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Theorem tanord 26505

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 1546	. 2 ⊢ ⊤
2		fveq2 6833	. . 3 ⊢ (𝑥 = 𝑦 → (tan‘𝑥) = (tan‘𝑦))
3		fveq2 6833	. . 3 ⊢ (𝑥 = 𝐴 → (tan‘𝑥) = (tan‘𝐴))
4		fveq2 6833	. . 3 ⊢ (𝑥 = 𝐵 → (tan‘𝑥) = (tan‘𝐵))
5		ioossre 13325	. . 3 ⊢ (-(π / 2)(,)(π / 2)) ⊆ ℝ
6		elioore 13293	. . . . 5 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → 𝑥 ∈ ℝ)
7	6	recnd 11162	. . . . . 6 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → 𝑥 ∈ ℂ)
8	6	rered 15149	. . . . . . 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 26496	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ (ℜ‘𝑥) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝑥) ≠ 0)
12	7, 10, 11	syl2anc 585	. . . . 5 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (cos‘𝑥) ≠ 0)
13	6, 12	retancld 16072	. . . 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 1134	. . . . . . . . . . . 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 11162	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑥 ∈ ℂ)
18	17	negnegd 11485	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → --𝑥 = 𝑥)
19	18	fveq2d 6837	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘--𝑥) = (tan‘𝑥))
20	17	negcld 11481	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → -𝑥 ∈ ℂ)
21		cosneg 16074	. . . . . . . . . . 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 1193	. . . . . . . . . . 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 2998	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (cos‘-𝑥) ≠ 0)
26		tanneg 16075	. . . . . . . . 9 ⊢ ((-𝑥 ∈ ℂ ∧ (cos‘-𝑥) ≠ 0) → (tan‘--𝑥) = -(tan‘-𝑥))
27	20, 25, 26	syl2anc 585	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘--𝑥) = -(tan‘-𝑥))
28	19, 27	eqtr3d 2772	. . . . . . 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 11566	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -𝑥 ∈ ℝ)
31	25	adantrr 718	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (cos‘-𝑥) ≠ 0)
32	30, 31	retancld 16072	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘-𝑥) ∈ ℝ)
33	32	renegcld 11566	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(tan‘-𝑥) ∈ ℝ)
34		0red 11137	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 ∈ ℝ)
35		simp2 1138	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
36	5, 35	sselid 3930	. . . . . . . . . . . 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 1194	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
39		elioore 13293	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → 𝑦 ∈ ℝ)
40	39	recnd 11162	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → 𝑦 ∈ ℂ)
41	39	rered 15149	. . . . . . . . . . . . . 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 26496	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℂ ∧ (ℜ‘𝑦) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝑦) ≠ 0)
45	40, 43, 44	syl2anc 585	. . . . . . . . . . . 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 16072	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘𝑦) ∈ ℝ)
48		simprl 771	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑥 < 0)
49	29	lt0neg1d 11708	. . . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑥 ∈ (-(π / 2)(,)(π / 2)))
52		eliooord 13323	. . . . . . . . . . . . . . . . 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 26431	. . . . . . . . . . . . . . . 16 ⊢ (π / 2) ∈ ℝ
56		ltnegcon1 11640	. . . . . . . . . . . . . . . 16 ⊢ (((π / 2) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (-(π / 2) < 𝑥 ↔ -𝑥 < (π / 2)))
57	55, 29, 56	sylancr 588	. . . . . . . . . . . . . . 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 11181	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ^*
60	55	rexri 11192	. . . . . . . . . . . . . . 15 ⊢ (π / 2) ∈ ℝ^*
61		elioo2 13304	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (-𝑥 ∈ (0(,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 < -𝑥 ∧ -𝑥 < (π / 2))))
62	59, 60, 61	mp2an 693	. . . . . . . . . . . . . 14 ⊢ (-𝑥 ∈ (0(,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 < -𝑥 ∧ -𝑥 < (π / 2)))
63	30, 50, 58, 62	syl3anbrc 1345	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -𝑥 ∈ (0(,)(π / 2)))
64		tanrpcl 26471	. . . . . . . . . . . . 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 12954	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < (tan‘-𝑥))
67	32	lt0neg2d 11709	. . . . . . . . . . 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 773	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < 𝑦)
70		eliooord 13323	. . . . . . . . . . . . . . 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 13304	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (𝑦 ∈ (0(,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦 ∧ 𝑦 < (π / 2))))
74	59, 60, 73	mp2an 693	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦 ∧ 𝑦 < (π / 2)))
75	37, 69, 72, 74	syl3anbrc 1345	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 ∈ (0(,)(π / 2)))
76		tanrpcl 26471	. . . . . . . . . . . 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 12954	. . . . . . . . . 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 11296	. . . . . . . . 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 1195	. . . . . . . . . . . . 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 11717	. . . . . . . . . . . . 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 11566	. . . . . . . . . . . . . 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 11710	. . . . . . . . . . . . . . 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 1194	. . . . . . . . . . . . . . . . 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 11640	. . . . . . . . . . . . . . . 16 ⊢ (((π / 2) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (-(π / 2) < 𝑦 ↔ -𝑦 < (π / 2)))
94	55, 83, 93	sylancr 588	. . . . . . . . . . . . . . 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 11136	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
97		elico2 13328	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (-𝑦 ∈ (0[,)(π / 2)) ↔ (-𝑦 ∈ ℝ ∧ 0 ≤ -𝑦 ∧ -𝑦 < (π / 2))))
98	96, 60, 97	mp2an 693	. . . . . . . . . . . . . 14 ⊢ (-𝑦 ∈ (0[,)(π / 2)) ↔ (-𝑦 ∈ ℝ ∧ 0 ≤ -𝑦 ∧ -𝑦 < (π / 2)))
99	86, 89, 95, 98	syl3anbrc 1345	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 ∈ (0[,)(π / 2)))
100	82	renegcld 11566	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑥 ∈ ℝ)
101		simp3 1139	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
102		0red 11137	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 0 ∈ ℝ)
103		ltletr 11227	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → ((𝑥 < 𝑦 ∧ 𝑦 ≤ 0) → 𝑥 < 0))
104	15, 36, 102, 103	syl3anc 1374	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → ((𝑥 < 𝑦 ∧ 𝑦 ≤ 0) → 𝑥 < 0))
105	101, 104	mpand 696	. . . . . . . . . . . . . . . . 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 11223	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑥 < 0 → 𝑥 ≤ 0))
108	82, 96, 107	sylancl 587	. . . . . . . . . . . . . . . 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 11710	. . . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . . 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 13328	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (-𝑥 ∈ (0[,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 ≤ -𝑥 ∧ -𝑥 < (π / 2))))
118	96, 60, 117	mp2an 693	. . . . . . . . . . . . . 14 ⊢ (-𝑥 ∈ (0[,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 ≤ -𝑥 ∧ -𝑥 < (π / 2)))
119	100, 111, 116, 118	syl3anbrc 1345	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑥 ∈ (0[,)(π / 2)))
120		tanord1 26504	. . . . . . . . . . . . 13 ⊢ ((-𝑦 ∈ (0[,)(π / 2)) ∧ -𝑥 ∈ (0[,)(π / 2))) → (-𝑦 < -𝑥 ↔ (tan‘-𝑦) < (tan‘-𝑥)))
121	99, 119, 120	syl2anc 585	. . . . . . . . . . . 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 11162	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ∈ ℂ)
124		cosneg 16074	. . . . . . . . . . . . . . 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 2998	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘-𝑦) ≠ 0)
128	86, 127	retancld 16072	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘-𝑦) ∈ ℝ)
129	106, 25	syldan 592	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘-𝑥) ≠ 0)
130	100, 129	retancld 16072	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘-𝑥) ∈ ℝ)
131	128, 130	ltnegd 11717	. . . . . . . . . . 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 11485	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → --𝑦 = 𝑦)
134	133	fveq2d 6837	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘--𝑦) = (tan‘𝑦))
135	123	negcld 11481	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 ∈ ℂ)
136		tanneg 16075	. . . . . . . . . . . 12 ⊢ ((-𝑦 ∈ ℂ ∧ (cos‘-𝑦) ≠ 0) → (tan‘--𝑦) = -(tan‘-𝑦))
137	135, 127, 136	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘--𝑦) = -(tan‘-𝑦))
138	134, 137	eqtr3d 2772	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘𝑦) = -(tan‘-𝑦))
139	132, 138	breqtrrd 5125	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(tan‘-𝑥) < (tan‘𝑦))
140	139	adantlr 716	. . . . . . . 8 ⊢ ((((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) ∧ 𝑦 ≤ 0) → -(tan‘-𝑥) < (tan‘𝑦))
141		0red 11137	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 0 ∈ ℝ)
142		simpl2 1194	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
143	5, 142	sselid 3930	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑦 ∈ ℝ)
144	80, 140, 141, 143	ltlecasei 11243	. . . . . . 7 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → -(tan‘-𝑥) < (tan‘𝑦))
145	28, 144	eqbrtrd 5119	. . . . . 6 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘𝑥) < (tan‘𝑦))
146		simpl3 1195	. . . . . . 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 1193	. . . . . . . . . . 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 13328	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (𝑥 ∈ (0[,)(π / 2)) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < (π / 2))))
153	96, 60, 152	mp2an 693	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,)(π / 2)) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < (π / 2)))
154	147, 148, 151, 153	syl3anbrc 1345	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ∈ (0[,)(π / 2)))
155		simpl2 1194	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
156	5, 155	sselid 3930	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ ℝ)
157		0red 11137	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 0 ∈ ℝ)
158	147, 156, 146	ltled 11283	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ≤ 𝑦)
159	157, 147, 156, 148, 158	letrd 11292	. . . . . . . . 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 13328	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (𝑦 ∈ (0[,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ∧ 𝑦 < (π / 2))))
163	96, 60, 162	mp2an 693	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ∧ 𝑦 < (π / 2)))
164	156, 159, 161, 163	syl3anbrc 1345	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ (0[,)(π / 2)))
165		tanord1 26504	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,)(π / 2)) ∧ 𝑦 ∈ (0[,)(π / 2))) → (𝑥 < 𝑦 ↔ (tan‘𝑥) < (tan‘𝑦)))
166	154, 164, 165	syl2anc 585	. . . . . . 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 11243	. . . . 5 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → (tan‘𝑥) < (tan‘𝑦))
169	168	3expia 1122	. . . 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 11665	. 2 ⊢ ((⊤ ∧ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐵 ∈ (-(π / 2)(,)(π / 2)))) → (𝐴 < 𝐵 ↔ (tan‘𝐴) < (tan‘𝐵)))
172	1, 171	mpan 691	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 1087 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ≠ wne 2931 class class class wbr 5097 ‘cfv 6491 (class class class)co 7358 ℂcc 11026 ℝcr 11027 0cc0 11028 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169 -cneg 11367 / cdiv 11796 2c2 12202 ℝ⁺crp 12907 (,)cioo 13263 [,)cico 13265 ℜcre 15022 cosccos 15989 tanctan 15990 πcpi 15991

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-inf2 9552 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3349 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4902 df-iun 4947 df-iin 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8767 df-pm 8768 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fsupp 9267 df-fi 9316 df-sup 9347 df-inf 9348 df-oi 9417 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-ioo 13267 df-ioc 13268 df-ico 13269 df-icc 13270 df-fz 13426 df-fzo 13573 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-hash 14256 df-shft 14992 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-limsup 15396 df-clim 15413 df-rlim 15414 df-sum 15612 df-ef 15992 df-sin 15994 df-cos 15995 df-tan 15996 df-pi 15997 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17425 df-qtop 17430 df-imas 17431 df-xps 17433 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19248 df-cmn 19713 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-fbas 21308 df-fg 21309 df-cnfld 21312 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22892 df-cld 22965 df-ntr 22966 df-cls 22967 df-nei 23044 df-lp 23082 df-perf 23083 df-cn 23173 df-cnp 23174 df-haus 23261 df-tx 23508 df-hmeo 23701 df-fil 23792 df-fm 23884 df-flim 23885 df-flf 23886 df-xms 24266 df-ms 24267 df-tms 24268 df-cncf 24829 df-limc 25825 df-dv 25826

This theorem is referenced by: atanlogsublem 26883 atanord 26895 basellem4 27052

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