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Theorem atanbndlem 26968

Description: Lemma for atanbnd 26969. (Contributed by Mario Carneiro, 5-Apr-2015.)

**Assertion**
Ref	Expression
atanbndlem	⊢ (𝐴 ∈ ℝ⁺ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))

**Proof of Theorem atanbndlem**
Step	Hyp	Ref	Expression
1		rpre 13043	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		atanrecl 26954	. . 3 ⊢ (𝐴 ∈ ℝ → (arctan‘𝐴) ∈ ℝ)
3	1, 2	syl 17	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (arctan‘𝐴) ∈ ℝ)
4		picn 26501	. . . 4 ⊢ π ∈ ℂ
5		2cn 12341	. . . 4 ⊢ 2 ∈ ℂ
6		2ne0 12370	. . . 4 ⊢ 2 ≠ 0
7		divneg 11959	. . . 4 ⊢ ((π ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(π / 2) = (-π / 2))
8	4, 5, 6, 7	mp3an 1463	. . 3 ⊢ -(π / 2) = (-π / 2)
9		ax-1cn 11213	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
10		ax-icn 11214	. . . . . . . . . . . . 13 ⊢ i ∈ ℂ
11	1	recnd 11289	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
12		mulcl 11239	. . . . . . . . . . . . 13 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
13	10, 11, 12	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (i · 𝐴) ∈ ℂ)
14		addcl 11237	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
15	9, 13, 14	sylancr 587	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (i · 𝐴)) ∈ ℂ)
16		atanre 26928	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ dom arctan)
17	1, 16	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ dom arctan)
18		atandm2 26920	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
19	17, 18	sylib 218	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
20	19	simp3d 1145	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (i · 𝐴)) ≠ 0)
21	15, 20	logcld 26612	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(1 + (i · 𝐴))) ∈ ℂ)
22		subcl 11507	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
23	9, 13, 22	sylancr 587	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1 − (i · 𝐴)) ∈ ℂ)
24	19	simp2d 1144	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1 − (i · 𝐴)) ≠ 0)
25	23, 24	logcld 26612	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(1 − (i · 𝐴))) ∈ ℂ)
26	21, 25	subcld 11620	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ℂ)
27		imre 15147	. . . . . . . . 9 ⊢ (((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ℂ → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) = (ℜ‘(-i · ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))))))
28	26, 27	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) = (ℜ‘(-i · ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))))))
29		atanval 26927	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
30	17, 29	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
31	30	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (2 · ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
32	10, 5, 6	divcan2i 12010	. . . . . . . . . . . . . 14 ⊢ (2 · (i / 2)) = i
33	32	oveq1i 7441	. . . . . . . . . . . . 13 ⊢ ((2 · (i / 2)) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = (i · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
34		2re 12340	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
35	34	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ⁺ → 2 ∈ ℝ)
36	35	recnd 11289	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → 2 ∈ ℂ)
37		halfcl 12491	. . . . . . . . . . . . . . 15 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
38	10, 37	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → (i / 2) ∈ ℂ)
39	25, 21	subcld 11620	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) ∈ ℂ)
40	36, 38, 39	mulassd 11284	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → ((2 · (i / 2)) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = (2 · ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
41	33, 40	eqtr3id 2791	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (i · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = (2 · ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
42	31, 41	eqtr4d 2780	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (i · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
43	21, 25	negsubdi2d 11636	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) = ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
44	43	oveq2d 7447	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (i · -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) = (i · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
45	42, 44	eqtr4d 2780	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (i · -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
46		mulneg12 11701	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ℂ) → (-i · ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) = (i · -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
47	10, 26, 46	sylancr 587	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (-i · ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) = (i · -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
48	45, 47	eqtr4d 2780	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (-i · ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
49	48	fveq2d 6910	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (ℜ‘(2 · (arctan‘𝐴))) = (ℜ‘(-i · ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))))))
50		remulcl 11240	. . . . . . . . . 10 ⊢ ((2 ∈ ℝ ∧ (arctan‘𝐴) ∈ ℝ) → (2 · (arctan‘𝐴)) ∈ ℝ)
51	34, 3, 50	sylancr 587	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) ∈ ℝ)
52	51	rered 15263	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (ℜ‘(2 · (arctan‘𝐴))) = (2 · (arctan‘𝐴)))
53	28, 49, 52	3eqtr2rd 2784	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
54		rpgt0 13047	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
55	1	rered 15263	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (ℜ‘𝐴) = 𝐴)
56	54, 55	breqtrrd 5171	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 0 < (ℜ‘𝐴))
57		atanlogsublem 26958	. . . . . . . 8 ⊢ ((𝐴 ∈ dom arctan ∧ 0 < (ℜ‘𝐴)) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
58	17, 56, 57	syl2anc 584	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
59	53, 58	eqeltrd 2841	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) ∈ (-π(,)π))
60		eliooord 13446	. . . . . 6 ⊢ ((2 · (arctan‘𝐴)) ∈ (-π(,)π) → (-π < (2 · (arctan‘𝐴)) ∧ (2 · (arctan‘𝐴)) < π))
61	59, 60	syl 17	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (-π < (2 · (arctan‘𝐴)) ∧ (2 · (arctan‘𝐴)) < π))
62	61	simpld 494	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → -π < (2 · (arctan‘𝐴)))
63		pire 26500	. . . . . . 7 ⊢ π ∈ ℝ
64	63	renegcli 11570	. . . . . 6 ⊢ -π ∈ ℝ
65	64	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → -π ∈ ℝ)
66		2pos 12369	. . . . . 6 ⊢ 0 < 2
67	66	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 2)
68		ltdivmul 12143	. . . . 5 ⊢ ((-π ∈ ℝ ∧ (arctan‘𝐴) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((-π / 2) < (arctan‘𝐴) ↔ -π < (2 · (arctan‘𝐴))))
69	65, 3, 35, 67, 68	syl112anc 1376	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((-π / 2) < (arctan‘𝐴) ↔ -π < (2 · (arctan‘𝐴))))
70	62, 69	mpbird 257	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (-π / 2) < (arctan‘𝐴))
71	8, 70	eqbrtrid 5178	. 2 ⊢ (𝐴 ∈ ℝ⁺ → -(π / 2) < (arctan‘𝐴))
72	61	simprd 495	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) < π)
73	63	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → π ∈ ℝ)
74		ltmuldiv2 12142	. . . 4 ⊢ (((arctan‘𝐴) ∈ ℝ ∧ π ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((2 · (arctan‘𝐴)) < π ↔ (arctan‘𝐴) < (π / 2)))
75	3, 73, 35, 67, 74	syl112anc 1376	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((2 · (arctan‘𝐴)) < π ↔ (arctan‘𝐴) < (π / 2)))
76	72, 75	mpbid 232	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (arctan‘𝐴) < (π / 2))
77		halfpire 26506	. . . . 5 ⊢ (π / 2) ∈ ℝ
78	77	renegcli 11570	. . . 4 ⊢ -(π / 2) ∈ ℝ
79	78	rexri 11319	. . 3 ⊢ -(π / 2) ∈ ℝ^*
80	77	rexri 11319	. . 3 ⊢ (π / 2) ∈ ℝ^*
81		elioo2 13428	. . 3 ⊢ ((-(π / 2) ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → ((arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)) ↔ ((arctan‘𝐴) ∈ ℝ ∧ -(π / 2) < (arctan‘𝐴) ∧ (arctan‘𝐴) < (π / 2))))
82	79, 80, 81	mp2an 692	. 2 ⊢ ((arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)) ↔ ((arctan‘𝐴) ∈ ℝ ∧ -(π / 2) < (arctan‘𝐴) ∧ (arctan‘𝐴) < (π / 2)))
83	3, 71, 76, 82	syl3anbrc 1344	1 ⊢ (𝐴 ∈ ℝ⁺ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 ici 11157 + caddc 11158 · cmul 11160 ℝ^*cxr 11294 < clt 11295 − cmin 11492 -cneg 11493 / cdiv 11920 2c2 12321 ℝ⁺crp 13034 (,)cioo 13387 ℜcre 15136 ℑcim 15137 πcpi 16102 logclog 26596 arctancatan 26907

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-tan 16107 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 df-atan 26910

This theorem is referenced by: atanbnd 26969

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