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Theorem atanlogsublem 26653
Description: Lemma for atanlogsub 26654. (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 11171 . . . . . 6 1 ∈ β„‚
2 ax-icn 11172 . . . . . . 7 i ∈ β„‚
3 simpl 482 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ 𝐴 ∈ dom arctan)
4 atandm2 26615 . . . . . . . . 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 1141 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ 𝐴 ∈ β„‚)
7 mulcl 11197 . . . . . . 7 ((i ∈ β„‚ ∧ 𝐴 ∈ β„‚) β†’ (i Β· 𝐴) ∈ β„‚)
82, 6, 7sylancr 586 . . . . . 6 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (i Β· 𝐴) ∈ β„‚)
9 addcl 11195 . . . . . 6 ((1 ∈ β„‚ ∧ (i Β· 𝐴) ∈ β„‚) β†’ (1 + (i Β· 𝐴)) ∈ β„‚)
101, 8, 9sylancr 586 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (1 + (i Β· 𝐴)) ∈ β„‚)
115simp3d 1143 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (1 + (i Β· 𝐴)) β‰  0)
1210, 11logcld 26312 . . . 4 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (logβ€˜(1 + (i Β· 𝐴))) ∈ β„‚)
13 subcl 11464 . . . . . 6 ((1 ∈ β„‚ ∧ (i Β· 𝐴) ∈ β„‚) β†’ (1 βˆ’ (i Β· 𝐴)) ∈ β„‚)
141, 8, 13sylancr 586 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (1 βˆ’ (i Β· 𝐴)) ∈ β„‚)
155simp2d 1142 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (1 βˆ’ (i Β· 𝐴)) β‰  0)
1614, 15logcld 26312 . . . 4 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (logβ€˜(1 βˆ’ (i Β· 𝐴))) ∈ β„‚)
1712, 16imsubd 15169 . . 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 11675 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (i Β· (𝐴 βˆ’ i)) = ((i Β· 𝐴) βˆ’ (i Β· i)))
20 ixi 11848 . . . . . . . . . . 11 (i Β· i) = -1
2120oveq2i 7423 . . . . . . . . . 10 ((i Β· 𝐴) βˆ’ (i Β· i)) = ((i Β· 𝐴) βˆ’ -1)
22 subneg 11514 . . . . . . . . . . 11 (((i Β· 𝐴) ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((i Β· 𝐴) βˆ’ -1) = ((i Β· 𝐴) + 1))
238, 1, 22sylancl 585 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((i Β· 𝐴) βˆ’ -1) = ((i Β· 𝐴) + 1))
2421, 23eqtrid 2783 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((i Β· 𝐴) βˆ’ (i Β· i)) = ((i Β· 𝐴) + 1))
25 addcom 11405 . . . . . . . . . 10 (((i Β· 𝐴) ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((i Β· 𝐴) + 1) = (1 + (i Β· 𝐴)))
268, 1, 25sylancl 585 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((i Β· 𝐴) + 1) = (1 + (i Β· 𝐴)))
2719, 24, 263eqtrd 2775 . . . . . . . 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 11464 . . . . . . . . 9 ((𝐴 ∈ β„‚ ∧ i ∈ β„‚) β†’ (𝐴 βˆ’ i) ∈ β„‚)
306, 2, 29sylancl 585 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (𝐴 βˆ’ i) ∈ β„‚)
31 resub 15079 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ i ∈ β„‚) β†’ (β„œβ€˜(𝐴 βˆ’ i)) = ((β„œβ€˜π΄) βˆ’ (β„œβ€˜i)))
326, 2, 31sylancl 585 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜(𝐴 βˆ’ i)) = ((β„œβ€˜π΄) βˆ’ (β„œβ€˜i)))
33 rei 15108 . . . . . . . . . . . . 13 (β„œβ€˜i) = 0
3433oveq2i 7423 . . . . . . . . . . . 12 ((β„œβ€˜π΄) βˆ’ (β„œβ€˜i)) = ((β„œβ€˜π΄) βˆ’ 0)
356recld 15146 . . . . . . . . . . . . . 14 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜π΄) ∈ ℝ)
3635recnd 11247 . . . . . . . . . . . . 13 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜π΄) ∈ β„‚)
3736subid1d 11565 . . . . . . . . . . . 12 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„œβ€˜π΄) βˆ’ 0) = (β„œβ€˜π΄))
3834, 37eqtrid 2783 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„œβ€˜π΄) βˆ’ (β„œβ€˜i)) = (β„œβ€˜π΄))
3932, 38eqtrd 2771 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜(𝐴 βˆ’ i)) = (β„œβ€˜π΄))
40 gt0ne0 11684 . . . . . . . . . . 11 (((β„œβ€˜π΄) ∈ ℝ ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜π΄) β‰  0)
4135, 40sylancom 587 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜π΄) β‰  0)
4239, 41eqnetrd 3007 . . . . . . . . 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 15104 . . . . . . . . . . 11 (β„œβ€˜0) = 0
4543, 44eqtrdi 2787 . . . . . . . . . 10 ((𝐴 βˆ’ i) = 0 β†’ (β„œβ€˜(𝐴 βˆ’ i)) = 0)
4645necon3i 2972 . . . . . . . . 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 484 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ 0 < (β„œβ€˜π΄))
49 0re 11221 . . . . . . . . . . 11 0 ∈ ℝ
50 ltle 11307 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ (β„œβ€˜π΄) ∈ ℝ) β†’ (0 < (β„œβ€˜π΄) β†’ 0 ≀ (β„œβ€˜π΄)))
5149, 35, 50sylancr 586 . . . . . . . . . 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 26355 . . . . . . . 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 1370 . . . . . . 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 2773 . . . . . 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 26312 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (logβ€˜(𝐴 βˆ’ i)) ∈ β„‚)
59 halfpire 26207 . . . . . . . . 9 (Ο€ / 2) ∈ ℝ
6059recni 11233 . . . . . . . 8 (Ο€ / 2) ∈ β„‚
612, 60mulcli 11226 . . . . . . 7 (i Β· (Ο€ / 2)) ∈ β„‚
62 imadd 15086 . . . . . . 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 585 . . . . . 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 15061 . . . . . . . . 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 15074 . . . . . . . . 9 ((Ο€ / 2) ∈ ℝ β†’ (β„œβ€˜(Ο€ / 2)) = (Ο€ / 2))
6759, 66ax-mp 5 . . . . . . . 8 (β„œβ€˜(Ο€ / 2)) = (Ο€ / 2)
6865, 67eqtr3i 2761 . . . . . . 7 (β„‘β€˜(i Β· (Ο€ / 2))) = (Ο€ / 2)
6968oveq2i 7423 . . . . . 6 ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) + (β„‘β€˜(i Β· (Ο€ / 2)))) = ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) + (Ο€ / 2))
7063, 69eqtrdi 2787 . . . . 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 2771 . . . 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 11195 . . . . . . . . . 10 ((𝐴 ∈ β„‚ ∧ i ∈ β„‚) β†’ (𝐴 + i) ∈ β„‚)
736, 2, 72sylancl 585 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (𝐴 + i) ∈ β„‚)
74 mulcl 11197 . . . . . . . . 9 ((i ∈ β„‚ ∧ (𝐴 + i) ∈ β„‚) β†’ (i Β· (𝐴 + i)) ∈ β„‚)
752, 73, 74sylancr 586 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (i Β· (𝐴 + i)) ∈ β„‚)
76 reim 15061 . . . . . . . . . . 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 15078 . . . . . . . . . . . 12 ((𝐴 ∈ β„‚ ∧ i ∈ β„‚) β†’ (β„œβ€˜(𝐴 + i)) = ((β„œβ€˜π΄) + (β„œβ€˜i)))
796, 2, 78sylancl 585 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜(𝐴 + i)) = ((β„œβ€˜π΄) + (β„œβ€˜i)))
8033oveq2i 7423 . . . . . . . . . . . 12 ((β„œβ€˜π΄) + (β„œβ€˜i)) = ((β„œβ€˜π΄) + 0)
8136addridd 11419 . . . . . . . . . . . 12 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„œβ€˜π΄) + 0) = (β„œβ€˜π΄))
8280, 81eqtrid 2783 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„œβ€˜π΄) + (β„œβ€˜i)) = (β„œβ€˜π΄))
8379, 82eqtrd 2771 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„œβ€˜(𝐴 + i)) = (β„œβ€˜π΄))
8477, 83eqtr3d 2773 . . . . . . . . 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 26356 . . . . . . . 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 583 . . . . . . 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 11243 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (i Β· (𝐴 + i)) = ((i Β· 𝐴) + (i Β· i)))
8920oveq2i 7423 . . . . . . . . . . . 12 ((i Β· 𝐴) + (i Β· i)) = ((i Β· 𝐴) + -1)
90 negsub 11513 . . . . . . . . . . . . 13 (((i Β· 𝐴) ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((i Β· 𝐴) + -1) = ((i Β· 𝐴) βˆ’ 1))
918, 1, 90sylancl 585 . . . . . . . . . . . 12 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((i Β· 𝐴) + -1) = ((i Β· 𝐴) βˆ’ 1))
9289, 91eqtrid 2783 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((i Β· 𝐴) + (i Β· i)) = ((i Β· 𝐴) βˆ’ 1))
9388, 92eqtrd 2771 . . . . . . . . . 10 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (i Β· (𝐴 + i)) = ((i Β· 𝐴) βˆ’ 1))
9493negeqd 11459 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ -(i Β· (𝐴 + i)) = -((i Β· 𝐴) βˆ’ 1))
95 negsubdi2 11524 . . . . . . . . . 10 (((i Β· 𝐴) ∈ β„‚ ∧ 1 ∈ β„‚) β†’ -((i Β· 𝐴) βˆ’ 1) = (1 βˆ’ (i Β· 𝐴)))
968, 1, 95sylancl 585 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ -((i Β· 𝐴) βˆ’ 1) = (1 βˆ’ (i Β· 𝐴)))
9794, 96eqtrd 2771 . . . . . . . 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 3007 . . . . . . . . . 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 2787 . . . . . . . . . . 11 ((𝐴 + i) = 0 β†’ (β„œβ€˜(𝐴 + i)) = 0)
102101necon3i 2972 . . . . . . . . . 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 26312 . . . . . . . 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 26202 . . . . . . . . . 10 Ο€ ∈ β„‚
1072, 106mulcli 11226 . . . . . . . . 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 26355 . . . . . . . . . 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 1370 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (logβ€˜(i Β· (𝐴 + i))) = ((logβ€˜(𝐴 + i)) + (i Β· (Ο€ / 2))))
112111oveq1d 7427 . . . . . . . 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 11633 . . . . . . 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 2779 . . . . . 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 11541 . . . . . . 7 ((i Β· (Ο€ / 2)) βˆ’ (i Β· Ο€)) ∈ β„‚
117 imadd 15086 . . . . . . 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 585 . . . . . 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 15087 . . . . . . . . 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 689 . . . . . . . 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 15061 . . . . . . . . . . 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 26201 . . . . . . . . . . 11 Ο€ ∈ ℝ
124 rere 15074 . . . . . . . . . . 11 (Ο€ ∈ ℝ β†’ (β„œβ€˜Ο€) = Ο€)
125123, 124ax-mp 5 . . . . . . . . . 10 (β„œβ€˜Ο€) = Ο€
126122, 125eqtr3i 2761 . . . . . . . . 9 (β„‘β€˜(i Β· Ο€)) = Ο€
12768, 126oveq12i 7424 . . . . . . . 8 ((β„‘β€˜(i Β· (Ο€ / 2))) βˆ’ (β„‘β€˜(i Β· Ο€))) = ((Ο€ / 2) βˆ’ Ο€)
12860negcli 11533 . . . . . . . . 9 -(Ο€ / 2) ∈ β„‚
129106, 60negsubi 11543 . . . . . . . . . 10 (Ο€ + -(Ο€ / 2)) = (Ο€ βˆ’ (Ο€ / 2))
130 pidiv2halves 26210 . . . . . . . . . . 11 ((Ο€ / 2) + (Ο€ / 2)) = Ο€
131106, 60, 60, 130subaddrii 11554 . . . . . . . . . 10 (Ο€ βˆ’ (Ο€ / 2)) = (Ο€ / 2)
132129, 131eqtri 2759 . . . . . . . . 9 (Ο€ + -(Ο€ / 2)) = (Ο€ / 2)
13360, 106, 128, 132subaddrii 11554 . . . . . . . 8 ((Ο€ / 2) βˆ’ Ο€) = -(Ο€ / 2)
134120, 127, 1333eqtri 2763 . . . . . . 7 (β„‘β€˜((i Β· (Ο€ / 2)) βˆ’ (i Β· Ο€))) = -(Ο€ / 2)
135134oveq2i 7423 . . . . . 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 2787 . . . . 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 2771 . . . 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 7430 . . 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 15147 . . . . . 6 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) ∈ ℝ)
140139recnd 11247 . . . . 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 15147 . . . . . 6 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(𝐴 + i))) ∈ ℝ)
143142recnd 11247 . . . . 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 11623 . . . 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 11544 . . . . . 6 ((Ο€ / 2) βˆ’ -(Ο€ / 2)) = ((Ο€ / 2) + (Ο€ / 2))
147146, 130eqtri 2759 . . . . 5 ((Ο€ / 2) βˆ’ -(Ο€ / 2)) = Ο€
148147oveq2i 7423 . . . 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 2787 . . 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 2775 . 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 11647 . . . 4 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) ∈ ℝ)
152 readdcl 11196 . . . 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 585 . . 3 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) βˆ’ (β„‘β€˜(logβ€˜(𝐴 + i)))) + Ο€) ∈ ℝ)
154123renegcli 11526 . . . . . . 7 -Ο€ ∈ ℝ
155154recni 11233 . . . . . 6 -Ο€ ∈ β„‚
156155, 106negsubi 11543 . . . . 5 (-Ο€ + -Ο€) = (-Ο€ βˆ’ Ο€)
157154a1i 11 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ -Ο€ ∈ ℝ)
158142renegcld 11646 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ -(β„‘β€˜(logβ€˜(𝐴 + i))) ∈ ℝ)
15930, 47logimcld 26313 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (-Ο€ < (β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) ∧ (β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) ≀ Ο€))
160159simpld 494 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ -Ο€ < (β„‘β€˜(logβ€˜(𝐴 βˆ’ i))))
16173, 103logimcld 26313 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (-Ο€ < (β„‘β€˜(logβ€˜(𝐴 + i))) ∧ (β„‘β€˜(logβ€˜(𝐴 + i))) ≀ Ο€))
162161simprd 495 . . . . . . . 8 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(𝐴 + i))) ≀ Ο€)
163 leneg 11722 . . . . . . . . 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 585 . . . . . . . 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 11840 . . . . . 6 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (-Ο€ + -Ο€) < ((β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) + -(β„‘β€˜(logβ€˜(𝐴 + i)))))
167140, 143negsubd 11582 . . . . . 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 11815 . . . 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 11222 . . . . 5 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ 0 ∈ ℝ)
1746imcld 15147 . . . . . . . . . . . . 13 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜π΄) ∈ ℝ)
175 peano2rem 11532 . . . . . . . . . . . . 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 11392 . . . . . . . . . . . . 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 12151 . . . . . . . . . . . 12 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„‘β€˜π΄) βˆ’ 1) < (β„‘β€˜π΄))
180174ltp1d 12149 . . . . . . . . . . . 12 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜π΄) < ((β„‘β€˜π΄) + 1))
181176, 174, 178, 179, 180lttrd 11380 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ ((β„‘β€˜π΄) βˆ’ 1) < ((β„‘β€˜π΄) + 1))
182 ltdiv1 12083 . . . . . . . . . . . 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 1373 . . . . . . . . . . 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 15087 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ i ∈ β„‚) β†’ (β„‘β€˜(𝐴 βˆ’ i)) = ((β„‘β€˜π΄) βˆ’ (β„‘β€˜i)))
1866, 2, 185sylancl 585 . . . . . . . . . . . 12 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(𝐴 βˆ’ i)) = ((β„‘β€˜π΄) βˆ’ (β„‘β€˜i)))
187 imi 15109 . . . . . . . . . . . . 13 (β„‘β€˜i) = 1
188187oveq2i 7423 . . . . . . . . . . . 12 ((β„‘β€˜π΄) βˆ’ (β„‘β€˜i)) = ((β„‘β€˜π΄) βˆ’ 1)
189186, 188eqtrdi 2787 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(𝐴 βˆ’ i)) = ((β„‘β€˜π΄) βˆ’ 1))
190189, 39oveq12d 7430 . . . . . . . . . 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 15086 . . . . . . . . . . . . 13 ((𝐴 ∈ β„‚ ∧ i ∈ β„‚) β†’ (β„‘β€˜(𝐴 + i)) = ((β„‘β€˜π΄) + (β„‘β€˜i)))
1926, 2, 191sylancl 585 . . . . . . . . . . . 12 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(𝐴 + i)) = ((β„‘β€˜π΄) + (β„‘β€˜i)))
193187oveq2i 7423 . . . . . . . . . . . 12 ((β„‘β€˜π΄) + (β„‘β€˜i)) = ((β„‘β€˜π΄) + 1)
194192, 193eqtrdi 2787 . . . . . . . . . . 11 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(𝐴 + i)) = ((β„‘β€˜π΄) + 1))
195194, 83oveq12d 7430 . . . . . . . . . 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 26360 . . . . . . . . . 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 583 . . . . . . . . 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 26360 . . . . . . . . . 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 583 . . . . . . . . 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 26351 . . . . . . . . . 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 583 . . . . . . . . 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 26351 . . . . . . . . . 10 (((𝐴 + i) ∈ β„‚ ∧ 0 < (β„œβ€˜(𝐴 + i))) β†’ (β„‘β€˜(logβ€˜(𝐴 + i))) ∈ (-(Ο€ / 2)(,)(Ο€ / 2)))
20773, 205, 206syl2anc 583 . . . . . . . . 9 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(𝐴 + i))) ∈ (-(Ο€ / 2)(,)(Ο€ / 2)))
208 tanord 26280 . . . . . . . . 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 583 . . . . . . . 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 256 . . . . . . 7 ((𝐴 ∈ dom arctan ∧ 0 < (β„œβ€˜π΄)) β†’ (β„‘β€˜(logβ€˜(𝐴 βˆ’ i))) < (β„‘β€˜(logβ€˜(𝐴 + i))))
211143addlidd 11420 . . . . . . 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 11815 . . . . . 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 256 . . . . 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 11830 . . . 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 11407 . . . 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 11277 . . . 4 -Ο€ ∈ ℝ*
219123rexri 11277 . . . 4 Ο€ ∈ ℝ*
220 elioo2 13370 . . . 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 689 . . 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 1342 . 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 2832 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939   class class class wbr 5149  dom cdm 5677  β€˜cfv 6544  (class class class)co 7412  β„‚cc 11111  β„cr 11112  0cc0 11113  1c1 11114  ici 11115   + caddc 11116   Β· cmul 11118  β„*cxr 11252   < clt 11253   ≀ cle 11254   βˆ’ cmin 11449  -cneg 11450   / cdiv 11876  2c2 12272  (,)cioo 13329  β„œcre 15049  β„‘cim 15050  tanctan 16014  Ο€cpi 16015  logclog 26296  arctancatan 26602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-inf2 9639  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190  ax-pre-sup 11191  ax-addf 11192  ax-mulf 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7673  df-om 7859  df-1st 7978  df-2nd 7979  df-supp 8150  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-2o 8470  df-er 8706  df-map 8825  df-pm 8826  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fsupp 9365  df-fi 9409  df-sup 9440  df-inf 9441  df-oi 9508  df-card 9937  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12478  df-z 12564  df-dec 12683  df-uz 12828  df-q 12938  df-rp 12980  df-xneg 13097  df-xadd 13098  df-xmul 13099  df-ioo 13333  df-ioc 13334  df-ico 13335  df-icc 13336  df-fz 13490  df-fzo 13633  df-fl 13762  df-mod 13840  df-seq 13972  df-exp 14033  df-fac 14239  df-bc 14268  df-hash 14296  df-shft 15019  df-cj 15051  df-re 15052  df-im 15053  df-sqrt 15187  df-abs 15188  df-limsup 15420  df-clim 15437  df-rlim 15438  df-sum 15638  df-ef 16016  df-sin 16018  df-cos 16019  df-tan 16020  df-pi 16021  df-struct 17085  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-mulr 17216  df-starv 17217  df-sca 17218  df-vsca 17219  df-ip 17220  df-tset 17221  df-ple 17222  df-ds 17224  df-unif 17225  df-hom 17226  df-cco 17227  df-rest 17373  df-topn 17374  df-0g 17392  df-gsum 17393  df-topgen 17394  df-pt 17395  df-prds 17398  df-xrs 17453  df-qtop 17458  df-imas 17459  df-xps 17461  df-mre 17535  df-mrc 17536  df-acs 17538  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-submnd 18707  df-mulg 18988  df-cntz 19223  df-cmn 19692  df-psmet 21137  df-xmet 21138  df-met 21139  df-bl 21140  df-mopn 21141  df-fbas 21142  df-fg 21143  df-cnfld 21146  df-top 22617  df-topon 22634  df-topsp 22656  df-bases 22670  df-cld 22744  df-ntr 22745  df-cls 22746  df-nei 22823  df-lp 22861  df-perf 22862  df-cn 22952  df-cnp 22953  df-haus 23040  df-tx 23287  df-hmeo 23480  df-fil 23571  df-fm 23663  df-flim 23664  df-flf 23665  df-xms 24047  df-ms 24048  df-tms 24049  df-cncf 24619  df-limc 25616  df-dv 25617  df-log 26298  df-atan 26605
This theorem is referenced by:  atanlogsub  26654  atanbndlem  26663
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