tanabsge - Metamath Proof Explorer

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Theorem tanabsge 26892

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 13546	. . . . . 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 11782	. . . 4 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ∈ ℝ)
4	1	lt0neg1d 11925	. . . . . . . . . 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 13576	. . . . . . . . . . . 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 26850	. . . . . . . . . . 11 ⊢ (π / 2) ∈ ℝ
10		ltnegcon1 11857	. . . . . . . . . . 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 11395	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
14	9	rexri 11406	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℝ^*
15		elioo2 13557	. . . . . . . . . 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 26884	. . . . . . . 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 11920	. . . . 5 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (cos‘ -𝐴) ≠ 0)
22	3, 21	retancld 16357	. . . 4 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘ -𝐴) ∈ ℝ)
23		tangtx 26891	. . . . 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 11497	. . 3 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ≤ (tan‘ -𝐴))
26		0re 11349	. . . . . 6 ⊢ 0 ∈ ℝ
27		ltle 11437	. . . . . 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 15625	. . 3 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘𝐴) = -𝐴)
31	1	recnd 11376	. . . . . . . . 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 11699	. . . . . . 7 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → - -𝐴 = 𝐴)
34	33	fveq2d 6920	. . . . . 6 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘ - -𝐴) = (tan‘𝐴))
35	32	negcld 11695	. . . . . . 7 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ∈ ℂ)
36		tanneg 16360	. . . . . . 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 2800	. . . . 5 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘𝐴) = -(tan‘ -𝐴))
39	38	fveq2d 6920	. . . 4 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘𝐴)) = (abs‘ -(tan‘ -𝐴)))
40	22	recnd 11376	. . . . 5 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘ -𝐴) ∈ ℂ)
41	40	absnegd 15663	. . . 4 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘ -(tan‘ -𝐴)) = (abs‘(tan‘ -𝐴)))
42		0red 11350	. . . . . 6 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 ∈ ℝ)
43		ltle 11437	. . . . . . . 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 11506	. . . . 5 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 ≤ (tan‘ -𝐴))
47	22, 46	absidd 15634	. . . 4 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘ -𝐴)) = (tan‘ -𝐴))
48	39, 41, 47	3eqtrd 2802	. . 3 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘𝐴)) = (tan‘ -𝐴))
49	25, 30, 48	3brtr4d 5143	. 2 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))
50		abs0 15495	. . . . . 6 ⊢ (abs‘0) = 0
51	50, 26	eqeltri 2860	. . . . 5 ⊢ (abs‘0) ∈ ℝ
52	51	leidi 11890	. . . 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 6920	. . 3 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘𝐴) = (abs‘0))
56	54	fveq2d 6920	. . . . 5 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (tan‘𝐴) = (tan‘0))
57		tan0 16363	. . . . 5 ⊢ (tan‘0) = 0
58	56, 57	eqtrdi 2814	. . . 4 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (tan‘𝐴) = 0)
59	58	fveq2d 6920	. . 3 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘(tan‘𝐴)) = (abs‘0))
60	53, 55, 59	3brtr4d 5143	. 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 13557	. . . . . . . . . 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 26884	. . . . . . . 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 11920	. . . . 5 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (cos‘𝐴) ≠ 0)
72	61, 71	retancld 16357	. . . 4 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (tan‘𝐴) ∈ ℝ)
73		tangtx 26891	. . . . 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 11497	. . 3 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 ≤ (tan‘𝐴))
76		ltle 11437	. . . . . 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 15634	. . 3 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘𝐴) = 𝐴)
80		0red 11350	. . . . 5 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 ∈ ℝ)
81	80, 61, 72, 78, 75	letrd 11506	. . . 4 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 ≤ (tan‘𝐴))
82	72, 81	absidd 15634	. . 3 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘(tan‘𝐴)) = (tan‘𝐴))
83	75, 79, 82	3brtr4d 5143	. 2 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))
84		lttri4 11433	. . 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 1460	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 2147 ≠ wne 2959 class class class wbr 5108 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 0cc0 11239 ℝ^*cxr 11381 < clt 11382 ≤ cle 11383 -cneg 11581 / cdiv 12013 2c2 12437 (,)cioo 13516 abscabs 15444 sincsin 16273 cosccos 16274 tanctan 16275 πcpi 16276

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ioc 13521 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-seq 14185 df-exp 14245 df-fac 14459 df-bc 14488 df-hash 14516 df-shft 15263 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-limsup 15682 df-clim 15699 df-rlim 15700 df-sum 15898 df-ef 16277 df-sin 16279 df-cos 16280 df-tan 16281 df-pi 16282 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-mulg 19328 df-cntz 19582 df-cmn 20047 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-haus 23688 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-limc 26241 df-dv 26242

This theorem is used by: logcnlem4 27030

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