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Theorem atanbndlem 27119

Description: Lemma for atanbnd 27120. (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 13036	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		atanrecl 27105	. . 3 ⊢ (𝐴 ∈ ℝ → (arctan‘𝐴) ∈ ℝ)
3	1, 2	syl 18	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (arctan‘𝐴) ∈ ℝ)
4		picn 26650	. . . 4 ⊢ π ∈ ℂ
5		2cn 12327	. . . 4 ⊢ 2 ∈ ℂ
6		2ne0 12358	. . . 4 ⊢ 2 ≠ 0
7		divneg 11917	. . . 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 11169	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
10		ax-icn 11170	. . . . . . . . . . . . 13 ⊢ i ∈ ℂ
11	1	recnd 11248	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
12		mulcl 11195	. . . . . . . . . . . . 13 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
13	10, 11, 12	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (i · 𝐴) ∈ ℂ)
14		addcl 11193	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
15	9, 13, 14	sylancr 599	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (i · 𝐴)) ∈ ℂ)
16		atanre 27079	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ dom arctan)
17	1, 16	syl 18	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ dom arctan)
18		atandm2 27071	. . . . . . . . . . . . 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 26764	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(1 + (i · 𝐴))) ∈ ℂ)
22		subcl 11467	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
23	9, 13, 22	sylancr 599	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1 − (i · 𝐴)) ∈ ℂ)
24	19	simp2d 1161	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1 − (i · 𝐴)) ≠ 0)
25	23, 24	logcld 26764	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(1 − (i · 𝐴))) ∈ ℂ)
26	21, 25	subcld 11580	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ℂ)
27		imre 15178	. . . . . . . . 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 27078	. . . . . . . . . . . . . 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 7432	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (2 · ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
32	10, 5, 6	divcan2i 11969	. . . . . . . . . . . . . 14 ⊢ (2 · (i / 2)) = i
33	32	oveq1i 7426	. . . . . . . . . . . . 13 ⊢ ((2 · (i / 2)) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = (i · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
34		2re 12326	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
35	34	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ⁺ → 2 ∈ ℝ)
36	35	recnd 11248	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → 2 ∈ ℂ)
37		halfcl 12481	. . . . . . . . . . . . . . 15 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
38	10, 37	mp1i 14	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → (i / 2) ∈ ℂ)
39	25, 21	subcld 11580	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) ∈ ℂ)
40	36, 38, 39	mulassd 11243	. . . . . . . . . . . . 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 2814	. . . . . . . . . . . 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 2803	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (i · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
43	21, 25	negsubdi2d 11596	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) = ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
44	43	oveq2d 7432	. . . . . . . . . . 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 2803	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (i · -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
46		mulneg12 11663	. . . . . . . . . . 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 599	. . . . . . . . . 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 2803	. . . . . . . . 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 11196	. . . . . . . . . 10 ⊢ ((2 ∈ ℝ ∧ (arctan‘𝐴) ∈ ℝ) → (2 · (arctan‘𝐴)) ∈ ℝ)
51	34, 3, 50	sylancr 599	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) ∈ ℝ)
52	51	rered 15294	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (ℜ‘(2 · (arctan‘𝐴))) = (2 · (arctan‘𝐴)))
53	28, 49, 52	3eqtr2rd 2807	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) = (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
54		rpgt0 13040	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
55	1	rered 15294	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (ℜ‘𝐴) = 𝐴)
56	54, 55	breqtrrd 5141	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 0 < (ℜ‘𝐴))
57		atanlogsublem 27109	. . . . . . . 8 ⊢ ((𝐴 ∈ dom arctan ∧ 0 < (ℜ‘𝐴)) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
58	17, 56, 57	syl2anc 596	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
59	53, 58	eqeltrd 2865	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) ∈ (-π(,)π))
60		eliooord 13443	. . . . . 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 500	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → -π < (2 · (arctan‘𝐴)))
63		pire 26648	. . . . . . 7 ⊢ π ∈ ℝ
64	63	renegcli 11530	. . . . . 6 ⊢ -π ∈ ℝ
65	64	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → -π ∈ ℝ)
66		2pos 12356	. . . . . 6 ⊢ 0 < 2
67	66	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 2)
68		ltdivmul 12101	. . . . 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 5148	. 2 ⊢ (𝐴 ∈ ℝ⁺ → -(π / 2) < (arctan‘𝐴))
72	61	simprd 501	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (2 · (arctan‘𝐴)) < π)
73	63	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → π ∈ ℝ)
74		ltmuldiv2 12100	. . . 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 26658	. . . . 5 ⊢ (π / 2) ∈ ℝ
78	77	renegcli 11530	. . . 4 ⊢ -(π / 2) ∈ ℝ
79	78	rexri 11278	. . 3 ⊢ -(π / 2) ∈ ℝ^*
80	77	rexri 11278	. . 3 ⊢ (π / 2) ∈ ℝ^*
81		elioo2 13424	. . 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 705	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 class class class wbr 5111 dom cdm 5663 ‘cfv 6540 (class class class)co 7416 ℂcc 11109 ℝcr 11110 0cc0 11111 1c1 11112 ici 11113 + caddc 11114 · cmul 11116 ℝ^*cxr 11253 < clt 11254 − cmin 11452 -cneg 11453 / cdiv 11882 2c2 12306 ℝ⁺crp 13027 (,)cioo 13383 ℜcre 15167 ℑcim 15168 πcpi 16137 logclog 26748 arctancatan 27058

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-inf2 9613 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-pre-sup 11189 ax-addf 11190

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 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 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9325 df-fi 9374 df-sup 9405 df-inf 9406 df-oi 9475 df-card 9937 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-div 11883 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-7 12319 df-8 12320 df-9 12321 df-n0 12516 df-z 12603 df-dec 12723 df-uz 12874 df-q 12984 df-rp 13028 df-xneg 13148 df-xadd 13149 df-xmul 13150 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13547 df-fzo 13695 df-fl 13838 df-mod 13916 df-seq 14051 df-exp 14111 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15123 df-cj 15169 df-re 15170 df-im 15171 df-sqrt 15305 df-abs 15306 df-limsup 15541 df-clim 15558 df-rlim 15559 df-sum 15757 df-ef 16138 df-sin 16140 df-cos 16141 df-tan 16142 df-pi 16143 df-struct 17224 df-sets 17241 df-slot 17259 df-ndx 17271 df-base 17287 df-ress 17308 df-plusg 17340 df-mulr 17341 df-starv 17342 df-sca 17343 df-vsca 17344 df-ip 17345 df-tset 17346 df-ple 17347 df-ds 17349 df-unif 17350 df-hom 17351 df-cco 17352 df-rest 17492 df-topn 17493 df-0g 17511 df-gsum 17512 df-topgen 17513 df-pt 17514 df-prds 17517 df-xrs 17573 df-qtop 17578 df-imas 17579 df-xps 17581 df-mre 17655 df-mrc 17656 df-acs 17658 df-mgm 18715 df-sgrp 18798 df-mnd 18814 df-submnd 18865 df-mulg 19157 df-cntz 19410 df-cmn 19875 df-psmet 21543 df-xmet 21544 df-met 21545 df-bl 21546 df-mopn 21547 df-fbas 21548 df-fg 21549 df-cnfld 21552 df-top 23080 df-topon 23097 df-topsp 23119 df-bases 23132 df-cld 23205 df-ntr 23206 df-cls 23207 df-nei 23284 df-lp 23322 df-perf 23323 df-cn 23413 df-cnp 23414 df-haus 23501 df-tx 23748 df-hmeo 23941 df-fil 24032 df-fm 24124 df-flim 24125 df-flf 24126 df-xms 24506 df-ms 24507 df-tms 24508 df-cncf 25066 df-limc 26054 df-dv 26055 df-log 26750 df-atan 27061

This theorem is used by: atanbnd 27120

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