tanord - Metamath Proof Explorer

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

Theorem tanord 26714

Description: The tangent function is strictly increasing on its principal domain. (Contributed by Mario Carneiro, 4-Apr-2015.)

**Assertion**
Ref	Expression
tanord	⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐵 ∈ (-(π / 2)(,)(π / 2))) → (𝐴 < 𝐵 ↔ (tan‘𝐴) < (tan‘𝐵)))

Proof of Theorem tanord Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tru 1573	. 2 ⊢ ⊤
2		fveq2 6881	. . 3 ⊢ (𝑥 = 𝑦 → (tan‘𝑥) = (tan‘𝑦))
3		fveq2 6881	. . 3 ⊢ (𝑥 = 𝐴 → (tan‘𝑥) = (tan‘𝐴))
4		fveq2 6881	. . 3 ⊢ (𝑥 = 𝐵 → (tan‘𝑥) = (tan‘𝐵))
5		ioossre 13440	. . 3 ⊢ (-(π / 2)(,)(π / 2)) ⊆ ℝ
6		elioore 13408	. . . . 5 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → 𝑥 ∈ ℝ)
7	6	recnd 11243	. . . . . 6 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → 𝑥 ∈ ℂ)
8	6	rered 15282	. . . . . . 7 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝑥) = 𝑥)
9		id 23	. . . . . . 7 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → 𝑥 ∈ (-(π / 2)(,)(π / 2)))
10	8, 9	eqeltrd 2862	. . . . . 6 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝑥) ∈ (-(π / 2)(,)(π / 2)))
11		cosne0 26705	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ (ℜ‘𝑥) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝑥) ≠ 0)
12	7, 10, 11	syl2anc 595	. . . . 5 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (cos‘𝑥) ≠ 0)
13	6, 12	retancld 16207	. . . 4 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (tan‘𝑥) ∈ ℝ)
14	13	adantl 486	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (-(π / 2)(,)(π / 2))) → (tan‘𝑥) ∈ ℝ)
15	6	3ad2ant1 1150	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ)
16	15	adantr 485	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑥 ∈ ℝ)
17	16	recnd 11243	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑥 ∈ ℂ)
18	17	negnegd 11566	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → --𝑥 = 𝑥)
19	18	fveq2d 6885	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘--𝑥) = (tan‘𝑥))
20	17	negcld 11562	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → -𝑥 ∈ ℂ)
21		cosneg 16209	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → (cos‘-𝑥) = (cos‘𝑥))
22	17, 21	syl 18	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (cos‘-𝑥) = (cos‘𝑥))
23		simpl1 1209	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑥 ∈ (-(π / 2)(,)(π / 2)))
24	23, 12	syl 18	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (cos‘𝑥) ≠ 0)
25	22, 24	eqnetrd 3024	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (cos‘-𝑥) ≠ 0)
26		tanneg 16210	. . . . . . . . 9 ⊢ ((-𝑥 ∈ ℂ ∧ (cos‘-𝑥) ≠ 0) → (tan‘--𝑥) = -(tan‘-𝑥))
27	20, 25, 26	syl2anc 595	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘--𝑥) = -(tan‘-𝑥))
28	19, 27	eqtr3d 2799	. . . . . . 7 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘𝑥) = -(tan‘-𝑥))
29	15	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑥 ∈ ℝ)
30	29	renegcld 11647	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -𝑥 ∈ ℝ)
31	25	adantrr 729	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (cos‘-𝑥) ≠ 0)
32	30, 31	retancld 16207	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘-𝑥) ∈ ℝ)
33	32	renegcld 11647	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(tan‘-𝑥) ∈ ℝ)
34		0red 11217	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 ∈ ℝ)
35		simp2 1154	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
36	5, 35	sselid 3934	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ)
37	36	adantr 485	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 ∈ ℝ)
38		simpl2 1210	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
39		elioore 13408	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → 𝑦 ∈ ℝ)
40	39	recnd 11243	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → 𝑦 ∈ ℂ)
41	39	rered 15282	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝑦) = 𝑦)
42		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
43	41, 42	eqeltrd 2862	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝑦) ∈ (-(π / 2)(,)(π / 2)))
44		cosne0 26705	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℂ ∧ (ℜ‘𝑦) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝑦) ≠ 0)
45	40, 43, 44	syl2anc 595	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → (cos‘𝑦) ≠ 0)
46	38, 45	syl 18	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (cos‘𝑦) ≠ 0)
47	37, 46	retancld 16207	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘𝑦) ∈ ℝ)
48		simprl 782	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑥 < 0)
49	29	lt0neg1d 11789	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (𝑥 < 0 ↔ 0 < -𝑥))
50	48, 49	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < -𝑥)
51		simpl1 1209	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑥 ∈ (-(π / 2)(,)(π / 2)))
52		eliooord 13438	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (-(π / 2) < 𝑥 ∧ 𝑥 < (π / 2)))
53	51, 52	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (-(π / 2) < 𝑥 ∧ 𝑥 < (π / 2)))
54	53	simpld 499	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(π / 2) < 𝑥)
55		halfpire 26640	. . . . . . . . . . . . . . . 16 ⊢ (π / 2) ∈ ℝ
56		ltnegcon1 11721	. . . . . . . . . . . . . . . 16 ⊢ (((π / 2) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (-(π / 2) < 𝑥 ↔ -𝑥 < (π / 2)))
57	55, 29, 56	sylancr 598	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (-(π / 2) < 𝑥 ↔ -𝑥 < (π / 2)))
58	54, 57	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -𝑥 < (π / 2))
59		0xr 11262	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ^*
60	55	rexri 11273	. . . . . . . . . . . . . . 15 ⊢ (π / 2) ∈ ℝ^*
61		elioo2 13419	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (-𝑥 ∈ (0(,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 < -𝑥 ∧ -𝑥 < (π / 2))))
62	59, 60, 61	mp2an 704	. . . . . . . . . . . . . 14 ⊢ (-𝑥 ∈ (0(,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 < -𝑥 ∧ -𝑥 < (π / 2)))
63	30, 50, 58, 62	syl3anbrc 1361	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -𝑥 ∈ (0(,)(π / 2)))
64		tanrpcl 26680	. . . . . . . . . . . . 13 ⊢ (-𝑥 ∈ (0(,)(π / 2)) → (tan‘-𝑥) ∈ ℝ⁺)
65	63, 64	syl 18	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘-𝑥) ∈ ℝ⁺)
66	65	rpgt0d 13069	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < (tan‘-𝑥))
67	32	lt0neg2d 11790	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (0 < (tan‘-𝑥) ↔ -(tan‘-𝑥) < 0))
68	66, 67	mpbid 235	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(tan‘-𝑥) < 0)
69		simprr 784	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < 𝑦)
70		eliooord 13438	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → (-(π / 2) < 𝑦 ∧ 𝑦 < (π / 2)))
71	38, 70	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (-(π / 2) < 𝑦 ∧ 𝑦 < (π / 2)))
72	71	simprd 500	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 < (π / 2))
73		elioo2 13419	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (𝑦 ∈ (0(,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦 ∧ 𝑦 < (π / 2))))
74	59, 60, 73	mp2an 704	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦 ∧ 𝑦 < (π / 2)))
75	37, 69, 72, 74	syl3anbrc 1361	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 ∈ (0(,)(π / 2)))
76		tanrpcl 26680	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(,)(π / 2)) → (tan‘𝑦) ∈ ℝ⁺)
77	75, 76	syl 18	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘𝑦) ∈ ℝ⁺)
78	77	rpgt0d 13069	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < (tan‘𝑦))
79	33, 34, 47, 68, 78	lttrd 11377	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(tan‘-𝑥) < (tan‘𝑦))
80	79	anassrs 472	. . . . . . . 8 ⊢ ((((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) ∧ 0 < 𝑦) → -(tan‘-𝑥) < (tan‘𝑦))
81		simpl3 1211	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 < 𝑦)
82	15	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 ∈ ℝ)
83	36	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ∈ ℝ)
84	82, 83	ltnegd 11798	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (𝑥 < 𝑦 ↔ -𝑦 < -𝑥))
85	81, 84	mpbid 235	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 < -𝑥)
86	83	renegcld 11647	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 ∈ ℝ)
87		simpr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ≤ 0)
88	83	le0neg1d 11791	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (𝑦 ≤ 0 ↔ 0 ≤ -𝑦))
89	87, 88	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 0 ≤ -𝑦)
90		simpl2 1210	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
91	90, 70	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (-(π / 2) < 𝑦 ∧ 𝑦 < (π / 2)))
92	91	simpld 499	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(π / 2) < 𝑦)
93		ltnegcon1 11721	. . . . . . . . . . . . . . . 16 ⊢ (((π / 2) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (-(π / 2) < 𝑦 ↔ -𝑦 < (π / 2)))
94	55, 83, 93	sylancr 598	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (-(π / 2) < 𝑦 ↔ -𝑦 < (π / 2)))
95	92, 94	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 < (π / 2))
96		0re 11216	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
97		elico2 13443	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (-𝑦 ∈ (0[,)(π / 2)) ↔ (-𝑦 ∈ ℝ ∧ 0 ≤ -𝑦 ∧ -𝑦 < (π / 2))))
98	96, 60, 97	mp2an 704	. . . . . . . . . . . . . 14 ⊢ (-𝑦 ∈ (0[,)(π / 2)) ↔ (-𝑦 ∈ ℝ ∧ 0 ≤ -𝑦 ∧ -𝑦 < (π / 2)))
99	86, 89, 95, 98	syl3anbrc 1361	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 ∈ (0[,)(π / 2)))
100	82	renegcld 11647	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑥 ∈ ℝ)
101		simp3 1155	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
102		0red 11217	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 0 ∈ ℝ)
103		ltletr 11308	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → ((𝑥 < 𝑦 ∧ 𝑦 ≤ 0) → 𝑥 < 0))
104	15, 36, 102, 103	syl3anc 1397	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → ((𝑥 < 𝑦 ∧ 𝑦 ≤ 0) → 𝑥 < 0))
105	101, 104	mpand 707	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → (𝑦 ≤ 0 → 𝑥 < 0))
106	105	imp 411	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 < 0)
107		ltle 11304	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑥 < 0 → 𝑥 ≤ 0))
108	82, 96, 107	sylancl 597	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (𝑥 < 0 → 𝑥 ≤ 0))
109	106, 108	mpd 16	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 ≤ 0)
110	82	le0neg1d 11791	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (𝑥 ≤ 0 ↔ 0 ≤ -𝑥))
111	109, 110	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 0 ≤ -𝑥)
112		simpl1 1209	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 ∈ (-(π / 2)(,)(π / 2)))
113	112, 52	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (-(π / 2) < 𝑥 ∧ 𝑥 < (π / 2)))
114	113	simpld 499	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(π / 2) < 𝑥)
115	55, 82, 56	sylancr 598	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (-(π / 2) < 𝑥 ↔ -𝑥 < (π / 2)))
116	114, 115	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑥 < (π / 2))
117		elico2 13443	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (-𝑥 ∈ (0[,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 ≤ -𝑥 ∧ -𝑥 < (π / 2))))
118	96, 60, 117	mp2an 704	. . . . . . . . . . . . . 14 ⊢ (-𝑥 ∈ (0[,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 ≤ -𝑥 ∧ -𝑥 < (π / 2)))
119	100, 111, 116, 118	syl3anbrc 1361	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑥 ∈ (0[,)(π / 2)))
120		tanord1 26713	. . . . . . . . . . . . 13 ⊢ ((-𝑦 ∈ (0[,)(π / 2)) ∧ -𝑥 ∈ (0[,)(π / 2))) → (-𝑦 < -𝑥 ↔ (tan‘-𝑦) < (tan‘-𝑥)))
121	99, 119, 120	syl2anc 595	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (-𝑦 < -𝑥 ↔ (tan‘-𝑦) < (tan‘-𝑥)))
122	85, 121	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘-𝑦) < (tan‘-𝑥))
123	83	recnd 11243	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ∈ ℂ)
124		cosneg 16209	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℂ → (cos‘-𝑦) = (cos‘𝑦))
125	123, 124	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘-𝑦) = (cos‘𝑦))
126	90, 45	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘𝑦) ≠ 0)
127	125, 126	eqnetrd 3024	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘-𝑦) ≠ 0)
128	86, 127	retancld 16207	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘-𝑦) ∈ ℝ)
129	106, 25	syldan 602	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘-𝑥) ≠ 0)
130	100, 129	retancld 16207	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘-𝑥) ∈ ℝ)
131	128, 130	ltnegd 11798	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → ((tan‘-𝑦) < (tan‘-𝑥) ↔ -(tan‘-𝑥) < -(tan‘-𝑦)))
132	122, 131	mpbid 235	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(tan‘-𝑥) < -(tan‘-𝑦))
133	123	negnegd 11566	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → --𝑦 = 𝑦)
134	133	fveq2d 6885	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘--𝑦) = (tan‘𝑦))
135	123	negcld 11562	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 ∈ ℂ)
136		tanneg 16210	. . . . . . . . . . . 12 ⊢ ((-𝑦 ∈ ℂ ∧ (cos‘-𝑦) ≠ 0) → (tan‘--𝑦) = -(tan‘-𝑦))
137	135, 127, 136	syl2anc 595	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘--𝑦) = -(tan‘-𝑦))
138	134, 137	eqtr3d 2799	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘𝑦) = -(tan‘-𝑦))
139	132, 138	breqtrrd 5138	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(tan‘-𝑥) < (tan‘𝑦))
140	139	adantlr 727	. . . . . . . 8 ⊢ ((((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) ∧ 𝑦 ≤ 0) → -(tan‘-𝑥) < (tan‘𝑦))
141		0red 11217	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 0 ∈ ℝ)
142		simpl2 1210	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
143	5, 142	sselid 3934	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑦 ∈ ℝ)
144	80, 140, 141, 143	ltlecasei 11324	. . . . . . 7 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → -(tan‘-𝑥) < (tan‘𝑦))
145	28, 144	eqbrtrd 5132	. . . . . 6 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘𝑥) < (tan‘𝑦))
146		simpl3 1211	. . . . . . 7 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 < 𝑦)
147	15	adantr 485	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ∈ ℝ)
148		simpr 489	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 0 ≤ 𝑥)
149		simpl1 1209	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ∈ (-(π / 2)(,)(π / 2)))
150	149, 52	syl 18	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → (-(π / 2) < 𝑥 ∧ 𝑥 < (π / 2)))
151	150	simprd 500	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 < (π / 2))
152		elico2 13443	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (𝑥 ∈ (0[,)(π / 2)) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < (π / 2))))
153	96, 60, 152	mp2an 704	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,)(π / 2)) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < (π / 2)))
154	147, 148, 151, 153	syl3anbrc 1361	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ∈ (0[,)(π / 2)))
155		simpl2 1210	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
156	5, 155	sselid 3934	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ ℝ)
157		0red 11217	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 0 ∈ ℝ)
158	147, 156, 146	ltled 11364	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ≤ 𝑦)
159	157, 147, 156, 148, 158	letrd 11373	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 0 ≤ 𝑦)
160	155, 70	syl 18	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → (-(π / 2) < 𝑦 ∧ 𝑦 < (π / 2)))
161	160	simprd 500	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 < (π / 2))
162		elico2 13443	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (𝑦 ∈ (0[,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ∧ 𝑦 < (π / 2))))
163	96, 60, 162	mp2an 704	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ∧ 𝑦 < (π / 2)))
164	156, 159, 161, 163	syl3anbrc 1361	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ (0[,)(π / 2)))
165		tanord1 26713	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,)(π / 2)) ∧ 𝑦 ∈ (0[,)(π / 2))) → (𝑥 < 𝑦 ↔ (tan‘𝑥) < (tan‘𝑦)))
166	154, 164, 165	syl2anc 595	. . . . . . 7 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → (𝑥 < 𝑦 ↔ (tan‘𝑥) < (tan‘𝑦)))
167	146, 166	mpbid 235	. . . . . 6 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → (tan‘𝑥) < (tan‘𝑦))
168	145, 167, 15, 102	ltlecasei 11324	. . . . 5 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → (tan‘𝑥) < (tan‘𝑦))
169	168	3expia 1138	. . . 4 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2))) → (𝑥 < 𝑦 → (tan‘𝑥) < (tan‘𝑦)))
170	169	adantl 486	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)))) → (𝑥 < 𝑦 → (tan‘𝑥) < (tan‘𝑦)))
171	2, 3, 4, 5, 14, 170	ltord1 11746	. 2 ⊢ ((⊤ ∧ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐵 ∈ (-(π / 2)(,)(π / 2)))) → (𝐴 < 𝐵 ↔ (tan‘𝐴) < (tan‘𝐵)))
172	1, 171	mpan 702	1 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐵 ∈ (-(π / 2)(,)(π / 2))) → (𝐴 < 𝐵 ↔ (tan‘𝐴) < (tan‘𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1102 = wceq 1569 ⊤wtru 1570 ∈ wcel 2142 ≠ wne 2957 class class class wbr 5108 ‘cfv 6536 (class class class)co 7412 ℂcc 11104 ℝcr 11105 0cc0 11106 ℝ^*cxr 11248 < clt 11249 ≤ cle 11250 -cneg 11448 / cdiv 11877 2c2 12301 ℝ⁺crp 13022 (,)cioo 13378 [,)cico 13380 ℜcre 15155 cosccos 16124 tanctan 16125 πcpi 16126

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 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 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-er 8692 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-fl 13832 df-mod 13910 df-seq 14045 df-exp 14105 df-fac 14317 df-bc 14346 df-hash 14374 df-shft 15111 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-limsup 15529 df-clim 15546 df-rlim 15547 df-sum 15745 df-ef 16127 df-sin 16129 df-cos 16130 df-tan 16131 df-pi 16132 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-rest 17481 df-topn 17482 df-0g 17500 df-gsum 17501 df-topgen 17502 df-pt 17503 df-prds 17506 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-submnd 18848 df-mulg 19140 df-cntz 19393 df-cmn 19858 df-psmet 21525 df-xmet 21526 df-met 21527 df-bl 21528 df-mopn 21529 df-fbas 21530 df-fg 21531 df-cnfld 21534 df-top 23062 df-topon 23079 df-topsp 23101 df-bases 23114 df-cld 23187 df-ntr 23188 df-cls 23189 df-nei 23266 df-lp 23304 df-perf 23305 df-cn 23395 df-cnp 23396 df-haus 23483 df-tx 23730 df-hmeo 23923 df-fil 24014 df-fm 24106 df-flim 24107 df-flf 24108 df-xms 24488 df-ms 24489 df-tms 24490 df-cncf 25048 df-limc 26036 df-dv 26037

This theorem is used by: atanlogsublem 27091 atanord 27103 basellem4 27259

W3C validator