tanabsge - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > tanabsge		Structured version Visualization version GIF version

Theorem tanabsge 24258

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 12205	. . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ)
2	1	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 𝐴 ∈ ℝ)
3	2	renegcld 10457	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ∈ ℝ)
4	1	lt0neg1d 10597	. . . . . . . . . 10 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (𝐴 < 0 ↔ 0 < -𝐴))
5	4	biimpa 501	. . . . . . . . 9 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 < -𝐴)
6		eliooord 12233	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (-(π / 2) < 𝐴 ∧ 𝐴 < (π / 2)))
7	6	simpld 475	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → -(π / 2) < 𝐴)
8	7	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -(π / 2) < 𝐴)
9		halfpire 24216	. . . . . . . . . . 11 ⊢ (π / 2) ∈ ℝ
10		ltnegcon1 10529	. . . . . . . . . . 11 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → (-(π / 2) < 𝐴 ↔ -𝐴 < (π / 2)))
11	9, 2, 10	sylancr 695	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (-(π / 2) < 𝐴 ↔ -𝐴 < (π / 2)))
12	8, 11	mpbid 222	. . . . . . . . 9 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 < (π / 2))
13		0xr 10086	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
14	9	rexri 10097	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℝ^*
15		elioo2 12216	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (-𝐴 ∈ (0(,)(π / 2)) ↔ (-𝐴 ∈ ℝ ∧ 0 < -𝐴 ∧ -𝐴 < (π / 2))))
16	13, 14, 15	mp2an 708	. . . . . . . . 9 ⊢ (-𝐴 ∈ (0(,)(π / 2)) ↔ (-𝐴 ∈ ℝ ∧ 0 < -𝐴 ∧ -𝐴 < (π / 2)))
17	3, 5, 12, 16	syl3anbrc 1246	. . . . . . . 8 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ∈ (0(,)(π / 2)))
18		sincosq1sgn 24250	. . . . . . . 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 479	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 < (cos‘-𝐴))
21	20	gt0ne0d 10592	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (cos‘-𝐴) ≠ 0)
22	3, 21	retancld 14875	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘-𝐴) ∈ ℝ)
23		tangtx 24257	. . . . 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 10185	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ≤ (tan‘-𝐴))
26		0re 10040	. . . . . 6 ⊢ 0 ∈ ℝ
27		ltle 10126	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 < 0 → 𝐴 ≤ 0))
28	1, 26, 27	sylancl 694	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (𝐴 < 0 → 𝐴 ≤ 0))
29	28	imp 445	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 𝐴 ≤ 0)
30	2, 29	absnidd 14152	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘𝐴) = -𝐴)
31	1	recnd 10068	. . . . . . . . 9 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℂ)
32	31	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 𝐴 ∈ ℂ)
33	32	negnegd 10383	. . . . . . 7 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → --𝐴 = 𝐴)
34	33	fveq2d 6195	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘--𝐴) = (tan‘𝐴))
35	32	negcld 10379	. . . . . . 7 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ∈ ℂ)
36		tanneg 14878	. . . . . . 7 ⊢ ((-𝐴 ∈ ℂ ∧ (cos‘-𝐴) ≠ 0) → (tan‘--𝐴) = -(tan‘-𝐴))
37	35, 21, 36	syl2anc 693	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘--𝐴) = -(tan‘-𝐴))
38	34, 37	eqtr3d 2658	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘𝐴) = -(tan‘-𝐴))
39	38	fveq2d 6195	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘𝐴)) = (abs‘-(tan‘-𝐴)))
40	22	recnd 10068	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘-𝐴) ∈ ℂ)
41	40	absnegd 14188	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘-(tan‘-𝐴)) = (abs‘(tan‘-𝐴)))
42		0red 10041	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 ∈ ℝ)
43		ltle 10126	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ -𝐴 ∈ ℝ) → (0 < -𝐴 → 0 ≤ -𝐴))
44	26, 3, 43	sylancr 695	. . . . . . 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 10194	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 ≤ (tan‘-𝐴))
47	22, 46	absidd 14161	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘-𝐴)) = (tan‘-𝐴))
48	39, 41, 47	3eqtrd 2660	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘𝐴)) = (tan‘-𝐴))
49	25, 30, 48	3brtr4d 4685	. 2 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))
50		abs0 14025	. . . . . 6 ⊢ (abs‘0) = 0
51	50, 26	eqeltri 2697	. . . . 5 ⊢ (abs‘0) ∈ ℝ
52	51	leidi 10562	. . . 4 ⊢ (abs‘0) ≤ (abs‘0)
53	52	a1i 11	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘0) ≤ (abs‘0))
54		simpr 477	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → 𝐴 = 0)
55	54	fveq2d 6195	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘𝐴) = (abs‘0))
56	54	fveq2d 6195	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (tan‘𝐴) = (tan‘0))
57		tan0 14881	. . . . 5 ⊢ (tan‘0) = 0
58	56, 57	syl6eq 2672	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (tan‘𝐴) = 0)
59	58	fveq2d 6195	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘(tan‘𝐴)) = (abs‘0))
60	53, 55, 59	3brtr4d 4685	. 2 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))
61	1	adantr 481	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 ∈ ℝ)
62		simpr 477	. . . . . . . . 9 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 < 𝐴)
63	6	simprd 479	. . . . . . . . . 10 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 < (π / 2))
64	63	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 < (π / 2))
65		elioo2 12216	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (𝐴 ∈ (0(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2))))
66	13, 14, 65	mp2an 708	. . . . . . . . 9 ⊢ (𝐴 ∈ (0(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)))
67	61, 62, 64, 66	syl3anbrc 1246	. . . . . . . 8 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 ∈ (0(,)(π / 2)))
68		sincosq1sgn 24250	. . . . . . . 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 479	. . . . . 6 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 < (cos‘𝐴))
71	70	gt0ne0d 10592	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (cos‘𝐴) ≠ 0)
72	61, 71	retancld 14875	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (tan‘𝐴) ∈ ℝ)
73		tangtx 24257	. . . . 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 10185	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 ≤ (tan‘𝐴))
76		ltle 10126	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 → 0 ≤ 𝐴))
77	26, 1, 76	sylancr 695	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (0 < 𝐴 → 0 ≤ 𝐴))
78	77	imp 445	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 ≤ 𝐴)
79	61, 78	absidd 14161	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘𝐴) = 𝐴)
80		0red 10041	. . . . 5 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 ∈ ℝ)
81	80, 61, 72, 78, 75	letrd 10194	. . . 4 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 ≤ (tan‘𝐴))
82	72, 81	absidd 14161	. . 3 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘(tan‘𝐴)) = (tan‘𝐴))
83	75, 79, 82	3brtr4d 4685	. 2 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))
84		lttri4 10122	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴))
85	1, 26, 84	sylancl 694	. 2 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (𝐴 < 0 ∨ 𝐴 = 0 ∨ 0 < 𝐴))
86	49, 60, 83, 85	mpjao3dan 1395	1 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∨ w3o 1036 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 -cneg 10267 / cdiv 10684 2c2 11070 (,)cioo 12175 abscabs 13974 sincsin 14794 cosccos 14795 tanctan 14796 πcpi 14797

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-tan 14802 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631

This theorem is referenced by: logcnlem4 24391

W3C validator