tanabsge - Metamath Proof Explorer

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Theorem tanabsge 26842

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 13511	. . . . . 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 11748	. . . 4 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ∈ ℝ)
4	1	lt0neg1d 11890	. . . . . . . . . 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 13541	. . . . . . . . . . . 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 26800	. . . . . . . . . . 11 ⊢ (π / 2) ∈ ℝ
10		ltnegcon1 11822	. . . . . . . . . . 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 11361	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
14	9	rexri 11372	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℝ^*
15		elioo2 13522	. . . . . . . . . 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 26834	. . . . . . . 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 11885	. . . . 5 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (cos‘ -𝐴) ≠ 0)
22	3, 21	retancld 16320	. . . 4 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘ -𝐴) ∈ ℝ)
23		tangtx 26841	. . . . 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 11463	. . 3 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ≤ (tan‘ -𝐴))
26		0re 11315	. . . . . 6 ⊢ 0 ∈ ℝ
27		ltle 11403	. . . . . 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 15588	. . 3 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘𝐴) = -𝐴)
31	1	recnd 11342	. . . . . . . . 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 11665	. . . . . . 7 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → - -𝐴 = 𝐴)
34	33	fveq2d 6892	. . . . . 6 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘ - -𝐴) = (tan‘𝐴))
35	32	negcld 11661	. . . . . . 7 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → -𝐴 ∈ ℂ)
36		tanneg 16323	. . . . . . 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 2797	. . . . 5 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘𝐴) = -(tan‘ -𝐴))
39	38	fveq2d 6892	. . . 4 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘𝐴)) = (abs‘ -(tan‘ -𝐴)))
40	22	recnd 11342	. . . . 5 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (tan‘ -𝐴) ∈ ℂ)
41	40	absnegd 15626	. . . 4 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘ -(tan‘ -𝐴)) = (abs‘(tan‘ -𝐴)))
42		0red 11316	. . . . . 6 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 ∈ ℝ)
43		ltle 11403	. . . . . . . 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 11472	. . . . 5 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → 0 ≤ (tan‘ -𝐴))
47	22, 46	absidd 15597	. . . 4 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘ -𝐴)) = (tan‘ -𝐴))
48	39, 41, 47	3eqtrd 2799	. . 3 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘(tan‘𝐴)) = (tan‘ -𝐴))
49	25, 30, 48	3brtr4d 5136	. 2 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 < 0) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))
50		abs0 15459	. . . . . 6 ⊢ (abs‘0) = 0
51	50, 26	eqeltri 2856	. . . . 5 ⊢ (abs‘0) ∈ ℝ
52	51	leidi 11855	. . . 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 6892	. . 3 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘𝐴) = (abs‘0))
56	54	fveq2d 6892	. . . . 5 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (tan‘𝐴) = (tan‘0))
57		tan0 16326	. . . . 5 ⊢ (tan‘0) = 0
58	56, 57	eqtrdi 2811	. . . 4 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (tan‘𝐴) = 0)
59	58	fveq2d 6892	. . 3 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐴 = 0) → (abs‘(tan‘𝐴)) = (abs‘0))
60	53, 55, 59	3brtr4d 5136	. 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 13522	. . . . . . . . . 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 26834	. . . . . . . 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 11885	. . . . 5 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (cos‘𝐴) ≠ 0)
72	61, 71	retancld 16320	. . . 4 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (tan‘𝐴) ∈ ℝ)
73		tangtx 26841	. . . . 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 11463	. . 3 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 𝐴 ≤ (tan‘𝐴))
76		ltle 11403	. . . . . 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 15597	. . 3 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘𝐴) = 𝐴)
80		0red 11316	. . . . 5 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 ∈ ℝ)
81	80, 61, 72, 78, 75	letrd 11472	. . . 4 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → 0 ≤ (tan‘𝐴))
82	72, 81	absidd 15597	. . 3 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘(tan‘𝐴)) = (tan‘𝐴))
83	75, 79, 82	3brtr4d 5136	. 2 ⊢ ((𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 0 < 𝐴) → (abs‘𝐴) ≤ (abs‘(tan‘𝐴)))
84		lttri4 11399	. . 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 1569 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 ‘cfv 6541 (class class class)co 7422 ℂcc 11203 ℝcr 11204 0cc0 11205 ℝ^*cxr 11347 < clt 11348 ≤ cle 11349 -cneg 11547 / cdiv 11978 2c2 12402 (,)cioo 13481 abscabs 15408 sincsin 16236 cosccos 16237 tanctan 16238 πcpi 16239

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-inf2 9647 ax-cnex 11261 ax-resscn 11262 ax-1cn 11263 ax-icn 11264 ax-addcl 11265 ax-addrcl 11266 ax-mulcl 11267 ax-mulrcl 11268 ax-mulcom 11269 ax-addass 11270 ax-mulass 11271 ax-distr 11272 ax-i2m1 11273 ax-1ne0 11274 ax-1rid 11275 ax-rnegex 11276 ax-rrecex 11277 ax-cnre 11278 ax-pre-lttri 11279 ax-pre-lttrn 11280 ax-pre-ltadd 11281 ax-pre-mulgt0 11282 ax-pre-sup 11283 ax-addf 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7379 df-ov 7425 df-oprab 7426 df-mpo 7427 df-of 7695 df-om 7880 df-1st 8003 df-2nd 8004 df-supp 8182 df-frecs 8304 df-wrecs 8335 df-recs 8384 df-rdg 8423 df-1o 8481 df-2o 8482 df-er 8722 df-map 8854 df-pm 8855 df-ixp 8931 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-fsupp 9359 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9509 df-card 10025 df-pnf 11350 df-mnf 11351 df-xr 11352 df-ltxr 11353 df-le 11354 df-sub 11548 df-neg 11549 df-div 11979 df-nn 12341 df-2 12410 df-3 12411 df-4 12412 df-5 12413 df-6 12414 df-7 12415 df-8 12416 df-9 12417 df-n0 12612 df-z 12699 df-dec 12820 df-uz 12971 df-q 13081 df-rp 13126 df-xneg 13246 df-xadd 13247 df-xmul 13248 df-ioo 13485 df-ioc 13486 df-ico 13487 df-icc 13488 df-fz 13645 df-fzo 13794 df-fl 13937 df-seq 14150 df-exp 14210 df-fac 14423 df-bc 14452 df-hash 14480 df-shft 15227 df-cj 15273 df-re 15274 df-im 15275 df-sqrt 15409 df-abs 15410 df-limsup 15645 df-clim 15662 df-rlim 15663 df-sum 15861 df-ef 16240 df-sin 16242 df-cos 16243 df-tan 16244 df-pi 16245 df-struct 17332 df-sets 17349 df-slot 17367 df-ndx 17379 df-base 17395 df-ress 17416 df-plusg 17448 df-mulr 17449 df-starv 17450 df-sca 17451 df-vsca 17452 df-ip 17453 df-tset 17454 df-ple 17455 df-ds 17457 df-unif 17458 df-hom 17459 df-cco 17460 df-rest 17600 df-topn 17601 df-0g 17619 df-gsum 17620 df-topgen 17621 df-pt 17622 df-prds 17625 df-xrs 17681 df-qtop 17686 df-imas 17687 df-xps 17689 df-mre 17763 df-mrc 17764 df-acs 17766 df-mgm 18823 df-sgrp 18915 df-mnd 18931 df-submnd 18986 df-mulg 19285 df-cntz 19538 df-cmn 20003 df-psmet 21677 df-xmet 21678 df-met 21679 df-bl 21680 df-mopn 21681 df-fbas 21682 df-fg 21683 df-cnfld 21686 df-top 23219 df-topon 23236 df-topsp 23258 df-bases 23271 df-cld 23344 df-ntr 23345 df-cls 23346 df-nei 23423 df-lp 23461 df-perf 23462 df-cn 23552 df-cnp 23553 df-haus 23640 df-tx 23888 df-hmeo 24081 df-fil 24172 df-fm 24264 df-flim 24265 df-flf 24266 df-xms 24646 df-ms 24647 df-tms 24648 df-cncf 25206 df-limc 26193 df-dv 26194

This theorem is used by: logcnlem4 26980

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