tanabsge - Metamath Proof Explorer

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Theorem tanabsge 26566

Description: The tangent function is greater than or equal to its argument in absolute value. (Contributed by Mario Carneiro, 25-Feb-2015.)

**Assertion**
Ref	Expression
tanabsge	⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))

**Proof of Theorem tanabsge**
Step	Hyp	Ref	Expression
1		elioore 13437	. . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ)
2	1	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 𝐴 ∈ ℝ)
3	2	renegcld 11717	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ∈ ℝ)
4	1	lt0neg1d 11859	. . . . . . . . . 10 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (𝐴 < 0 ↔ 0 < -𝐴))
5	4	biimpa 476	. . . . . . . . 9 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 < -𝐴)
6		eliooord 13466	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (-(π / 2) < 𝐴 ∧ 𝐴 < (π / 2)))
7	6	simpld 494	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → -(π / 2) < 𝐴)
8	7	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -(π / 2) < 𝐴)
9		halfpire 26524	. . . . . . . . . . 11 ⊢ (π / 2) ∈ ℝ
10		ltnegcon1 11791	. . . . . . . . . . 11 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → (-(π / 2) < 𝐴 ↔ -𝐴 < (π / 2)))
11	9, 2, 10	sylancr 586	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (-(π / 2) < 𝐴 ↔ -𝐴 < (π / 2)))
12	8, 11	mpbid 232	. . . . . . . . 9 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 < (π / 2))
13		0xr 11337	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
14	9	rexri 11348	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℝ^*
15		elioo2 13448	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (-𝐴 ∈ (0(,)(π / 2)) ↔ (-𝐴 ∈ ℝ ∧ 0 < -𝐴 ∧ -𝐴 < (π / 2))))
16	13, 14, 15	mp2an 691	. . . . . . . . 9 ⊢ (-𝐴 ∈ (0(,)(π / 2)) ↔ (-𝐴 ∈ ℝ ∧ 0 < -𝐴 ∧ -𝐴 < (π / 2)))
17	3, 5, 12, 16	syl3anbrc 1343	. . . . . . . 8 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ∈ (0(,)(π / 2)))
18		sincosq1sgn 26558	. . . . . . . 8 ⊢ (-𝐴 ∈ (0(,)(π / 2)) → (0 < (sin‘-𝐴) ∧ 0 < (cos‘-𝐴)))
19	17, 18	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (0 < (sin‘-𝐴) ∧ 0 < (cos‘-𝐴)))
20	19	simprd 495	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 < (cos‘-𝐴))
21	20	gt0ne0d 11854	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (cos‘-𝐴) ≠ 0)
22	3, 21	retancld 16193	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘-𝐴) ∈ ℝ)
23		tangtx 26565	. . . . 5 ⊢ (-𝐴 ∈ (0(,)(π / 2)) → -𝐴 < (tan‘-𝐴))
24	17, 23	syl 17	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 < (tan‘-𝐴))
25	3, 22, 24	ltled 11438	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ≤ (tan‘-𝐴))
26		0re 11292	. . . . . 6 ⊢ 0 ∈ ℝ
27		ltle 11378	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 < 0 → 𝐴 ≤ 0))
28	1, 26, 27	sylancl 585	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (𝐴 < 0 → 𝐴 ≤ 0))
29	28	imp 406	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 𝐴 ≤ 0)
30	2, 29	absnidd 15462	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘𝐴) = -𝐴)
31	1	recnd 11318	. . . . . . . . 9 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℂ)
32	31	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 𝐴 ∈ ℂ)
33	32	negnegd 11638	. . . . . . 7 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → --𝐴 = 𝐴)
34	33	fveq2d 6924	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘--𝐴) = (tan‘𝐴))
35	32	negcld 11634	. . . . . . 7 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ∈ ℂ)
36		tanneg 16196	. . . . . . 7 ⊢ ((-𝐴 ∈ ℂ ∧ (cos‘-𝐴) ≠ 0) → (tan‘--𝐴) = -(tan‘-𝐴))
37	35, 21, 36	syl2anc 583	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘--𝐴) = -(tan‘-𝐴))
38	34, 37	eqtr3d 2782	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘𝐴) = -(tan‘-𝐴))
39	38	fveq2d 6924	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘𝐴)) = (abs‘-(tan‘-𝐴)))
40	22	recnd 11318	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘-𝐴) ∈ ℂ)
41	40	absnegd 15498	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘-(tan‘-𝐴)) = (abs‘(tan‘-𝐴)))
42		0red 11293	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 ∈ ℝ)
43		ltle 11378	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ -𝐴 ∈ ℝ) → (0 < -𝐴 → 0 ≤ -𝐴))
44	26, 3, 43	sylancr 586	. . . . . . 7 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (0 < -𝐴 → 0 ≤ -𝐴))
45	5, 44	mpd 15	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 ≤ -𝐴)
46	42, 3, 22, 45, 25	letrd 11447	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 ≤ (tan‘-𝐴))
47	22, 46	absidd 15471	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘-𝐴)) = (tan‘-𝐴))
48	39, 41, 47	3eqtrd 2784	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘𝐴)) = (tan‘-𝐴))
49	25, 30, 48	3brtr4d 5198	. 2 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))
50		abs0 15334	. . . . . 6 ⊢ (abs‘0) = 0
51	50, 26	eqeltri 2840	. . . . 5 ⊢ (abs‘0) ∈ ℝ
52	51	leidi 11824	. . . 4 ⊢ (abs‘0) ≤ (abs‘0)
53	52	a1i 11	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘0) ≤ (abs‘0))
54		simpr 484	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → 𝐴 = 0)
55	54	fveq2d 6924	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘𝐴) = (abs‘0))
56	54	fveq2d 6924	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (tan‘𝐴) = (tan‘0))
57		tan0 16199	. . . . 5 ⊢ (tan‘0) = 0
58	56, 57	eqtrdi 2796	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (tan‘𝐴) = 0)
59	58	fveq2d 6924	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘(tan‘𝐴)) = (abs‘0))
60	53, 55, 59	3brtr4d 5198	. 2 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))
61	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 ∈ ℝ)
62		simpr 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 < 𝐴)
63	6	simprd 495	. . . . . . . . . 10 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 < (π / 2))
64	63	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 < (π / 2))
65		elioo2 13448	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (𝐴 ∈ (0(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2))))
66	13, 14, 65	mp2an 691	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)))
67	61, 62, 64, 66	syl3anbrc 1343	. . . . . . . 8 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 ∈ (0(,)(π / 2)))
68		sincosq1sgn 26558	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(π / 2)) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴)))
69	67, 68	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴)))
70	69	simprd 495	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 < (cos‘𝐴))
71	70	gt0ne0d 11854	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (cos‘𝐴) ≠ 0)
72	61, 71	retancld 16193	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (tan‘𝐴) ∈ ℝ)
73		tangtx 26565	. . . . 5 ⊢ (𝐴 ∈ (0(,)(π / 2)) → 𝐴 < (tan‘𝐴))
74	67, 73	syl 17	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 < (tan‘𝐴))
75	61, 72, 74	ltled 11438	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 ≤ (tan‘𝐴))
76		ltle 11378	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 → 0 ≤ 𝐴))
77	26, 1, 76	sylancr 586	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (0 < 𝐴 → 0 ≤ 𝐴))
78	77	imp 406	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 ≤ 𝐴)
79	61, 78	absidd 15471	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘𝐴) = 𝐴)
80		0red 11293	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 ∈ ℝ)
81	80, 61, 72, 78, 75	letrd 11447	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 ≤ (tan‘𝐴))
82	72, 81	absidd 15471	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘(tan‘𝐴)) = (tan‘𝐴))
83	75, 79, 82	3brtr4d 5198	. 2 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))
84		lttri4 11374	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴))
85	1, 26, 84	sylancl 585	. 2 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴))
86	49, 60, 83, 85	mpjao3dan 1432	1 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1086 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 -cneg 11521 / cdiv 11947 2c2 12348 (,)cioo 13407 abscabs 15283 sincsin 16111 cosccos 16112 tanctan 16113 πcpi 16114

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-tan 16119 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 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 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922

This theorem is referenced by: logcnlem4 26705

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