tanabsge - Metamath Proof Explorer

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Theorem tanabsge 26751

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 13432	. . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ)
2	1	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 𝐴 ∈ ℝ)
3	2	renegcld 11669	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ∈ ℝ)
4	1	lt0neg1d 11811	. . . . . . . . . 10 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (𝐴 < 0 ↔ 0 < -𝐴))
5	4	biimpa 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 < -𝐴)
6		eliooord 13462	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (-(π / 2) < 𝐴 ∧ 𝐴 < (π / 2)))
7	6	simpld 500	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → -(π / 2) < 𝐴)
8	7	adantr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -(π / 2) < 𝐴)
9		halfpire 26709	. . . . . . . . . . 11 ⊢ (π / 2) ∈ ℝ
10		ltnegcon1 11743	. . . . . . . . . . 11 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → (-(π / 2) < 𝐴 ↔ -𝐴 < (π / 2)))
11	9, 2, 10	sylancr 599	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (-(π / 2) < 𝐴 ↔ -𝐴 < (π / 2)))
12	8, 11	mpbid 235	. . . . . . . . 9 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 < (π / 2))
13		0xr 11284	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
14	9	rexri 11295	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℝ^*
15		elioo2 13443	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (-𝐴 ∈ (0(,)(π / 2)) ↔ (-𝐴 ∈ ℝ ∧ 0 < -𝐴 ∧ -𝐴 < (π / 2))))
16	13, 14, 15	mp2an 705	. . . . . . . . 9 ⊢ (-𝐴 ∈ (0(,)(π / 2)) ↔ (-𝐴 ∈ ℝ ∧ 0 < -𝐴 ∧ -𝐴 < (π / 2)))
17	3, 5, 12, 16	syl3anbrc 1362	. . . . . . . 8 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ∈ (0(,)(π / 2)))
18		sincosq1sgn 26743	. . . . . . . 8 ⊢ (-𝐴 ∈ (0(,)(π / 2)) → (0 < (sin‘-𝐴) ∧ 0 < (cos‘-𝐴)))
19	17, 18	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (0 < (sin‘-𝐴) ∧ 0 < (cos‘-𝐴)))
20	19	simprd 501	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 < (cos‘-𝐴))
21	20	gt0ne0d 11806	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (cos‘-𝐴) ≠ 0)
22	3, 21	retancld 16239	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘-𝐴) ∈ ℝ)
23		tangtx 26750	. . . . 5 ⊢ (-𝐴 ∈ (0(,)(π / 2)) → -𝐴 < (tan‘-𝐴))
24	17, 23	syl 18	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 < (tan‘-𝐴))
25	3, 22, 24	ltled 11386	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ≤ (tan‘-𝐴))
26		0re 11238	. . . . . 6 ⊢ 0 ∈ ℝ
27		ltle 11326	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 < 0 → 𝐴 ≤ 0))
28	1, 26, 27	sylancl 598	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (𝐴 < 0 → 𝐴 ≤ 0))
29	28	imp 412	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 𝐴 ≤ 0)
30	2, 29	absnidd 15505	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘𝐴) = -𝐴)
31	1	recnd 11265	. . . . . . . . 9 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℂ)
32	31	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 𝐴 ∈ ℂ)
33	32	negnegd 11588	. . . . . . 7 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → --𝐴 = 𝐴)
34	33	fveq2d 6886	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘--𝐴) = (tan‘𝐴))
35	32	negcld 11584	. . . . . . 7 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ∈ ℂ)
36		tanneg 16242	. . . . . . 7 ⊢ ((-𝐴 ∈ ℂ ∧ (cos‘-𝐴) ≠ 0) → (tan‘--𝐴) = -(tan‘-𝐴))
37	35, 21, 36	syl2anc 596	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘--𝐴) = -(tan‘-𝐴))
38	34, 37	eqtr3d 2799	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘𝐴) = -(tan‘-𝐴))
39	38	fveq2d 6886	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘𝐴)) = (abs‘-(tan‘-𝐴)))
40	22	recnd 11265	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘-𝐴) ∈ ℂ)
41	40	absnegd 15543	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘-(tan‘-𝐴)) = (abs‘(tan‘-𝐴)))
42		0red 11239	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 ∈ ℝ)
43		ltle 11326	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ -𝐴 ∈ ℝ) → (0 < -𝐴 → 0 ≤ -𝐴))
44	26, 3, 43	sylancr 599	. . . . . . 7 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (0 < -𝐴 → 0 ≤ -𝐴))
45	5, 44	mpd 16	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 ≤ -𝐴)
46	42, 3, 22, 45, 25	letrd 11395	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 ≤ (tan‘-𝐴))
47	22, 46	absidd 15514	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘-𝐴)) = (tan‘-𝐴))
48	39, 41, 47	3eqtrd 2801	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘𝐴)) = (tan‘-𝐴))
49	25, 30, 48	3brtr4d 5141	. 2 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))
50		abs0 15376	. . . . . 6 ⊢ (abs‘0) = 0
51	50, 26	eqeltri 2858	. . . . 5 ⊢ (abs‘0) ∈ ℝ
52	51	leidi 11776	. . . 4 ⊢ (abs‘0) ≤ (abs‘0)
53	52	a1i 11	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘0) ≤ (abs‘0))
54		simpr 490	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → 𝐴 = 0)
55	54	fveq2d 6886	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘𝐴) = (abs‘0))
56	54	fveq2d 6886	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (tan‘𝐴) = (tan‘0))
57		tan0 16245	. . . . 5 ⊢ (tan‘0) = 0
58	56, 57	eqtrdi 2813	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (tan‘𝐴) = 0)
59	58	fveq2d 6886	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘(tan‘𝐴)) = (abs‘0))
60	53, 55, 59	3brtr4d 5141	. 2 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))
61	1	adantr 486	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 ∈ ℝ)
62		simpr 490	. . . . . . . . 9 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 < 𝐴)
63	6	simprd 501	. . . . . . . . . 10 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 < (π / 2))
64	63	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 < (π / 2))
65		elioo2 13443	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (𝐴 ∈ (0(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2))))
66	13, 14, 65	mp2an 705	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)))
67	61, 62, 64, 66	syl3anbrc 1362	. . . . . . . 8 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 ∈ (0(,)(π / 2)))
68		sincosq1sgn 26743	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)(π / 2)) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴)))
69	67, 68	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴)))
70	69	simprd 501	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 < (cos‘𝐴))
71	70	gt0ne0d 11806	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (cos‘𝐴) ≠ 0)
72	61, 71	retancld 16239	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (tan‘𝐴) ∈ ℝ)
73		tangtx 26750	. . . . 5 ⊢ (𝐴 ∈ (0(,)(π / 2)) → 𝐴 < (tan‘𝐴))
74	67, 73	syl 18	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 < (tan‘𝐴))
75	61, 72, 74	ltled 11386	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 ≤ (tan‘𝐴))
76		ltle 11326	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 → 0 ≤ 𝐴))
77	26, 1, 76	sylancr 599	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (0 < 𝐴 → 0 ≤ 𝐴))
78	77	imp 412	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 ≤ 𝐴)
79	61, 78	absidd 15514	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘𝐴) = 𝐴)
80		0red 11239	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 ∈ ℝ)
81	80, 61, 72, 78, 75	letrd 11395	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 ≤ (tan‘𝐴))
82	72, 81	absidd 15514	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘(tan‘𝐴)) = (tan‘𝐴))
83	75, 79, 82	3brtr4d 5141	. 2 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))
84		lttri4 11322	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴))
85	1, 26, 84	sylancl 598	. 2 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴))
86	49, 60, 83, 85	mpjao3dan 1459	1 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 class class class wbr 5107 ‘cfv 6537 (class class class)co 7417 ℂcc 11126 ℝcr 11127 0cc0 11128 ℝ^*cxr 11270 < clt 11271 ≤ cle 11272 -cneg 11470 / cdiv 11899 2c2 12323 (,)cioo 13402 abscabs 15325 sincsin 16155 cosccos 16156 tanctan 16157 πcpi 16158

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-inf2 9624 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206 ax-addf 11207

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 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 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8909 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fsupp 9336 df-fi 9385 df-sup 9416 df-inf 9417 df-oi 9486 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-z 12620 df-dec 12741 df-uz 12892 df-q 13002 df-rp 13047 df-xneg 13167 df-xadd 13168 df-xmul 13169 df-ioo 13406 df-ioc 13407 df-ico 13408 df-icc 13409 df-fz 13566 df-fzo 13714 df-fl 13857 df-seq 14070 df-exp 14130 df-fac 14342 df-bc 14371 df-hash 14399 df-shft 15144 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-limsup 15562 df-clim 15579 df-rlim 15580 df-sum 15778 df-ef 16159 df-sin 16161 df-cos 16162 df-tan 16163 df-pi 16164 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-starv 17363 df-sca 17364 df-vsca 17365 df-ip 17366 df-tset 17367 df-ple 17368 df-ds 17370 df-unif 17371 df-hom 17372 df-cco 17373 df-rest 17513 df-topn 17514 df-0g 17532 df-gsum 17533 df-topgen 17534 df-pt 17535 df-prds 17538 df-xrs 17594 df-qtop 17599 df-imas 17600 df-xps 17602 df-mre 17676 df-mrc 17677 df-acs 17679 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-submnd 18898 df-mulg 19197 df-cntz 19450 df-cmn 19915 df-psmet 21583 df-xmet 21584 df-met 21585 df-bl 21586 df-mopn 21587 df-fbas 21588 df-fg 21589 df-cnfld 21592 df-top 23125 df-topon 23142 df-topsp 23164 df-bases 23177 df-cld 23250 df-ntr 23251 df-cls 23252 df-nei 23329 df-lp 23367 df-perf 23368 df-cn 23458 df-cnp 23459 df-haus 23546 df-tx 23794 df-hmeo 23987 df-fil 24078 df-fm 24170 df-flim 24171 df-flf 24172 df-xms 24552 df-ms 24553 df-tms 24554 df-cncf 25112 df-limc 26100 df-dv 26101

This theorem is used by: logcnlem4 26890

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