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Theorem tanord 26504

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 1544	. 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 13429	. . 3 ⊢ (-(π / 2)(,)(π / 2)) ⊆ ℝ
6		elioore 13397	. . . . 5 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → 𝑥 ∈ ℝ)
7	6	recnd 11268	. . . . . 6 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → 𝑥 ∈ ℂ)
8	6	rered 15248	. . . . . . 7 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝑥) = 𝑥)
9		id 22	. . . . . . 7 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → 𝑥 ∈ (-(π / 2)(,)(π / 2)))
10	8, 9	eqeltrd 2835	. . . . . 6 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝑥) ∈ (-(π / 2)(,)(π / 2)))
11		cosne0 26495	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ (ℜ‘𝑥) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝑥) ≠ 0)
12	7, 10, 11	syl2anc 584	. . . . 5 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (cos‘𝑥) ≠ 0)
13	6, 12	retancld 16168	. . . 4 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (tan‘𝑥) ∈ ℝ)
14	13	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (-(π / 2)(,)(π / 2))) → (tan‘𝑥) ∈ ℝ)
15	6	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ)
16	15	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑥 ∈ ℝ)
17	16	recnd 11268	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑥 ∈ ℂ)
18	17	negnegd 11590	. . . . . . . . 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 11586	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → -𝑥 ∈ ℂ)
21		cosneg 16170	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → (cos‘-𝑥) = (cos‘𝑥))
22	17, 21	syl 17	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (cos‘-𝑥) = (cos‘𝑥))
23		simpl1 1192	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑥 ∈ (-(π / 2)(,)(π / 2)))
24	23, 12	syl 17	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (cos‘𝑥) ≠ 0)
25	22, 24	eqnetrd 3000	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (cos‘-𝑥) ≠ 0)
26		tanneg 16171	. . . . . . . . 9 ⊢ ((-𝑥 ∈ ℂ ∧ (cos‘-𝑥) ≠ 0) → (tan‘--𝑥) = -(tan‘-𝑥))
27	20, 25, 26	syl2anc 584	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘--𝑥) = -(tan‘-𝑥))
28	19, 27	eqtr3d 2773	. . . . . . 7 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘𝑥) = -(tan‘-𝑥))
29	15	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑥 ∈ ℝ)
30	29	renegcld 11669	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -𝑥 ∈ ℝ)
31	25	adantrr 717	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (cos‘-𝑥) ≠ 0)
32	30, 31	retancld 16168	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘-𝑥) ∈ ℝ)
33	32	renegcld 11669	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(tan‘-𝑥) ∈ ℝ)
34		0red 11243	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 ∈ ℝ)
35		simp2 1137	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
36	5, 35	sselid 3961	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ)
37	36	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 ∈ ℝ)
38		simpl2 1193	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
39		elioore 13397	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → 𝑦 ∈ ℝ)
40	39	recnd 11268	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → 𝑦 ∈ ℂ)
41	39	rered 15248	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝑦) = 𝑦)
42		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
43	41, 42	eqeltrd 2835	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝑦) ∈ (-(π / 2)(,)(π / 2)))
44		cosne0 26495	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℂ ∧ (ℜ‘𝑦) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝑦) ≠ 0)
45	40, 43, 44	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → (cos‘𝑦) ≠ 0)
46	38, 45	syl 17	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (cos‘𝑦) ≠ 0)
47	37, 46	retancld 16168	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘𝑦) ∈ ℝ)
48		simprl 770	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑥 < 0)
49	29	lt0neg1d 11811	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < -𝑥)
51		simpl1 1192	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑥 ∈ (-(π / 2)(,)(π / 2)))
52		eliooord 13427	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (-(π / 2) < 𝑥 ∧ 𝑥 < (π / 2)))
53	51, 52	syl 17	. . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(π / 2) < 𝑥)
55		halfpire 26430	. . . . . . . . . . . . . . . 16 ⊢ (π / 2) ∈ ℝ
56		ltnegcon1 11743	. . . . . . . . . . . . . . . 16 ⊢ (((π / 2) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (-(π / 2) < 𝑥 ↔ -𝑥 < (π / 2)))
57	55, 29, 56	sylancr 587	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -𝑥 < (π / 2))
59		0xr 11287	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ^*
60	55	rexri 11298	. . . . . . . . . . . . . . 15 ⊢ (π / 2) ∈ ℝ^*
61		elioo2 13408	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (-𝑥 ∈ (0(,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 < -𝑥 ∧ -𝑥 < (π / 2))))
62	59, 60, 61	mp2an 692	. . . . . . . . . . . . . 14 ⊢ (-𝑥 ∈ (0(,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 < -𝑥 ∧ -𝑥 < (π / 2)))
63	30, 50, 58, 62	syl3anbrc 1344	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -𝑥 ∈ (0(,)(π / 2)))
64		tanrpcl 26470	. . . . . . . . . . . . 13 ⊢ (-𝑥 ∈ (0(,)(π / 2)) → (tan‘-𝑥) ∈ ℝ⁺)
65	63, 64	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘-𝑥) ∈ ℝ⁺)
66	65	rpgt0d 13059	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < (tan‘-𝑥))
67	32	lt0neg2d 11812	. . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(tan‘-𝑥) < 0)
69		simprr 772	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < 𝑦)
70		eliooord 13427	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → (-(π / 2) < 𝑦 ∧ 𝑦 < (π / 2)))
71	38, 70	syl 17	. . . . . . . . . . . . . 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 495	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 < (π / 2))
73		elioo2 13408	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (𝑦 ∈ (0(,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦 ∧ 𝑦 < (π / 2))))
74	59, 60, 73	mp2an 692	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦 ∧ 𝑦 < (π / 2)))
75	37, 69, 72, 74	syl3anbrc 1344	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 ∈ (0(,)(π / 2)))
76		tanrpcl 26470	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0(,)(π / 2)) → (tan‘𝑦) ∈ ℝ⁺)
77	75, 76	syl 17	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘𝑦) ∈ ℝ⁺)
78	77	rpgt0d 13059	. . . . . . . . . 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 11401	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(tan‘-𝑥) < (tan‘𝑦))
80	79	anassrs 467	. . . . . . . 8 ⊢ ((((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) ∧ 0 < 𝑦) → -(tan‘-𝑥) < (tan‘𝑦))
81		simpl3 1194	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 < 𝑦)
82	15	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 ∈ ℝ)
83	36	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ∈ ℝ)
84	82, 83	ltnegd 11820	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (𝑥 < 𝑦 ↔ -𝑦 < -𝑥))
85	81, 84	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 < -𝑥)
86	83	renegcld 11669	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 ∈ ℝ)
87		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ≤ 0)
88	83	le0neg1d 11813	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (𝑦 ≤ 0 ↔ 0 ≤ -𝑦))
89	87, 88	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 0 ≤ -𝑦)
90		simpl2 1193	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
91	90, 70	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (-(π / 2) < 𝑦 ∧ 𝑦 < (π / 2)))
92	91	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(π / 2) < 𝑦)
93		ltnegcon1 11743	. . . . . . . . . . . . . . . 16 ⊢ (((π / 2) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (-(π / 2) < 𝑦 ↔ -𝑦 < (π / 2)))
94	55, 83, 93	sylancr 587	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 < (π / 2))
96		0re 11242	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
97		elico2 13432	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (-𝑦 ∈ (0[,)(π / 2)) ↔ (-𝑦 ∈ ℝ ∧ 0 ≤ -𝑦 ∧ -𝑦 < (π / 2))))
98	96, 60, 97	mp2an 692	. . . . . . . . . . . . . 14 ⊢ (-𝑦 ∈ (0[,)(π / 2)) ↔ (-𝑦 ∈ ℝ ∧ 0 ≤ -𝑦 ∧ -𝑦 < (π / 2)))
99	86, 89, 95, 98	syl3anbrc 1344	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 ∈ (0[,)(π / 2)))
100	82	renegcld 11669	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑥 ∈ ℝ)
101		simp3 1138	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
102		0red 11243	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 0 ∈ ℝ)
103		ltletr 11332	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → ((𝑥 < 𝑦 ∧ 𝑦 ≤ 0) → 𝑥 < 0))
104	15, 36, 102, 103	syl3anc 1373	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → ((𝑥 < 𝑦 ∧ 𝑦 ≤ 0) → 𝑥 < 0))
105	101, 104	mpand 695	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → (𝑦 ≤ 0 → 𝑥 < 0))
106	105	imp 406	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 < 0)
107		ltle 11328	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑥 < 0 → 𝑥 ≤ 0))
108	82, 96, 107	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (𝑥 < 0 → 𝑥 ≤ 0))
109	106, 108	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 ≤ 0)
110	82	le0neg1d 11813	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (𝑥 ≤ 0 ↔ 0 ≤ -𝑥))
111	109, 110	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 0 ≤ -𝑥)
112		simpl1 1192	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 ∈ (-(π / 2)(,)(π / 2)))
113	112, 52	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (-(π / 2) < 𝑥 ∧ 𝑥 < (π / 2)))
114	113	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(π / 2) < 𝑥)
115	55, 82, 56	sylancr 587	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑥 < (π / 2))
117		elico2 13432	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (-𝑥 ∈ (0[,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 ≤ -𝑥 ∧ -𝑥 < (π / 2))))
118	96, 60, 117	mp2an 692	. . . . . . . . . . . . . 14 ⊢ (-𝑥 ∈ (0[,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 ≤ -𝑥 ∧ -𝑥 < (π / 2)))
119	100, 111, 116, 118	syl3anbrc 1344	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑥 ∈ (0[,)(π / 2)))
120		tanord1 26503	. . . . . . . . . . . . 13 ⊢ ((-𝑦 ∈ (0[,)(π / 2)) ∧ -𝑥 ∈ (0[,)(π / 2))) → (-𝑦 < -𝑥 ↔ (tan‘-𝑦) < (tan‘-𝑥)))
121	99, 119, 120	syl2anc 584	. . . . . . . . . . . 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 232	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘-𝑦) < (tan‘-𝑥))
123	83	recnd 11268	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ∈ ℂ)
124		cosneg 16170	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℂ → (cos‘-𝑦) = (cos‘𝑦))
125	123, 124	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘-𝑦) = (cos‘𝑦))
126	90, 45	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘𝑦) ≠ 0)
127	125, 126	eqnetrd 3000	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘-𝑦) ≠ 0)
128	86, 127	retancld 16168	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘-𝑦) ∈ ℝ)
129	106, 25	syldan 591	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘-𝑥) ≠ 0)
130	100, 129	retancld 16168	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘-𝑥) ∈ ℝ)
131	128, 130	ltnegd 11820	. . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(tan‘-𝑥) < -(tan‘-𝑦))
133	123	negnegd 11590	. . . . . . . . . . . 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 11586	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 ∈ ℂ)
136		tanneg 16171	. . . . . . . . . . . 12 ⊢ ((-𝑦 ∈ ℂ ∧ (cos‘-𝑦) ≠ 0) → (tan‘--𝑦) = -(tan‘-𝑦))
137	135, 127, 136	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘--𝑦) = -(tan‘-𝑦))
138	134, 137	eqtr3d 2773	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘𝑦) = -(tan‘-𝑦))
139	132, 138	breqtrrd 5152	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(tan‘-𝑥) < (tan‘𝑦))
140	139	adantlr 715	. . . . . . . 8 ⊢ ((((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) ∧ 𝑦 ≤ 0) → -(tan‘-𝑥) < (tan‘𝑦))
141		0red 11243	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 0 ∈ ℝ)
142		simpl2 1193	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
143	5, 142	sselid 3961	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑦 ∈ ℝ)
144	80, 140, 141, 143	ltlecasei 11348	. . . . . . 7 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → -(tan‘-𝑥) < (tan‘𝑦))
145	28, 144	eqbrtrd 5146	. . . . . 6 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘𝑥) < (tan‘𝑦))
146		simpl3 1194	. . . . . . 7 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 < 𝑦)
147	15	adantr 480	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ∈ ℝ)
148		simpr 484	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 0 ≤ 𝑥)
149		simpl1 1192	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ∈ (-(π / 2)(,)(π / 2)))
150	149, 52	syl 17	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → (-(π / 2) < 𝑥 ∧ 𝑥 < (π / 2)))
151	150	simprd 495	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 < (π / 2))
152		elico2 13432	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (𝑥 ∈ (0[,)(π / 2)) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < (π / 2))))
153	96, 60, 152	mp2an 692	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,)(π / 2)) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < (π / 2)))
154	147, 148, 151, 153	syl3anbrc 1344	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ∈ (0[,)(π / 2)))
155		simpl2 1193	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
156	5, 155	sselid 3961	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ ℝ)
157		0red 11243	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 0 ∈ ℝ)
158	147, 156, 146	ltled 11388	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ≤ 𝑦)
159	157, 147, 156, 148, 158	letrd 11397	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 0 ≤ 𝑦)
160	155, 70	syl 17	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → (-(π / 2) < 𝑦 ∧ 𝑦 < (π / 2)))
161	160	simprd 495	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 < (π / 2))
162		elico2 13432	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (𝑦 ∈ (0[,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ∧ 𝑦 < (π / 2))))
163	96, 60, 162	mp2an 692	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ∧ 𝑦 < (π / 2)))
164	156, 159, 161, 163	syl3anbrc 1344	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ (0[,)(π / 2)))
165		tanord1 26503	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,)(π / 2)) ∧ 𝑦 ∈ (0[,)(π / 2))) → (𝑥 < 𝑦 ↔ (tan‘𝑥) < (tan‘𝑦)))
166	154, 164, 165	syl2anc 584	. . . . . . 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 232	. . . . . 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 11348	. . . . 5 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → (tan‘𝑥) < (tan‘𝑦))
169	168	3expia 1121	. . . 4 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2))) → (𝑥 < 𝑦 → (tan‘𝑥) < (tan‘𝑦)))
170	169	adantl 481	. . 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 11768	. 2 ⊢ ((⊤ ∧ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐵 ∈ (-(π / 2)(,)(π / 2)))) → (𝐴 < 𝐵 ↔ (tan‘𝐴) < (tan‘𝐵)))
172	1, 171	mpan 690	1 ⊢ ((𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐵 ∈ (-(π / 2)(,)(π / 2))) → (𝐴 < 𝐵 ↔ (tan‘𝐴) < (tan‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ≠ wne 2933 class class class wbr 5124 ‘cfv 6536 (class class class)co 7410 ℂcc 11132 ℝcr 11133 0cc0 11134 ℝ^*cxr 11273 < clt 11274 ≤ cle 11275 -cneg 11472 / cdiv 11899 2c2 12300 ℝ⁺crp 13013 (,)cioo 13367 [,)cico 13369 ℜcre 15121 cosccos 16085 tanctan 16086 πcpi 16087

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 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 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 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 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-shft 15091 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-limsup 15492 df-clim 15509 df-rlim 15510 df-sum 15708 df-ef 16088 df-sin 16090 df-cos 16091 df-tan 16092 df-pi 16093 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-mulg 19056 df-cntz 19305 df-cmn 19768 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25824 df-dv 25825

This theorem is referenced by: atanlogsublem 26882 atanord 26894 basellem4 27051

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