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Theorem atanbndlem 25980

Description: Lemma for atanbnd 25981. (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 12667	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		atanrecl 25966	. . 3 ⊢ (𝐴 ∈ ℝ → (arctan‘𝐴) ∈ ℝ)
3	1, 2	syl 17	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (arctan‘𝐴) ∈ ℝ)
4		picn 25521	. . . 4 ⊢ π ∈ ℂ
5		2cn 11978	. . . 4 ⊢ 2 ∈ ℂ
6		2ne0 12007	. . . 4 ⊢ 2 ≠ 0
7		divneg 11597	. . . 4 ⊢ ((π ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(π / 2) = (-π / 2))
8	4, 5, 6, 7	mp3an 1459	. . 3 ⊢ -(π / 2) = (-π / 2)
9		ax-1cn 10860	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
10		ax-icn 10861	. . . . . . . . . . . . 13 ⊢ i ∈ ℂ
11	1	recnd 10934	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
12		mulcl 10886	. . . . . . . . . . . . 13 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
13	10, 11, 12	sylancr 586	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (i · 𝐴) ∈ ℂ)
14		addcl 10884	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
15	9, 13, 14	sylancr 586	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (i · 𝐴)) ∈ ℂ)
16		atanre 25940	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ dom arctan)
17	1, 16	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ dom arctan)
18		atandm2 25932	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
19	17, 18	sylib 217	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
20	19	simp3d 1142	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (i · 𝐴)) ≠ 0)
21	15, 20	logcld 25631	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(1 + (i · 𝐴))) ∈ ℂ)
22		subcl 11150	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
23	9, 13, 22	sylancr 586	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1 − (i · 𝐴)) ∈ ℂ)
24	19	simp2d 1141	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1 − (i · 𝐴)) ≠ 0)
25	23, 24	logcld 25631	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(1 − (i · 𝐴))) ∈ ℂ)
26	21, 25	subcld 11262	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ℂ)
27		imre 14747	. . . . . . . . 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 25939	. . . . . . . . . . . . . 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 7271	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (2 · ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
32	10, 5, 6	divcan2i 11648	. . . . . . . . . . . . . 14 ⊢ (2 · (i / 2)) = i
33	32	oveq1i 7265	. . . . . . . . . . . . 13 ⊢ ((2 · (i / 2)) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = (i · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
34		2re 11977	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
35	34	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ⁺ → 2 ∈ ℝ)
36	35	recnd 10934	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → 2 ∈ ℂ)
37		halfcl 12128	. . . . . . . . . . . . . . 15 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
38	10, 37	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → (i / 2) ∈ ℂ)
39	25, 21	subcld 11262	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) ∈ ℂ)
40	36, 38, 39	mulassd 10929	. . . . . . . . . . . . 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 2793	. . . . . . . . . . . 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 2781	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (i · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
43	21, 25	negsubdi2d 11278	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) = ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
44	43	oveq2d 7271	. . . . . . . . . . 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 2781	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (i · -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
46		mulneg12 11343	. . . . . . . . . . 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 586	. . . . . . . . . 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 2781	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (-i · ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
49	48	fveq2d 6760	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (ℜ‘(2 · (arctan‘𝐴))) = (ℜ‘(-i · ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))))))
50		remulcl 10887	. . . . . . . . . 10 ⊢ ((2 ∈ ℝ ∧ (arctan‘𝐴) ∈ ℝ) → (2 · (arctan‘𝐴)) ∈ ℝ)
51	34, 3, 50	sylancr 586	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) ∈ ℝ)
52	51	rered 14863	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (ℜ‘(2 · (arctan‘𝐴))) = (2 · (arctan‘𝐴)))
53	28, 49, 52	3eqtr2rd 2785	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
54		rpgt0 12671	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
55	1	rered 14863	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (ℜ‘𝐴) = 𝐴)
56	54, 55	breqtrrd 5098	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 0 < (ℜ‘𝐴))
57		atanlogsublem 25970	. . . . . . . 8 ⊢ ((𝐴 ∈ dom arctan ∧ 0 < (ℜ‘𝐴)) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
58	17, 56, 57	syl2anc 583	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
59	53, 58	eqeltrd 2839	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) ∈ (-π(,)π))
60		eliooord 13067	. . . . . 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 25520	. . . . . . 7 ⊢ π ∈ ℝ
64	63	renegcli 11212	. . . . . 6 ⊢ -π ∈ ℝ
65	64	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → -π ∈ ℝ)
66		2pos 12006	. . . . . 6 ⊢ 0 < 2
67	66	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 2)
68		ltdivmul 11780	. . . . 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 1372	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((-π / 2) < (arctan‘𝐴) ↔ -π < (2 · (arctan‘𝐴))))
70	62, 69	mpbird 256	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (-π / 2) < (arctan‘𝐴))
71	8, 70	eqbrtrid 5105	. 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 11779	. . . 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 1372	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((2 · (arctan‘𝐴)) < π ↔ (arctan‘𝐴) < (π / 2)))
76	72, 75	mpbid 231	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (arctan‘𝐴) < (π / 2))
77		halfpire 25526	. . . . 5 ⊢ (π / 2) ∈ ℝ
78	77	renegcli 11212	. . . 4 ⊢ -(π / 2) ∈ ℝ
79	78	rexri 10964	. . 3 ⊢ -(π / 2) ∈ ℝ^*
80	77	rexri 10964	. . 3 ⊢ (π / 2) ∈ ℝ^*
81		elioo2 13049	. . 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 688	. 2 ⊢ ((arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)) ↔ ((arctan‘𝐴) ∈ ℝ ∧ -(π / 2) < (arctan‘𝐴) ∧ (arctan‘𝐴) < (π / 2)))
83	3, 71, 76, 82	syl3anbrc 1341	1 ⊢ (𝐴 ∈ ℝ⁺ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 class class class wbr 5070 dom cdm 5580 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 ici 10804 + caddc 10805 · cmul 10807 ℝ^*cxr 10939 < clt 10940 − cmin 11135 -cneg 11136 / cdiv 11562 2c2 11958 ℝ⁺crp 12659 (,)cioo 13008 ℜcre 14736 ℑcim 14737 πcpi 15704 logclog 25615 arctancatan 25919

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-tan 15709 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-log 25617 df-atan 25922

This theorem is referenced by: atanbnd 25981

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