 11 (i Β· i) = -1
2120oveq2i 7420 . . . . . . . . . 10 ((i Β· 𝐴) βˆ’ (i Β· i)) = ((i Β· 𝐴) βˆ’ -1)
22 subneg 11509 . . . . . . . . . . 11 (((i Β· 𝐴) ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((i Β· 𝐴) βˆ’ -1) = ((i Β· 𝐴) + 1))
238, 1, 22sylancl 587 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((i Β· 𝐴) βˆ’ -1) = ((i Β· 𝐴) + 1))
2421, 23eqtrid 2785 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((i Β· 𝐴) βˆ’ (i Β· i)) = ((i Β· 𝐴) + 1))
25 addcom 11400 . . . . . . . . . 10 (((i Β· 𝐴) ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((i Β· 𝐴) + 1) = (1 + (i Β· 𝐴)))
268, 1, 25sylancl 587 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((i Β· 𝐴) + 1) = (1 + (i Β· 𝐴)))
2719, 24, 263eqtrd 2777 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (i Β· (𝐴 βˆ’ i)) = (1 + (i Β· 𝐴)))
2827fveq2d 6896 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (logβ€˜(i Β· (𝐴 βˆ’ i))) = (logβ€˜(1 + (i Β· 𝐴))))
29 subcl 11459 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ i ∈ β„‚) β†’ (𝐴 βˆ’ i) ∈ β„‚)
306, 2, 29sylancl 587 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (𝐴 βˆ’ i) ∈ β„‚)
31 resub 15074 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ i ∈ β„‚) β†’ (β„œβ€˜(𝐴 βˆ’ i)) = ((β„œβ€˜π΄) βˆ’ (β„œβ€˜i)))
326, 2, 31sylancl 587 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜(𝐴 βˆ’ i)) = ((β„œβ€˜π΄) βˆ’ (β„œβ€˜i)))
33 rei 15103 . . . . . . . . . . . . 13 (β„œβ€˜i) = 0
3433oveq2i 7420 . . . . . . . . . . . 12 ((β„œβ€˜π΄) βˆ’ (β„œβ€˜i)) = ((β„œβ€˜π΄) βˆ’ 0)
356recld 15141 . . . . . . . . . . . . . 14 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜π΄) ∈ ℝ)
3635recnd 11242 . . . . . . . . . . . . 13 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜π΄) ∈ β„‚)
3736subid1d 11560 . . . . . . . . . . . 12 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„œβ€˜π΄) βˆ’ 0) = (β„œβ€˜π΄))
3834, 37eqtrid 2785 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„œβ€˜π΄) βˆ’ (β„œβ€˜i)) = (β„œβ€˜π΄))
3932, 38eqtrd 2773 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜(𝐴 βˆ’ i)) = (β„œβ€˜π΄))
40 gt0ne0 11679 . . . . . . . . . . 11 (((β„œβ€˜π΄) ∈ ℝ ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜π΄) β‰  0)
4135, 40sylancom 589 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜π΄) β‰  0)
4239, 41eqnetrd 3009 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜(𝐴 βˆ’ i)) β‰  0)
43 fveq2 6892 . . . . . . . . . . 11 ((𝐴 βˆ’ i) = 0 β†’ (β„œβ€˜(𝐴 βˆ’ i)) = (β„œβ€˜0))
44 re0 15099 . . . . . . . . . . 11 (β„œβ€˜0) = 0
4543, 44eqtrdi 2789 . . . . . . . . . 10 ((𝐴 βˆ’ i) = 0 β†’ (β„œβ€˜(𝐴 βˆ’ i)) = 0)
4645necon3i 2974 . . . . . . . . 9 ((β„œβ€˜(𝐴 βˆ’ i)) β‰  0 β†’ (𝐴 βˆ’ i) β‰  0)
4742, 46syl 17 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (𝐴 βˆ’ i) β‰  0)
48 simpr 486 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ 0 < (β„œβ€˜π΄))
49 0re 11216 . . . . . . . . . . 11 0 ∈ ℝ
50 ltle 11302 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ (β„œβ€˜π΄) ∈ ℝ) β†’ (0 < (β„œβ€˜π΄) β†’ 0 ≀ (β„œβ€˜π΄)))
5149, 35, 50sylancr 588 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (0 < (β„œβ€˜π΄) β†’ 0 ≀ (β„œβ€˜π΄)))
5248, 51mpd 15 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ 0 ≀ (β„œβ€˜π΄))
5352, 39breqtrrd 5177 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ 0 ≀ (β„œβ€˜(𝐴 βˆ’ i)))
54 logimul 26122 . . . . . . . 8 (((𝐴 βˆ’ i) ∈ β„‚ ∧ (𝐴 βˆ’ i) β‰  0 ∧ 0 ≀ (β„œβ€˜(𝐴 βˆ’ i))) β†’ (logβ€˜(i Β· (𝐴 βˆ’ i))) = ((logβ€˜(𝐴 βˆ’ i)) + (i Β· (Ο€ / 2))))
5530, 47, 53, 54syl3anc 1372 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (logβ€˜(i Β· (𝐴 βˆ’ i))) = ((logβ€˜(𝐴 βˆ’ i)) + (i Β· (Ο€ / 2))))
5628, 55eqtr3d 2775 . . . . . 6 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (logβ€˜(1 + (i Β· 𝐴))) = ((logβ€˜(𝐴 βˆ’ i)) + (i Β· (Ο€ / 2))))
5756fveq2d 6896 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(1 + (i Β· 𝐴)))) = (β„‘β€˜((logβ€˜(𝐴 βˆ’ i)) + (i Β· (Ο€ / 2)))))
5830, 47logcld 26079 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (logβ€˜(𝐴 βˆ’ i)) ∈ β„‚)
59 halfpire 25974 . . . . . . . . 9 (Ο€ / 2) ∈ ℝ
6059recni 11228 . . . . . . . 8 (Ο€ / 2) ∈ β„‚
612, 60mulcli 11221 . . . . . . 7 (i Β· (Ο€ / 2)) ∈ β„‚
62 imadd 15081 . . . . . . 7 (((logβ€˜(𝐴 βˆ’ i)) ∈ β„‚ ∧ (i Β· (Ο€ / 2)) ∈ β„‚) β†’ (β„‘β€˜((logβ€˜(𝐴 βˆ’ i)) + (i Β· (Ο€ / 2)))) = ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) + (β„‘β€˜(i Β· (Ο€ / 2)))))
6358, 61, 62sylancl 587 . . . . . 6 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜((logβ€˜(𝐴 βˆ’ i)) + (i Β· (Ο€ / 2)))) = ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) + (β„‘β€˜(i Β· (Ο€ / 2)))))
64 reim 15056 . . . . . . . . 9 ((Ο€ / 2) ∈ β„‚ β†’ (β„œβ€˜(Ο€ / 2)) = (β„‘β€˜(i Β· (Ο€ / 2))))
6560, 64ax-mp 5 . . . . . . . 8 (β„œβ€˜(Ο€ / 2)) = (β„‘β€˜(i Β· (Ο€ / 2)))
66 rere 15069 . . . . . . . . 9 ((Ο€ / 2) ∈ ℝ β†’ (β„œβ€˜(Ο€ / 2)) = (Ο€ / 2))
6759, 66ax-mp 5 . . . . . . . 8 (β„œβ€˜(Ο€ / 2)) = (Ο€ / 2)
6865, 67eqtr3i 2763 . . . . . . 7 (β„‘β€˜(i Β· (Ο€ / 2))) = (Ο€ / 2)
6968oveq2i 7420 . . . . . 6 ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) + (β„‘β€˜(i Β· (Ο€ / 2)))) = ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) + (Ο€ / 2))
7063, 69eqtrdi 2789 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜((logβ€˜(𝐴 βˆ’ i)) + (i Β· (Ο€ / 2)))) = ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) + (Ο€ / 2)))
7157, 70eqtrd 2773 . . . 4 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(1 + (i Β· 𝐴)))) = ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) + (Ο€ / 2)))
72 addcl 11192 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ i ∈ β„‚) β†’ (𝐴 + i) ∈ β„‚)
736, 2, 72sylancl 587 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (𝐴 + i) ∈ β„‚)
74 mulcl 11194 . . . . . . . . 9 ((i ∈ β„‚ ∧ (𝐴 + i) ∈ β„‚) β†’ (i Β· (𝐴 + i)) ∈ β„‚)
752, 73, 74sylancr 588 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (i Β· (𝐴 + i)) ∈ β„‚)
76 reim 15056 . . . . . . . . . . 11 ((𝐴 + i) ∈ β„‚ β†’ (β„œβ€˜(𝐴 + i)) = (β„‘β€˜(i Β· (𝐴 + i))))
7773, 76syl 17 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜(𝐴 + i)) = (β„‘β€˜(i Β· (𝐴 + i))))
78 readd 15073 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ i ∈ β„‚) β†’ (β„œβ€˜(𝐴 + i)) = ((β„œβ€˜π΄) + (β„œβ€˜i)))
796, 2, 78sylancl 587 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜(𝐴 + i)) = ((β„œβ€˜π΄) + (β„œβ€˜i)))
8033oveq2i 7420 . . . . . . . . . . . 12 ((β„œβ€˜π΄) + (β„œβ€˜i)) = ((β„œβ€˜π΄) + 0)
8136addridd 11414 . . . . . . . . . . . 12 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„œβ€˜π΄) + 0) = (β„œβ€˜π΄))
8280, 81eqtrid 2785 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„œβ€˜π΄) + (β„œβ€˜i)) = (β„œβ€˜π΄))
8379, 82eqtrd 2773 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜(𝐴 + i)) = (β„œβ€˜π΄))
8477, 83eqtr3d 2775 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(i Β· (𝐴 + i))) = (β„œβ€˜π΄))
8548, 84breqtrrd 5177 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ 0 < (β„‘β€˜(i Β· (𝐴 + i))))
86 logneg2 26123 . . . . . . . 8 (((i Β· (𝐴 + i)) ∈ β„‚ ∧ 0 < (β„‘β€˜(i Β· (𝐴 + i)))) β†’ (logβ€˜-(i Β· (𝐴 + i))) = ((logβ€˜(i Β· (𝐴 + i))) βˆ’ (i Β· Ο€)))
8775, 85, 86syl2anc 585 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (logβ€˜-(i Β· (𝐴 + i))) = ((logβ€˜(i Β· (𝐴 + i))) βˆ’ (i Β· Ο€)))
8818, 6, 18adddid 11238 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (i Β· (𝐴 + i)) = ((i Β· 𝐴) + (i Β· i)))
8920oveq2i 7420 . . . . . . . . . . . 12 ((i Β· 𝐴) + (i Β· i)) = ((i Β· 𝐴) + -1)
90 negsub 11508 . . . . . . . . . . . . 13 (((i Β· 𝐴) ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((i Β· 𝐴) + -1) = ((i Β· 𝐴) βˆ’ 1))
918, 1, 90sylancl 587 . . . . . . . . . . . 12 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((i Β· 𝐴) + -1) = ((i Β· 𝐴) βˆ’ 1))
9289, 91eqtrid 2785 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((i Β· 𝐴) + (i Β· i)) = ((i Β· 𝐴) βˆ’ 1))
9388, 92eqtrd 2773 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (i Β· (𝐴 + i)) = ((i Β· 𝐴) βˆ’ 1))
9493negeqd 11454 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ -(i Β· (𝐴 + i)) = -((i Β· 𝐴) βˆ’ 1))
95 negsubdi2 11519 . . . . . . . . . 10 (((i Β· 𝐴) ∈ β„‚ ∧ 1 ∈ β„‚) β†’ -((i Β· 𝐴) βˆ’ 1) = (1 βˆ’ (i Β· 𝐴)))
968, 1, 95sylancl 587 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ -((i Β· 𝐴) βˆ’ 1) = (1 βˆ’ (i Β· 𝐴)))
9794, 96eqtrd 2773 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ -(i Β· (𝐴 + i)) = (1 βˆ’ (i Β· 𝐴)))
9897fveq2d 6896 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (logβ€˜-(i Β· (𝐴 + i))) = (logβ€˜(1 βˆ’ (i Β· 𝐴))))
9983, 41eqnetrd 3009 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜(𝐴 + i)) β‰  0)
100 fveq2 6892 . . . . . . . . . . . 12 ((𝐴 + i) = 0 β†’ (β„œβ€˜(𝐴 + i)) = (β„œβ€˜0))
101100, 44eqtrdi 2789 . . . . . . . . . . 11 ((𝐴 + i) = 0 β†’ (β„œβ€˜(𝐴 + i)) = 0)
102101necon3i 2974 . . . . . . . . . 10 ((β„œβ€˜(𝐴 + i)) β‰  0 β†’ (𝐴 + i) β‰  0)
10399, 102syl 17 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (𝐴 + i) β‰  0)
10473, 103logcld 26079 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (logβ€˜(𝐴 + i)) ∈ β„‚)
10561a1i 11 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (i Β· (Ο€ / 2)) ∈ β„‚)
106 picn 25969 . . . . . . . . . 10 Ο€ ∈ β„‚
1072, 106mulcli 11221 . . . . . . . . 9 (i Β· Ο€) ∈ β„‚
108107a1i 11 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (i Β· Ο€) ∈ β„‚)
10952, 83breqtrrd 5177 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ 0 ≀ (β„œβ€˜(𝐴 + i)))
110 logimul 26122 . . . . . . . . . 10 (((𝐴 + i) ∈ β„‚ ∧ (𝐴 + i) β‰  0 ∧ 0 ≀ (β„œβ€˜(𝐴 + i))) β†’ (logβ€˜(i Β· (𝐴 + i))) = ((logβ€˜(𝐴 + i)) + (i Β· (Ο€ / 2))))
11173, 103, 109, 110syl3anc 1372 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (logβ€˜(i Β· (𝐴 + i))) = ((logβ€˜(𝐴 + i)) + (i Β· (Ο€ / 2))))
112111oveq1d 7424 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((logβ€˜(i Β· (𝐴 + i))) βˆ’ (i Β· Ο€)) = (((logβ€˜(𝐴 + i)) + (i Β· (Ο€ / 2))) βˆ’ (i Β· Ο€)))
113104, 105, 108, 112assraddsubd 11628 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((logβ€˜(i Β· (𝐴 + i))) βˆ’ (i Β· Ο€)) = ((logβ€˜(𝐴 + i)) + ((i Β· (Ο€ / 2)) βˆ’ (i Β· Ο€))))
11487, 98, 1133eqtr3d 2781 . . . . . 6 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (logβ€˜(1 βˆ’ (i Β· 𝐴))) = ((logβ€˜(𝐴 + i)) + ((i Β· (Ο€ / 2)) βˆ’ (i Β· Ο€))))
115114fveq2d 6896 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(1 βˆ’ (i Β· 𝐴)))) = (β„‘β€˜((logβ€˜(𝐴 + i)) + ((i Β· (Ο€ / 2)) βˆ’ (i Β· Ο€)))))
11661, 107subcli 11536 . . . . . . 7 ((i Β· (Ο€ / 2)) βˆ’ (i Β· Ο€)) ∈ β„‚
117 imadd 15081 . . . . . . 7 (((logβ€˜(𝐴 + i)) ∈ β„‚ ∧ ((i Β· (Ο€ / 2)) βˆ’ (i Β· Ο€)) ∈ β„‚) β†’ (β„‘β€˜((logβ€˜(𝐴 + i)) + ((i Β· (Ο€ / 2)) βˆ’ (i Β· Ο€)))) = ((β„‘β€˜(logβ€˜(𝐴 + i))) + (β„‘β€˜((i Β· (Ο€ / 2)) βˆ’ (i Β· Ο€)))))
118104, 116, 117sylancl 587 . . . . . 6 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜((logβ€˜(𝐴 + i)) + ((i Β· (Ο€ / 2)) βˆ’ (i Β· Ο€)))) = ((β„‘β€˜(logβ€˜(𝐴 + i))) + (β„‘β€˜((i Β· (Ο€ / 2)) βˆ’ (i Β· Ο€)))))
119 imsub 15082 . . . . . . . . 9 (((i Β· (Ο€ / 2)) ∈ β„‚ ∧ (i Β· Ο€) ∈ β„‚) β†’ (β„‘β€˜((i Β· (Ο€ / 2)) βˆ’ (i Β· Ο€))) = ((β„‘β€˜(i Β· (Ο€ / 2))) βˆ’ (β„‘β€˜(i Β· Ο€))))
12061, 107, 119mp2an 691 . . . . . . . 8 (β„‘β€˜((i Β· (Ο€ / 2)) βˆ’ (i Β· Ο€))) = ((β„‘β€˜(i Β· (Ο€ / 2))) βˆ’ (β„‘β€˜(i Β· Ο€)))
121 reim 15056 . . . . . . . . . . 11 (Ο€ ∈ β„‚ β†’ (β„œβ€˜Ο€) = (β„‘β€˜(i Β· Ο€)))
122106, 121ax-mp 5 . . . . . . . . . 10 (β„œβ€˜Ο€) = (β„‘β€˜(i Β· Ο€))
123 pire 25968 . . . . . . . . . . 11 Ο€ ∈ ℝ
124 rere 15069 . . . . . . . . . . 11 (Ο€ ∈ ℝ β†’ (β„œβ€˜Ο€) = Ο€)
125123, 124ax-mp 5 . . . . . . . . . 10 (β„œβ€˜Ο€) = Ο€
126122, 125eqtr3i 2763 . . . . . . . . 9 (β„‘β€˜(i Β· Ο€)) = Ο€
12768, 126oveq12i 7421 . . . . . . . 8 ((β„‘β€˜(i Β· (Ο€ / 2))) βˆ’ (β„‘β€˜(i Β· Ο€))) = ((Ο€ / 2) βˆ’ Ο€)
12860negcli 11528 . . . . . . . . 9 -(Ο€ / 2) ∈ β„‚
129106, 60negsubi 11538 . . . . . . . . . 10 (Ο€ + -(Ο€ / 2)) = (Ο€ βˆ’ (Ο€ / 2))
130 pidiv2halves 25977 . . . . . . . . . . 11 ((Ο€ / 2) + (Ο€ / 2)) = Ο€
131106, 60, 60, 130subaddrii 11549 . . . . . . . . . 10 (Ο€ βˆ’ (Ο€ / 2)) = (Ο€ / 2)
132129, 131eqtri 2761 . . . . . . . . 9 (Ο€ + -(Ο€ / 2)) = (Ο€ / 2)
13360, 106, 128, 132subaddrii 11549 . . . . . . . 8 ((Ο€ / 2) βˆ’ Ο€) = -(Ο€ / 2)
134120, 127, 1333eqtri 2765 . . . . . . 7 (β„‘β€˜((i Β· (Ο€ / 2)) βˆ’ (i Β· Ο€))) = -(Ο€ / 2)
135134oveq2i 7420 . . . . . 6 ((β„‘β€˜(logβ€˜(𝐴 + i))) + (β„‘β€˜((i Β· (Ο€ / 2)) βˆ’ (i Β· Ο€)))) = ((β„‘β€˜(logβ€˜(𝐴 + i))) + -(Ο€ / 2))
136118, 135eqtrdi 2789 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜((logβ€˜(𝐴 + i)) + ((i Β· (Ο€ / 2)) βˆ’ (i Β· Ο€)))) = ((β„‘β€˜(logβ€˜(𝐴 + i))) + -(Ο€ / 2)))
137115, 136eqtrd 2773 . . . 4 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(1 βˆ’ (i Β· 𝐴)))) = ((β„‘β€˜(logβ€˜(𝐴 + i))) + -(Ο€ / 2)))
13871, 137oveq12d 7427 . . 3 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„‘β€˜(logβ€˜(1 + (i Β· 𝐴)))) βˆ’ (β„‘β€˜(logβ€˜(1 βˆ’ (i Β· 𝐴))))) = (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) + (Ο€ / 2)) βˆ’ ((β„‘β€˜(logβ€˜(𝐴 + i))) + -(Ο€ / 2))))
13958imcld 15142 . . . . . 6 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) ∈ ℝ)
140139recnd 11242 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) ∈ β„‚)
14160a1i 11 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (Ο€ / 2) ∈ β„‚)
142104imcld 15142 . . . . . 6 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(𝐴 + i))) ∈ ℝ)
143142recnd 11242 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(𝐴 + i))) ∈ β„‚)
144128a1i 11 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ -(Ο€ / 2) ∈ β„‚)
145140, 141, 143, 144addsub4d 11618 . . . 4 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) + (Ο€ / 2)) βˆ’ ((β„‘β€˜(logβ€˜(𝐴 + i))) + -(Ο€ / 2))) = (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + ((Ο€ / 2) βˆ’ -(Ο€ / 2))))
14660, 60subnegi 11539 . . . . . 6 ((Ο€ / 2) βˆ’ -(Ο€ / 2)) = ((Ο€ / 2) + (Ο€ / 2))
147146, 130eqtri 2761 . . . . 5 ((Ο€ / 2) βˆ’ -(Ο€ / 2)) = Ο€
148147oveq2i 7420 . . . 4 (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + ((Ο€ / 2) βˆ’ -(Ο€ / 2))) = (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€)
149145, 148eqtrdi 2789 . . 3 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) + (Ο€ / 2)) βˆ’ ((β„‘β€˜(logβ€˜(𝐴 + i))) + -(Ο€ / 2))) = (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€))
15017, 138, 1493eqtrd 2777 . 2 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜((logβ€˜(1 + (i Β· 𝐴))) βˆ’ (logβ€˜(1 βˆ’ (i Β· 𝐴))))) = (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€))
151139, 142resubcld 11642 . . . 4 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) ∈ ℝ)
152 readdcl 11193 . . . 4 ((((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) ∈ ℝ ∧ Ο€ ∈ ℝ) β†’ (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€) ∈ ℝ)
153151, 123, 152sylancl 587 . . 3 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€) ∈ ℝ)
154123renegcli 11521 . . . . . . 7 -Ο€ ∈ ℝ
155154recni 11228 . . . . . 6 -Ο€ ∈ β„‚
156155, 106negsubi 11538 . . . . 5 (-Ο€ + -Ο€) = (-Ο€ βˆ’ Ο€)
157154a1i 11 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ -Ο€ ∈ ℝ)
158142renegcld 11641 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ -(β„‘β€˜(logβ€˜(𝐴 + i))) ∈ ℝ)
15930, 47logimcld 26080 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (-Ο€ < (β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) ∧ (β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) ≀ Ο€))
160159simpld 496 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ -Ο€ < (β„‘β€˜(logβ€˜(𝐴 βˆ’ i))))
16173, 103logimcld 26080 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (-Ο€ < (β„‘β€˜(logβ€˜(𝐴 + i))) ∧ (β„‘β€˜(logβ€˜(𝐴 + i))) ≀ Ο€))
162161simprd 497 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(𝐴 + i))) ≀ Ο€)
163 leneg 11717 . . . . . . . . 9 (((β„‘β€˜(logβ€˜(𝐴 + i))) ∈ ℝ ∧ Ο€ ∈ ℝ) β†’ ((β„‘β€˜(logβ€˜(𝐴 + i))) ≀ Ο€ ↔ -Ο€ ≀ -(β„‘β€˜(logβ€˜(𝐴 + i)))))
164142, 123, 163sylancl 587 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„‘β€˜(logβ€˜(𝐴 + i))) ≀ Ο€ ↔ -Ο€ ≀ -(β„‘β€˜(logβ€˜(𝐴 + i)))))
165162, 164mpbid 231 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ -Ο€ ≀ -(β„‘β€˜(logβ€˜(𝐴 + i))))
166157, 157, 139, 158, 160, 165ltleaddd 11835 . . . . . 6 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (-Ο€ + -Ο€) < ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) + -(β„‘β€˜(logβ€˜(𝐴 + i)))))
167140, 143negsubd 11577 . . . . . 6 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) + -(β„‘β€˜(logβ€˜(𝐴 + i)))) = ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))))
168166, 167breqtrd 5175 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (-Ο€ + -Ο€) < ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))))
169156, 168eqbrtrrid 5185 . . . 4 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (-Ο€ βˆ’ Ο€) < ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))))
170123a1i 11 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ Ο€ ∈ ℝ)
171157, 170, 151ltsubaddd 11810 . . . 4 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((-Ο€ βˆ’ Ο€) < ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) ↔ -Ο€ < (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€)))
172169, 171mpbid 231 . . 3 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ -Ο€ < (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€))
173 0red 11217 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ 0 ∈ ℝ)
1746imcld 15142 . . . . . . . . . . . . 13 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜π΄) ∈ ℝ)
175 peano2rem 11527 . . . . . . . . . . . . 13 ((β„‘β€˜π΄) ∈ ℝ β†’ ((β„‘β€˜π΄) βˆ’ 1) ∈ ℝ)
176174, 175syl 17 . . . . . . . . . . . 12 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„‘β€˜π΄) βˆ’ 1) ∈ ℝ)
177 peano2re 11387 . . . . . . . . . . . . 13 ((β„‘β€˜π΄) ∈ ℝ β†’ ((β„‘β€˜π΄) + 1) ∈ ℝ)
178174, 177syl 17 . . . . . . . . . . . 12 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„‘β€˜π΄) + 1) ∈ ℝ)
179174ltm1d 12146 . . . . . . . . . . . 12 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„‘β€˜π΄) βˆ’ 1) < (β„‘β€˜π΄))
180174ltp1d 12144 . . . . . . . . . . . 12 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜π΄) < ((β„‘β€˜π΄) + 1))
181176, 174, 178, 179, 180lttrd 11375 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„‘β€˜π΄) βˆ’ 1) < ((β„‘β€˜π΄) + 1))
182 ltdiv1 12078 . . . . . . . . . . . 12 ((((β„‘β€˜π΄) βˆ’ 1) ∈ ℝ ∧ ((β„‘β€˜π΄) + 1) ∈ ℝ ∧ ((β„œβ€˜π΄) ∈ ℝ ∧ 0 < (β„œβ€˜π΄))) β†’ (((β„‘β€˜π΄) βˆ’ 1) < ((β„‘β€˜π΄) + 1) ↔ (((β„‘β€˜π΄) βˆ’ 1) / (β„œβ€˜π΄)) < (((β„‘β€˜π΄) + 1) / (β„œβ€˜π΄))))
183176, 178, 35, 48, 182syl112anc 1375 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (((β„‘β€˜π΄) βˆ’ 1) < ((β„‘β€˜π΄) + 1) ↔ (((β„‘β€˜π΄) βˆ’ 1) / (β„œβ€˜π΄)) < (((β„‘β€˜π΄) + 1) / (β„œβ€˜π΄))))
184181, 183mpbid 231 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (((β„‘β€˜π΄) βˆ’ 1) / (β„œβ€˜π΄)) < (((β„‘β€˜π΄) + 1) / (β„œβ€˜π΄)))
185 imsub 15082 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ i ∈ β„‚) β†’ (β„‘β€˜(𝐴 βˆ’ i)) = ((β„‘β€˜π΄) βˆ’ (β„‘β€˜i)))
1866, 2, 185sylancl 587 . . . . . . . . . . . 12 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(𝐴 βˆ’ i)) = ((β„‘β€˜π΄) βˆ’ (β„‘β€˜i)))
187 imi 15104 . . . . . . . . . . . . 13 (β„‘β€˜i) = 1
188187oveq2i 7420 . . . . . . . . . . . 12 ((β„‘β€˜π΄) βˆ’ (β„‘β€˜i)) = ((β„‘β€˜π΄) βˆ’ 1)
189186, 188eqtrdi 2789 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(𝐴 βˆ’ i)) = ((β„‘β€˜π΄) βˆ’ 1))
190189, 39oveq12d 7427 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„‘β€˜(𝐴 βˆ’ i)) / (β„œβ€˜(𝐴 βˆ’ i))) = (((β„‘β€˜π΄) βˆ’ 1) / (β„œβ€˜π΄)))
191 imadd 15081 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ i ∈ β„‚) β†’ (β„‘β€˜(𝐴 + i)) = ((β„‘β€˜π΄) + (β„‘β€˜i)))
1926, 2, 191sylancl 587 . . . . . . . . . . . 12 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(𝐴 + i)) = ((β„‘β€˜π΄) + (β„‘β€˜i)))
193187oveq2i 7420 . . . . . . . . . . . 12 ((β„‘β€˜π΄) + (β„‘β€˜i)) = ((β„‘β€˜π΄) + 1)
194192, 193eqtrdi 2789 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(𝐴 + i)) = ((β„‘β€˜π΄) + 1))
195194, 83oveq12d 7427 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„‘β€˜(𝐴 + i)) / (β„œβ€˜(𝐴 + i))) = (((β„‘β€˜π΄) + 1) / (β„œβ€˜π΄)))
196184, 190, 1953brtr4d 5181 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„‘β€˜(𝐴 βˆ’ i)) / (β„œβ€˜(𝐴 βˆ’ i))) < ((β„‘β€˜(𝐴 + i)) / (β„œβ€˜(𝐴 + i))))
197 tanarg 26127 . . . . . . . . . 10 (((𝐴 βˆ’ i) ∈ β„‚ ∧ (β„œβ€˜(𝐴 βˆ’ i)) β‰  0) β†’ (tanβ€˜(β„‘β€˜(logβ€˜(𝐴 βˆ’ i)))) = ((β„‘β€˜(𝐴 βˆ’ i)) / (β„œβ€˜(𝐴 βˆ’ i))))
19830, 42, 197syl2anc 585 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (tanβ€˜(β„‘β€˜(logβ€˜(𝐴 βˆ’ i)))) = ((β„‘β€˜(𝐴 βˆ’ i)) / (β„œβ€˜(𝐴 βˆ’ i))))
199 tanarg 26127 . . . . . . . . . 10 (((𝐴 + i) ∈ β„‚ ∧ (β„œβ€˜(𝐴 + i)) β‰  0) β†’ (tanβ€˜(β„‘β€˜(logβ€˜(𝐴 + i)))) = ((β„‘β€˜(𝐴 + i)) / (β„œβ€˜(𝐴 + i))))
20073, 99, 199syl2anc 585 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (tanβ€˜(β„‘β€˜(logβ€˜(𝐴 + i)))) = ((β„‘β€˜(𝐴 + i)) / (β„œβ€˜(𝐴 + i))))
201196, 198, 2003brtr4d 5181 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (tanβ€˜(β„‘β€˜(logβ€˜(𝐴 βˆ’ i)))) < (tanβ€˜(β„‘β€˜(logβ€˜(𝐴 + i)))))
20248, 39breqtrrd 5177 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ 0 < (β„œβ€˜(𝐴 βˆ’ i)))
203 argregt0 26118 . . . . . . . . . 10 (((𝐴 βˆ’ i) ∈ β„‚ ∧ 0 < (β„œβ€˜(𝐴 βˆ’ i))) β†’ (β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) ∈ (-(Ο€ / 2)(,)(Ο€ / 2)))
20430, 202, 203syl2anc 585 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) ∈ (-(Ο€ / 2)(,)(Ο€ / 2)))
20548, 83breqtrrd 5177 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ 0 < (β„œβ€˜(𝐴 + i)))
206 argregt0 26118 . . . . . . . . . 10 (((𝐴 + i) ∈ β„‚ ∧ 0 < (β„œβ€˜(𝐴 + i))) β†’ (β„‘β€˜(logβ€˜(𝐴 + i))) ∈ (-(Ο€ / 2)(,)(Ο€ / 2)))
20773, 205, 206syl2anc 585 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(𝐴 + i))) ∈ (-(Ο€ / 2)(,)(Ο€ / 2)))
208 tanord 26047 . . . . . . . . 9 (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) ∈ (-(Ο€ / 2)(,)(Ο€ / 2)) ∧ (β„‘β€˜(logβ€˜(𝐴 + i))) ∈ (-(Ο€ / 2)(,)(Ο€ / 2))) β†’ ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) < (β„‘β€˜(logβ€˜(𝐴 + i))) ↔ (tanβ€˜(β„‘β€˜(logβ€˜(𝐴 βˆ’ i)))) < (tanβ€˜(β„‘β€˜(logβ€˜(𝐴 + i))))))
209204, 207, 208syl2anc 585 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) < (β„‘β€˜(logβ€˜(𝐴 + i))) ↔ (tanβ€˜(β„‘β€˜(logβ€˜(𝐴 βˆ’ i)))) < (tanβ€˜(β„‘β€˜(logβ€˜(𝐴 + i))))))
210201, 209mpbird 257 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) < (β„‘β€˜(logβ€˜(𝐴 + i))))
211143addlidd 11415 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (0 + (β„‘β€˜(logβ€˜(𝐴 + i)))) = (β„‘β€˜(logβ€˜(𝐴 + i))))
212210, 211breqtrrd 5177 . . . . . 6 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) < (0 + (β„‘β€˜(logβ€˜(𝐴 + i)))))
213139, 142, 173ltsubaddd 11810 . . . . . 6 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) < 0 ↔ (β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) < (0 + (β„‘β€˜(logβ€˜(𝐴 + i))))))
214212, 213mpbird 257 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) < 0)
215151, 173, 170, 214ltadd1dd 11825 . . . 4 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€) < (0 + Ο€))
216106addlidi 11402 . . . 4 (0 + Ο€) = Ο€
217215, 216breqtrdi 5190 . . 3 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€) < Ο€)
218154rexri 11272 . . . 4 -Ο€ ∈ ℝ*
219123rexri 11272 . . . 4 Ο€ ∈ ℝ*
220 elioo2 13365 . . . 4 ((-Ο€ ∈ ℝ* ∧ Ο€ ∈ ℝ*) β†’ ((((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€) ∈ (-Ο€(,)Ο€) ↔ ((((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€) ∈ ℝ ∧ -Ο€ < (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€) ∧ (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€) < Ο€)))
221218, 219, 220mp2an 691 . . 3 ((((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€) ∈ (-Ο€(,)Ο€) ↔ ((((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€) ∈ ℝ ∧ -Ο€ < (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€) ∧ (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€) < Ο€))
222153, 172, 217, 221syl3anbrc 1344 . 2 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€) ∈ (-Ο€(,)Ο€))
223150, 222eqeltrd 2834 1 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜((logβ€˜(1 + (i Β· 𝐴))) βˆ’ (logβ€˜(1 βˆ’ (i Β· 𝐴))))) ∈ (-Ο€(,)Ο€))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941   class class class wbr 5149  dom cdm 5677  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  β„*cxr 11247   < clt 11248   ≀ cle 11249   βˆ’ cmin 11444  -cneg 11445   / cdiv 11871  2c2 12267  (,)cioo 13324  β„œcre 15044  β„‘cim 15045  tanctan 16009  Ο€cpi 16010  logclog 26063  arctancatan 26369
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-tan 16015  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-atan 26372
This theorem is referenced by:  atanlogsub  26421  atanbndlem  26430
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