tanabsge - Metamath Proof Explorer

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Theorem tanabsge 26471

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 13291	. . . . . 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 11564	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ∈ ℝ)
4	1	lt0neg1d 11706	. . . . . . . . . 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 13321	. . . . . . . . . . . 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 26429	. . . . . . . . . . 11 ⊢ (π / 2) ∈ ℝ
10		ltnegcon1 11638	. . . . . . . . . . 11 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → (-(π / 2) < 𝐴 ↔ -𝐴 < (π / 2)))
11	9, 2, 10	sylancr 587	. . . . . . . . . 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 11179	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
14	9	rexri 11190	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℝ^*
15		elioo2 13302	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (-𝐴 ∈ (0(,)(π / 2)) ↔ (-𝐴 ∈ ℝ ∧ 0 < -𝐴 ∧ -𝐴 < (π / 2))))
16	13, 14, 15	mp2an 692	. . . . . . . . 9 ⊢ (-𝐴 ∈ (0(,)(π / 2)) ↔ (-𝐴 ∈ ℝ ∧ 0 < -𝐴 ∧ -𝐴 < (π / 2)))
17	3, 5, 12, 16	syl3anbrc 1344	. . . . . . . 8 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ∈ (0(,)(π / 2)))
18		sincosq1sgn 26463	. . . . . . . 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 11701	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (cos‘-𝐴) ≠ 0)
22	3, 21	retancld 16070	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘-𝐴) ∈ ℝ)
23		tangtx 26470	. . . . 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 11281	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ≤ (tan‘-𝐴))
26		0re 11134	. . . . . 6 ⊢ 0 ∈ ℝ
27		ltle 11221	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 < 0 → 𝐴 ≤ 0))
28	1, 26, 27	sylancl 586	. . . . 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 15337	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘𝐴) = -𝐴)
31	1	recnd 11160	. . . . . . . . 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 11483	. . . . . . 7 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → --𝐴 = 𝐴)
34	33	fveq2d 6838	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘--𝐴) = (tan‘𝐴))
35	32	negcld 11479	. . . . . . 7 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ∈ ℂ)
36		tanneg 16073	. . . . . . 7 ⊢ ((-𝐴 ∈ ℂ ∧ (cos‘-𝐴) ≠ 0) → (tan‘--𝐴) = -(tan‘-𝐴))
37	35, 21, 36	syl2anc 584	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘--𝐴) = -(tan‘-𝐴))
38	34, 37	eqtr3d 2773	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘𝐴) = -(tan‘-𝐴))
39	38	fveq2d 6838	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘𝐴)) = (abs‘-(tan‘-𝐴)))
40	22	recnd 11160	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘-𝐴) ∈ ℂ)
41	40	absnegd 15375	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘-(tan‘-𝐴)) = (abs‘(tan‘-𝐴)))
42		0red 11135	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 ∈ ℝ)
43		ltle 11221	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ -𝐴 ∈ ℝ) → (0 < -𝐴 → 0 ≤ -𝐴))
44	26, 3, 43	sylancr 587	. . . . . . 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 11290	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 ≤ (tan‘-𝐴))
47	22, 46	absidd 15346	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘-𝐴)) = (tan‘-𝐴))
48	39, 41, 47	3eqtrd 2775	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘𝐴)) = (tan‘-𝐴))
49	25, 30, 48	3brtr4d 5130	. 2 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))
50		abs0 15208	. . . . . 6 ⊢ (abs‘0) = 0
51	50, 26	eqeltri 2832	. . . . 5 ⊢ (abs‘0) ∈ ℝ
52	51	leidi 11671	. . . 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 6838	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘𝐴) = (abs‘0))
56	54	fveq2d 6838	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (tan‘𝐴) = (tan‘0))
57		tan0 16076	. . . . 5 ⊢ (tan‘0) = 0
58	56, 57	eqtrdi 2787	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (tan‘𝐴) = 0)
59	58	fveq2d 6838	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘(tan‘𝐴)) = (abs‘0))
60	53, 55, 59	3brtr4d 5130	. 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 13302	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (𝐴 ∈ (0(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2))))
66	13, 14, 65	mp2an 692	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)))
67	61, 62, 64, 66	syl3anbrc 1344	. . . . . . . 8 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 ∈ (0(,)(π / 2)))
68		sincosq1sgn 26463	. . . . . . . 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 11701	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (cos‘𝐴) ≠ 0)
72	61, 71	retancld 16070	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (tan‘𝐴) ∈ ℝ)
73		tangtx 26470	. . . . 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 11281	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 ≤ (tan‘𝐴))
76		ltle 11221	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 → 0 ≤ 𝐴))
77	26, 1, 76	sylancr 587	. . . . 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 15346	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘𝐴) = 𝐴)
80		0red 11135	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 ∈ ℝ)
81	80, 61, 72, 78, 75	letrd 11290	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 ≤ (tan‘𝐴))
82	72, 81	absidd 15346	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘(tan‘𝐴)) = (tan‘𝐴))
83	75, 79, 82	3brtr4d 5130	. 2 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))
84		lttri4 11217	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴))
85	1, 26, 84	sylancl 586	. 2 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴))
86	49, 60, 83, 85	mpjao3dan 1434	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 1085 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 ℝcr 11025 0cc0 11026 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167 -cneg 11365 / cdiv 11794 2c2 12200 (,)cioo 13261 abscabs 15157 sincsin 15986 cosccos 15987 tanctan 15988 πcpi 15989

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-shft 14990 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15610 df-ef 15990 df-sin 15992 df-cos 15993 df-tan 15994 df-pi 15995 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25823 df-dv 25824

This theorem is referenced by: logcnlem4 26610

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