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Theorem tanord 24684

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 1661	. 2 ⊢ ⊤
2		fveq2 6433	. . 3 ⊢ (𝑥 = 𝑦 → (tan‘𝑥) = (tan‘𝑦))
3		fveq2 6433	. . 3 ⊢ (𝑥 = 𝐴 → (tan‘𝑥) = (tan‘𝐴))
4		fveq2 6433	. . 3 ⊢ (𝑥 = 𝐵 → (tan‘𝑥) = (tan‘𝐵))
5		ioossre 12523	. . 3 ⊢ (-(π / 2)(,)(π / 2)) ⊆ ℝ
6		elioore 12493	. . . . 5 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → 𝑥 ∈ ℝ)
7	6	recnd 10385	. . . . . 6 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → 𝑥 ∈ ℂ)
8	6	rered 14341	. . . . . . 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 2906	. . . . . 6 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝑥) ∈ (-(π / 2)(,)(π / 2)))
11		cosne0 24676	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ (ℜ‘𝑥) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝑥) ≠ 0)
12	7, 10, 11	syl2anc 579	. . . . 5 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (cos‘𝑥) ≠ 0)
13	6, 12	retancld 15247	. . . 4 ⊢ (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (tan‘𝑥) ∈ ℝ)
14	13	adantl 475	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (-(π / 2)(,)(π / 2))) → (tan‘𝑥) ∈ ℝ)
15	6	3ad2ant1 1167	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ)
16	15	adantr 474	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑥 ∈ ℝ)
17	16	recnd 10385	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑥 ∈ ℂ)
18	17	negnegd 10704	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → --𝑥 = 𝑥)
19	18	fveq2d 6437	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘--𝑥) = (tan‘𝑥))
20	17	negcld 10700	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → -𝑥 ∈ ℂ)
21		cosneg 15249	. . . . . . . . . . 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 1246	. . . . . . . . . . 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 3066	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (cos‘-𝑥) ≠ 0)
26		tanneg 15250	. . . . . . . . 9 ⊢ ((-𝑥 ∈ ℂ ∧ (cos‘-𝑥) ≠ 0) → (tan‘--𝑥) = -(tan‘-𝑥))
27	20, 25, 26	syl2anc 579	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘--𝑥) = -(tan‘-𝑥))
28	19, 27	eqtr3d 2863	. . . . . . 7 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘𝑥) = -(tan‘-𝑥))
29	15	adantr 474	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑥 ∈ ℝ)
30	29	renegcld 10781	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -𝑥 ∈ ℝ)
31	25	adantrr 708	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (cos‘-𝑥) ≠ 0)
32	30, 31	retancld 15247	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘-𝑥) ∈ ℝ)
33	32	renegcld 10781	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(tan‘-𝑥) ∈ ℝ)
34		0red 10360	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 ∈ ℝ)
35		simp2 1171	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
36	5, 35	sseldi 3825	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ)
37	36	adantr 474	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 ∈ ℝ)
38		simpl2 1248	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
39		elioore 12493	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → 𝑦 ∈ ℝ)
40	39	recnd 10385	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → 𝑦 ∈ ℂ)
41	39	rered 14341	. . . . . . . . . . . . . 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 2906	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝑦) ∈ (-(π / 2)(,)(π / 2)))
44		cosne0 24676	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℂ ∧ (ℜ‘𝑦) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝑦) ≠ 0)
45	40, 43, 44	syl2anc 579	. . . . . . . . . . . 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 15247	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘𝑦) ∈ ℝ)
48		simprl 787	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑥 < 0)
49	29	lt0neg1d 10921	. . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < -𝑥)
51		simpl1 1246	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑥 ∈ (-(π / 2)(,)(π / 2)))
52		eliooord 12521	. . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(π / 2) < 𝑥)
55		halfpire 24616	. . . . . . . . . . . . . . . 16 ⊢ (π / 2) ∈ ℝ
56		ltnegcon1 10853	. . . . . . . . . . . . . . . 16 ⊢ (((π / 2) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (-(π / 2) < 𝑥 ↔ -𝑥 < (π / 2)))
57	55, 29, 56	sylancr 581	. . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -𝑥 < (π / 2))
59		0xr 10403	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ^*
60	55	rexri 10415	. . . . . . . . . . . . . . 15 ⊢ (π / 2) ∈ ℝ^*
61		elioo2 12504	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (-𝑥 ∈ (0(,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 < -𝑥 ∧ -𝑥 < (π / 2))))
62	59, 60, 61	mp2an 683	. . . . . . . . . . . . . 14 ⊢ (-𝑥 ∈ (0(,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 < -𝑥 ∧ -𝑥 < (π / 2)))
63	30, 50, 58, 62	syl3anbrc 1447	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -𝑥 ∈ (0(,)(π / 2)))
64		tanrpcl 24656	. . . . . . . . . . . . 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 12159	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < (tan‘-𝑥))
67	32	lt0neg2d 10922	. . . . . . . . . . 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 224	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(tan‘-𝑥) < 0)
69		simprr 789	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < 𝑦)
70		eliooord 12521	. . . . . . . . . . . . . . 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 491	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 < (π / 2))
73		elioo2 12504	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (𝑦 ∈ (0(,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦 ∧ 𝑦 < (π / 2))))
74	59, 60, 73	mp2an 683	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦 ∧ 𝑦 < (π / 2)))
75	37, 69, 72, 74	syl3anbrc 1447	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 ∈ (0(,)(π / 2)))
76		tanrpcl 24656	. . . . . . . . . . . 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 12159	. . . . . . . . . 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 10517	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(tan‘-𝑥) < (tan‘𝑦))
80	79	anassrs 461	. . . . . . . 8 ⊢ ((((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) ∧ 0 < 𝑦) → -(tan‘-𝑥) < (tan‘𝑦))
81		simpl3 1250	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 < 𝑦)
82	15	adantr 474	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 ∈ ℝ)
83	36	adantr 474	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ∈ ℝ)
84	82, 83	ltnegd 10930	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (𝑥 < 𝑦 ↔ -𝑦 < -𝑥))
85	81, 84	mpbid 224	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 < -𝑥)
86	83	renegcld 10781	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 ∈ ℝ)
87		simpr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ≤ 0)
88	83	le0neg1d 10923	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (𝑦 ≤ 0 ↔ 0 ≤ -𝑦))
89	87, 88	mpbid 224	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 0 ≤ -𝑦)
90		simpl2 1248	. . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(π / 2) < 𝑦)
93		ltnegcon1 10853	. . . . . . . . . . . . . . . 16 ⊢ (((π / 2) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (-(π / 2) < 𝑦 ↔ -𝑦 < (π / 2)))
94	55, 83, 93	sylancr 581	. . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 < (π / 2))
96		0re 10358	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
97		elico2 12525	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (-𝑦 ∈ (0[,)(π / 2)) ↔ (-𝑦 ∈ ℝ ∧ 0 ≤ -𝑦 ∧ -𝑦 < (π / 2))))
98	96, 60, 97	mp2an 683	. . . . . . . . . . . . . 14 ⊢ (-𝑦 ∈ (0[,)(π / 2)) ↔ (-𝑦 ∈ ℝ ∧ 0 ≤ -𝑦 ∧ -𝑦 < (π / 2)))
99	86, 89, 95, 98	syl3anbrc 1447	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 ∈ (0[,)(π / 2)))
100	82	renegcld 10781	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑥 ∈ ℝ)
101		simp3 1172	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
102		0red 10360	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 0 ∈ ℝ)
103		ltletr 10448	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → ((𝑥 < 𝑦 ∧ 𝑦 ≤ 0) → 𝑥 < 0))
104	15, 36, 102, 103	syl3anc 1494	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → ((𝑥 < 𝑦 ∧ 𝑦 ≤ 0) → 𝑥 < 0))
105	101, 104	mpand 686	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → (𝑦 ≤ 0 → 𝑥 < 0))
106	105	imp 397	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 < 0)
107		ltle 10445	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑥 < 0 → 𝑥 ≤ 0))
108	82, 96, 107	sylancl 580	. . . . . . . . . . . . . . . 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 10923	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (𝑥 ≤ 0 ↔ 0 ≤ -𝑥))
111	109, 110	mpbid 224	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 0 ≤ -𝑥)
112		simpl1 1246	. . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(π / 2) < 𝑥)
115	55, 82, 56	sylancr 581	. . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑥 < (π / 2))
117		elico2 12525	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (-𝑥 ∈ (0[,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 ≤ -𝑥 ∧ -𝑥 < (π / 2))))
118	96, 60, 117	mp2an 683	. . . . . . . . . . . . . 14 ⊢ (-𝑥 ∈ (0[,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 ≤ -𝑥 ∧ -𝑥 < (π / 2)))
119	100, 111, 116, 118	syl3anbrc 1447	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑥 ∈ (0[,)(π / 2)))
120		tanord1 24683	. . . . . . . . . . . . 13 ⊢ ((-𝑦 ∈ (0[,)(π / 2)) ∧ -𝑥 ∈ (0[,)(π / 2))) → (-𝑦 < -𝑥 ↔ (tan‘-𝑦) < (tan‘-𝑥)))
121	99, 119, 120	syl2anc 579	. . . . . . . . . . . 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 224	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘-𝑦) < (tan‘-𝑥))
123	83	recnd 10385	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ∈ ℂ)
124		cosneg 15249	. . . . . . . . . . . . . . 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 3066	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘-𝑦) ≠ 0)
128	86, 127	retancld 15247	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘-𝑦) ∈ ℝ)
129	106, 25	syldan 585	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘-𝑥) ≠ 0)
130	100, 129	retancld 15247	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘-𝑥) ∈ ℝ)
131	128, 130	ltnegd 10930	. . . . . . . . . . 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 224	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(tan‘-𝑥) < -(tan‘-𝑦))
133	123	negnegd 10704	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → --𝑦 = 𝑦)
134	133	fveq2d 6437	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘--𝑦) = (tan‘𝑦))
135	123	negcld 10700	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 ∈ ℂ)
136		tanneg 15250	. . . . . . . . . . . 12 ⊢ ((-𝑦 ∈ ℂ ∧ (cos‘-𝑦) ≠ 0) → (tan‘--𝑦) = -(tan‘-𝑦))
137	135, 127, 136	syl2anc 579	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘--𝑦) = -(tan‘-𝑦))
138	134, 137	eqtr3d 2863	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘𝑦) = -(tan‘-𝑦))
139	132, 138	breqtrrd 4901	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(tan‘-𝑥) < (tan‘𝑦))
140	139	adantlr 706	. . . . . . . 8 ⊢ ((((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) ∧ 𝑦 ≤ 0) → -(tan‘-𝑥) < (tan‘𝑦))
141		0red 10360	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 0 ∈ ℝ)
142		simpl2 1248	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
143	5, 142	sseldi 3825	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑦 ∈ ℝ)
144	80, 140, 141, 143	ltlecasei 10464	. . . . . . 7 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → -(tan‘-𝑥) < (tan‘𝑦))
145	28, 144	eqbrtrd 4895	. . . . . 6 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘𝑥) < (tan‘𝑦))
146		simpl3 1250	. . . . . . 7 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 < 𝑦)
147	15	adantr 474	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ∈ ℝ)
148		simpr 479	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 0 ≤ 𝑥)
149		simpl1 1246	. . . . . . . . . . 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 491	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 < (π / 2))
152		elico2 12525	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (𝑥 ∈ (0[,)(π / 2)) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < (π / 2))))
153	96, 60, 152	mp2an 683	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,)(π / 2)) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < (π / 2)))
154	147, 148, 151, 153	syl3anbrc 1447	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ∈ (0[,)(π / 2)))
155		simpl2 1248	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
156	5, 155	sseldi 3825	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ ℝ)
157		0red 10360	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 0 ∈ ℝ)
158	147, 156, 146	ltled 10504	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ≤ 𝑦)
159	157, 147, 156, 148, 158	letrd 10513	. . . . . . . . 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 491	. . . . . . . . 9 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 < (π / 2))
162		elico2 12525	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (𝑦 ∈ (0[,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ∧ 𝑦 < (π / 2))))
163	96, 60, 162	mp2an 683	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ∧ 𝑦 < (π / 2)))
164	156, 159, 161, 163	syl3anbrc 1447	. . . . . . . 8 ⊢ (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ (0[,)(π / 2)))
165		tanord1 24683	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,)(π / 2)) ∧ 𝑦 ∈ (0[,)(π / 2))) → (𝑥 < 𝑦 ↔ (tan‘𝑥) < (tan‘𝑦)))
166	154, 164, 165	syl2anc 579	. . . . . . 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 224	. . . . . 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 10464	. . . . 5 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → (tan‘𝑥) < (tan‘𝑦))
169	168	3expia 1154	. . . 4 ⊢ ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2))) → (𝑥 < 𝑦 → (tan‘𝑥) < (tan‘𝑦)))
170	169	adantl 475	. . 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 10878	. 2 ⊢ ((⊤ ∧ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐵 ∈ (-(π / 2)(,)(π / 2)))) → (𝐴 < 𝐵 ↔ (tan‘𝐴) < (tan‘𝐵)))
172	1, 171	mpan 681	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 198 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ⊤wtru 1657 ∈ wcel 2164 ≠ wne 2999 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 ℂcc 10250 ℝcr 10251 0cc0 10252 ℝ^*cxr 10390 < clt 10391 ≤ cle 10392 -cneg 10586 / cdiv 11009 2c2 11406 ℝ⁺crp 12112 (,)cioo 12463 [,)cico 12465 ℜcre 14214 cosccos 15167 tanctan 15168 πcpi 15169

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 ax-addf 10331 ax-mulf 10332

This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-ixp 8176 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fsupp 8545 df-fi 8586 df-sup 8617 df-inf 8618 df-oi 8684 df-card 9078 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-xadd 12233 df-xmul 12234 df-ioo 12467 df-ioc 12468 df-ico 12469 df-icc 12470 df-fz 12620 df-fzo 12761 df-fl 12888 df-mod 12964 df-seq 13096 df-exp 13155 df-fac 13354 df-bc 13383 df-hash 13411 df-shft 14184 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-limsup 14579 df-clim 14596 df-rlim 14597 df-sum 14794 df-ef 15170 df-sin 15172 df-cos 15173 df-tan 15174 df-pi 15175 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-starv 16320 df-sca 16321 df-vsca 16322 df-ip 16323 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-hom 16329 df-cco 16330 df-rest 16436 df-topn 16437 df-0g 16455 df-gsum 16456 df-topgen 16457 df-pt 16458 df-prds 16461 df-xrs 16515 df-qtop 16520 df-imas 16521 df-xps 16523 df-mre 16599 df-mrc 16600 df-acs 16602 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-mulg 17895 df-cntz 18100 df-cmn 18548 df-psmet 20098 df-xmet 20099 df-met 20100 df-bl 20101 df-mopn 20102 df-fbas 20103 df-fg 20104 df-cnfld 20107 df-top 21069 df-topon 21086 df-topsp 21108 df-bases 21121 df-cld 21194 df-ntr 21195 df-cls 21196 df-nei 21273 df-lp 21311 df-perf 21312 df-cn 21402 df-cnp 21403 df-haus 21490 df-tx 21736 df-hmeo 21929 df-fil 22020 df-fm 22112 df-flim 22113 df-flf 22114 df-xms 22495 df-ms 22496 df-tms 22497 df-cncf 23051 df-limc 24029 df-dv 24030

This theorem is referenced by: atanlogsublem 25055 atanord 25067 basellem4 25223

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