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Theorem atanbndlem 26812

Description: Lemma for atanbnd 26813. (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 12988	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		atanrecl 26798	. . 3 ⊢ (𝐴 ∈ ℝ → (arctan‘𝐴) ∈ ℝ)
3	1, 2	syl 17	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (arctan‘𝐴) ∈ ℝ)
4		picn 26349	. . . 4 ⊢ π ∈ ℂ
5		2cn 12291	. . . 4 ⊢ 2 ∈ ℂ
6		2ne0 12320	. . . 4 ⊢ 2 ≠ 0
7		divneg 11910	. . . 4 ⊢ ((π ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -(π / 2) = (-π / 2))
8	4, 5, 6, 7	mp3an 1457	. . 3 ⊢ -(π / 2) = (-π / 2)
9		ax-1cn 11170	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
10		ax-icn 11171	. . . . . . . . . . . . 13 ⊢ i ∈ ℂ
11	1	recnd 11246	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
12		mulcl 11196	. . . . . . . . . . . . 13 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
13	10, 11, 12	sylancr 586	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (i · 𝐴) ∈ ℂ)
14		addcl 11194	. . . . . . . . . . . 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 26772	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ dom arctan)
17	1, 16	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ dom arctan)
18		atandm2 26764	. . . . . . . . . . . . 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 1141	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (i · 𝐴)) ≠ 0)
21	15, 20	logcld 26459	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(1 + (i · 𝐴))) ∈ ℂ)
22		subcl 11463	. . . . . . . . . . . 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 1140	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1 − (i · 𝐴)) ≠ 0)
25	23, 24	logcld 26459	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(1 − (i · 𝐴))) ∈ ℂ)
26	21, 25	subcld 11575	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ℂ)
27		imre 15061	. . . . . . . . 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 26771	. . . . . . . . . . . . . 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 7421	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (2 · ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
32	10, 5, 6	divcan2i 11961	. . . . . . . . . . . . . 14 ⊢ (2 · (i / 2)) = i
33	32	oveq1i 7415	. . . . . . . . . . . . 13 ⊢ ((2 · (i / 2)) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = (i · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
34		2re 12290	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
35	34	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ⁺ → 2 ∈ ℝ)
36	35	recnd 11246	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → 2 ∈ ℂ)
37		halfcl 12441	. . . . . . . . . . . . . . 15 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
38	10, 37	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → (i / 2) ∈ ℂ)
39	25, 21	subcld 11575	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) ∈ ℂ)
40	36, 38, 39	mulassd 11241	. . . . . . . . . . . . 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 2780	. . . . . . . . . . . 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 2769	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (i · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
43	21, 25	negsubdi2d 11591	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) = ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
44	43	oveq2d 7421	. . . . . . . . . . 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 2769	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (i · -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
46		mulneg12 11656	. . . . . . . . . . 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 2769	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (-i · ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
49	48	fveq2d 6889	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (ℜ‘(2 · (arctan‘𝐴))) = (ℜ‘(-i · ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))))))
50		remulcl 11197	. . . . . . . . . 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 15177	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (ℜ‘(2 · (arctan‘𝐴))) = (2 · (arctan‘𝐴)))
53	28, 49, 52	3eqtr2rd 2773	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
54		rpgt0 12992	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
55	1	rered 15177	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (ℜ‘𝐴) = 𝐴)
56	54, 55	breqtrrd 5169	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 0 < (ℜ‘𝐴))
57		atanlogsublem 26802	. . . . . . . 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 2827	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) ∈ (-π(,)π))
60		eliooord 13389	. . . . . 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 26348	. . . . . . 7 ⊢ π ∈ ℝ
64	63	renegcli 11525	. . . . . 6 ⊢ -π ∈ ℝ
65	64	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → -π ∈ ℝ)
66		2pos 12319	. . . . . 6 ⊢ 0 < 2
67	66	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 2)
68		ltdivmul 12093	. . . . 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 1371	. . . 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 5176	. 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 12092	. . . 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 1371	. . 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 26354	. . . . 5 ⊢ (π / 2) ∈ ℝ
78	77	renegcli 11525	. . . 4 ⊢ -(π / 2) ∈ ℝ
79	78	rexri 11276	. . 3 ⊢ -(π / 2) ∈ ℝ^*
80	77	rexri 11276	. . 3 ⊢ (π / 2) ∈ ℝ^*
81		elioo2 13371	. . 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 689	. 2 ⊢ ((arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)) ↔ ((arctan‘𝐴) ∈ ℝ ∧ -(π / 2) < (arctan‘𝐴) ∧ (arctan‘𝐴) < (π / 2)))
83	3, 71, 76, 82	syl3anbrc 1340	1 ⊢ (𝐴 ∈ ℝ⁺ → (arctan‘𝐴) ∈ (-(π / 2)(,)(π / 2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 class class class wbr 5141 dom cdm 5669 ‘cfv 6537 (class class class)co 7405 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 ici 11114 + caddc 11115 · cmul 11117 ℝ^*cxr 11251 < clt 11252 − cmin 11448 -cneg 11449 / cdiv 11875 2c2 12271 ℝ⁺crp 12980 (,)cioo 13330 ℜcre 15050 ℑcim 15051 πcpi 16016 logclog 26443 arctancatan 26751

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-ef 16017 df-sin 16019 df-cos 16020 df-tan 16021 df-pi 16022 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-lp 22995 df-perf 22996 df-cn 23086 df-cnp 23087 df-haus 23174 df-tx 23421 df-hmeo 23614 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-xms 24181 df-ms 24182 df-tms 24183 df-cncf 24753 df-limc 25750 df-dv 25751 df-log 26445 df-atan 26754

This theorem is referenced by: atanbnd 26813

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