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Theorem atanbndlem 27071

Description: Lemma for atanbnd 27072. (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 13026	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		atanrecl 27057	. . 3 ⊢ (𝐴 ∈ ℝ → (arctan‘𝐴) ∈ ℝ)
3	1, 2	syl 18	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (arctan‘𝐴) ∈ ℝ)
4		picn 26602	. . . 4 ⊢ π ∈ ℂ
5		2cn 12317	. . . 4 ⊢ 2 ∈ ℂ
6		2ne0 12348	. . . 4 ⊢ 2 ≠ 0
7		divneg 11907	. . . 4 ⊢ ((π ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(π / 2) = (-π / 2))
8	4, 5, 6, 7	mp3an 1490	. . 3 ⊢ -(π / 2) = (-π / 2)
9		ax-1cn 11159	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
10		ax-icn 11160	. . . . . . . . . . . . 13 ⊢ i ∈ ℂ
11	1	recnd 11238	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
12		mulcl 11185	. . . . . . . . . . . . 13 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
13	10, 11, 12	sylancr 598	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (i · 𝐴) ∈ ℂ)
14		addcl 11183	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
15	9, 13, 14	sylancr 598	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (i · 𝐴)) ∈ ℂ)
16		atanre 27031	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ dom arctan)
17	1, 16	syl 18	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ dom arctan)
18		atandm2 27023	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
19	17, 18	sylib 221	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
20	19	simp3d 1162	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (i · 𝐴)) ≠ 0)
21	15, 20	logcld 26716	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(1 + (i · 𝐴))) ∈ ℂ)
22		subcl 11457	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
23	9, 13, 22	sylancr 598	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1 − (i · 𝐴)) ∈ ℂ)
24	19	simp2d 1161	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1 − (i · 𝐴)) ≠ 0)
25	23, 24	logcld 26716	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(1 − (i · 𝐴))) ∈ ℂ)
26	21, 25	subcld 11570	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ℂ)
27		imre 15161	. . . . . . . . 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 18	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) = (ℜ‘(-i · ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))))))
29		atanval 27030	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
30	17, 29	syl 18	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
31	30	oveq2d 7428	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (2 · ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
32	10, 5, 6	divcan2i 11959	. . . . . . . . . . . . . 14 ⊢ (2 · (i / 2)) = i
33	32	oveq1i 7422	. . . . . . . . . . . . 13 ⊢ ((2 · (i / 2)) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = (i · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
34		2re 12316	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
35	34	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ⁺ → 2 ∈ ℝ)
36	35	recnd 11238	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → 2 ∈ ℂ)
37		halfcl 12471	. . . . . . . . . . . . . . 15 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
38	10, 37	mp1i 14	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → (i / 2) ∈ ℂ)
39	25, 21	subcld 11570	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) ∈ ℂ)
40	36, 38, 39	mulassd 11233	. . . . . . . . . . . . 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 2812	. . . . . . . . . . . 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 2801	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (i · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
43	21, 25	negsubdi2d 11586	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) = ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
44	43	oveq2d 7428	. . . . . . . . . . 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 2801	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (i · -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
46		mulneg12 11653	. . . . . . . . . . 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 598	. . . . . . . . . 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 2801	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (-i · ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
49	48	fveq2d 6887	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (ℜ‘(2 · (arctan‘𝐴))) = (ℜ‘(-i · ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))))))
50		remulcl 11186	. . . . . . . . . 10 ⊢ ((2 ∈ ℝ ∧ (arctan‘𝐴) ∈ ℝ) → (2 · (arctan‘𝐴)) ∈ ℝ)
51	34, 3, 50	sylancr 598	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) ∈ ℝ)
52	51	rered 15277	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (ℜ‘(2 · (arctan‘𝐴))) = (2 · (arctan‘𝐴)))
53	28, 49, 52	3eqtr2rd 2805	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
54		rpgt0 13030	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
55	1	rered 15277	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (ℜ‘𝐴) = 𝐴)
56	54, 55	breqtrrd 5140	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 0 < (ℜ‘𝐴))
57		atanlogsublem 27061	. . . . . . . 8 ⊢ ((𝐴 ∈ dom arctan ∧ 0 < (ℜ‘𝐴)) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
58	17, 56, 57	syl2anc 595	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
59	53, 58	eqeltrd 2863	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) ∈ (-π(,)π))
60		eliooord 13433	. . . . . 6 ⊢ ((2 · (arctan‘𝐴)) ∈ (-π(,)π) → (-π < (2 · (arctan‘𝐴)) ∧ (2 · (arctan‘𝐴)) < π))
61	59, 60	syl 18	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (-π < (2 · (arctan‘𝐴)) ∧ (2 · (arctan‘𝐴)) < π))
62	61	simpld 499	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → -π < (2 · (arctan‘𝐴)))
63		pire 26600	. . . . . . 7 ⊢ π ∈ ℝ
64	63	renegcli 11520	. . . . . 6 ⊢ -π ∈ ℝ
65	64	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → -π ∈ ℝ)
66		2pos 12346	. . . . . 6 ⊢ 0 < 2
67	66	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 2)
68		ltdivmul 12091	. . . . 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 1401	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((-π / 2) < (arctan‘𝐴) ↔ -π < (2 · (arctan‘𝐴))))
70	62, 69	mpbird 260	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (-π / 2) < (arctan‘𝐴))
71	8, 70	eqbrtrid 5147	. 2 ⊢ (𝐴 ∈ ℝ⁺ → -(π / 2) < (arctan‘𝐴))
72	61	simprd 500	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) < π)
73	63	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → π ∈ ℝ)
74		ltmuldiv2 12090	. . . 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 1401	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((2 · (arctan‘𝐴)) < π ↔ (arctan‘𝐴) < (π / 2)))
76	72, 75	mpbid 235	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (arctan‘𝐴) < (π / 2))
77		halfpire 26610	. . . . 5 ⊢ (π / 2) ∈ ℝ
78	77	renegcli 11520	. . . 4 ⊢ -(π / 2) ∈ ℝ
79	78	rexri 11268	. . 3 ⊢ -(π / 2) ∈ ℝ^*
80	77	rexri 11268	. . 3 ⊢ (π / 2) ∈ ℝ^*
81		elioo2 13414	. . 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 704	. 2 ⊢ ((arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)) ↔ ((arctan‘𝐴) ∈ ℝ ∧ -(π / 2) < (arctan‘𝐴) ∧ (arctan‘𝐴) < (π / 2)))
83	3, 71, 76, 82	syl3anbrc 1362	1 ⊢ (𝐴 ∈ ℝ⁺ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 class class class wbr 5110 dom cdm 5663 ‘cfv 6538 (class class class)co 7412 ℂcc 11099 ℝcr 11100 0cc0 11101 1c1 11102 ici 11103 + caddc 11104 · cmul 11106 ℝ^*cxr 11243 < clt 11244 − cmin 11442 -cneg 11443 / cdiv 11872 2c2 12296 ℝ⁺crp 13017 (,)cioo 13373 ℜcre 15150 ℑcim 15151 πcpi 16121 logclog 26700 arctancatan 27010

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179 ax-addf 11180

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8158 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-er 8695 df-map 8827 df-pm 8828 df-ixp 8897 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-fsupp 9323 df-fi 9372 df-sup 9403 df-inf 9404 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-z 12593 df-dec 12713 df-uz 12864 df-q 12974 df-rp 13018 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13377 df-ioc 13378 df-ico 13379 df-icc 13380 df-fz 13537 df-fzo 13685 df-fl 13827 df-mod 13905 df-seq 14040 df-exp 14100 df-fac 14312 df-bc 14341 df-hash 14369 df-shft 15106 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-limsup 15524 df-clim 15541 df-rlim 15542 df-sum 15740 df-ef 16122 df-sin 16124 df-cos 16125 df-tan 16126 df-pi 16127 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17476 df-topn 17477 df-0g 17495 df-gsum 17496 df-topgen 17497 df-pt 17498 df-prds 17501 df-xrs 17557 df-qtop 17562 df-imas 17563 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-submnd 18843 df-mulg 19135 df-cntz 19388 df-cmn 19853 df-psmet 21495 df-xmet 21496 df-met 21497 df-bl 21498 df-mopn 21499 df-fbas 21500 df-fg 21501 df-cnfld 21504 df-top 23032 df-topon 23049 df-topsp 23071 df-bases 23084 df-cld 23157 df-ntr 23158 df-cls 23159 df-nei 23236 df-lp 23274 df-perf 23275 df-cn 23365 df-cnp 23366 df-haus 23453 df-tx 23700 df-hmeo 23893 df-fil 23984 df-fm 24076 df-flim 24077 df-flf 24078 df-xms 24458 df-ms 24459 df-tms 24460 df-cncf 25018 df-limc 26006 df-dv 26007 df-log 26702 df-atan 27013

This theorem is referenced by: atanbnd 27072

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