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Theorem tanord 26867

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 1574	. 2 ⊢ ⊤
2		fveq2 6886	. . 3 ⊢ (𝑥 = 𝑦 → (tan‘𝑥) = (tan‘𝑦))
3		fveq2 6886	. . 3 ⊢ (𝑥 = 𝐴 → (tan‘𝑥) = (tan‘𝐴))
4		fveq2 6886	. . 3 ⊢ (𝑥 = 𝐵 → (tan‘𝑥) = (tan‘𝐵))
5		ioossre 13539	. . 3 ⊢ ( -(π / 2)(,)(π / 2)) ⊆ ℝ
6		elioore 13507	. . . . 5 ⊢ (𝑥 ∈ ( -(π / 2)(,)(π / 2)) → 𝑥 ∈ ℝ)
7	6	recnd 11338	. . . . . 6 ⊢ (𝑥 ∈ ( -(π / 2)(,)(π / 2)) → 𝑥 ∈ ℂ)
8	6	rered 15392	. . . . . . 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 2861	. . . . . 6 ⊢ (𝑥 ∈ ( -(π / 2)(,)(π / 2)) → (ℜ‘𝑥) ∈ ( -(π / 2)(,)(π / 2)))
11		cosne0 26858	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ (ℜ‘𝑥) ∈ ( -(π / 2)(,)(π / 2))) → (cos‘𝑥) ≠ 0)
12	7, 10, 11	syl2anc 596	. . . . 5 ⊢ (𝑥 ∈ ( -(π / 2)(,)(π / 2)) → (cos‘𝑥) ≠ 0)
13	6, 12	retancld 16314	. . . 4 ⊢ (𝑥 ∈ ( -(π / 2)(,)(π / 2)) → (tan‘𝑥) ∈ ℝ)
14	13	adantl 487	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ( -(π / 2)(,)(π / 2))) → (tan‘𝑥) ∈ ℝ)
15	6	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ)
16	15	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑥 ∈ ℝ)
17	16	recnd 11338	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑥 ∈ ℂ)
18	17	negnegd 11661	. . . . . . . . 9 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → - -𝑥 = 𝑥)
19	18	fveq2d 6890	. . . . . . . 8 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘ - -𝑥) = (tan‘𝑥))
20	17	negcld 11657	. . . . . . . . 9 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → -𝑥 ∈ ℂ)
21		cosneg 16316	. . . . . . . . . . 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 1210	. . . . . . . . . . 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 3023	. . . . . . . . 9 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (cos‘ -𝑥) ≠ 0)
26		tanneg 16317	. . . . . . . . 9 ⊢ (( -𝑥 ∈ ℂ ∧ (cos‘ -𝑥) ≠ 0) → (tan‘ - -𝑥) = -(tan‘ -𝑥))
27	20, 25, 26	syl2anc 596	. . . . . . . 8 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘ - -𝑥) = -(tan‘ -𝑥))
28	19, 27	eqtr3d 2798	. . . . . . 7 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘𝑥) = -(tan‘ -𝑥))
29	15	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑥 ∈ ℝ)
30	29	renegcld 11744	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -𝑥 ∈ ℝ)
31	25	adantrr 730	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (cos‘ -𝑥) ≠ 0)
32	30, 31	retancld 16314	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘ -𝑥) ∈ ℝ)
33	32	renegcld 11744	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(tan‘ -𝑥) ∈ ℝ)
34		0red 11312	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 ∈ ℝ)
35		simp2 1155	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ( -(π / 2)(,)(π / 2)))
36	5, 35	sselid 3927	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ)
37	36	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 ∈ ℝ)
38		simpl2 1211	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 ∈ ( -(π / 2)(,)(π / 2)))
39		elioore 13507	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ( -(π / 2)(,)(π / 2)) → 𝑦 ∈ ℝ)
40	39	recnd 11338	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ( -(π / 2)(,)(π / 2)) → 𝑦 ∈ ℂ)
41	39	rered 15392	. . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ( -(π / 2)(,)(π / 2)) → (ℜ‘𝑦) ∈ ( -(π / 2)(,)(π / 2)))
44		cosne0 26858	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℂ ∧ (ℜ‘𝑦) ∈ ( -(π / 2)(,)(π / 2))) → (cos‘𝑦) ≠ 0)
45	40, 43, 44	syl2anc 596	. . . . . . . . . . . 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 16314	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘𝑦) ∈ ℝ)
48		simprl 783	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑥 < 0)
49	29	lt0neg1d 11886	. . . . . . . . . . . . . . 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 1210	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑥 ∈ ( -(π / 2)(,)(π / 2)))
52		eliooord 13537	. . . . . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(π / 2) < 𝑥)
55		halfpire 26794	. . . . . . . . . . . . . . . 16 ⊢ (π / 2) ∈ ℝ
56		ltnegcon1 11818	. . . . . . . . . . . . . . . 16 ⊢ (((π / 2) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ( -(π / 2) < 𝑥 ↔ -𝑥 < (π / 2)))
57	55, 29, 56	sylancr 599	. . . . . . . . . . . . . . 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 11357	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ^*
60	55	rexri 11368	. . . . . . . . . . . . . . 15 ⊢ (π / 2) ∈ ℝ^*
61		elioo2 13518	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → ( -𝑥 ∈ (0(,)(π / 2)) ↔ ( -𝑥 ∈ ℝ ∧ 0 < -𝑥 ∧ -𝑥 < (π / 2))))
62	59, 60, 61	mp2an 705	. . . . . . . . . . . . . 14 ⊢ ( -𝑥 ∈ (0(,)(π / 2)) ↔ ( -𝑥 ∈ ℝ ∧ 0 < -𝑥 ∧ -𝑥 < (π / 2)))
63	30, 50, 58, 62	syl3anbrc 1362	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -𝑥 ∈ (0(,)(π / 2)))
64		tanrpcl 26834	. . . . . . . . . . . . 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 13168	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < (tan‘ -𝑥))
67	32	lt0neg2d 11887	. . . . . . . . . . 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 785	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < 𝑦)
70		eliooord 13537	. . . . . . . . . . . . . . 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 501	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 < (π / 2))
73		elioo2 13518	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (𝑦 ∈ (0(,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦 ∧ 𝑦 < (π / 2))))
74	59, 60, 73	mp2an 705	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0(,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦 ∧ 𝑦 < (π / 2)))
75	37, 69, 72, 74	syl3anbrc 1362	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 ∈ (0(,)(π / 2)))
76		tanrpcl 26834	. . . . . . . . . . . 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 13168	. . . . . . . . . 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 11472	. . . . . . . . 9 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(tan‘ -𝑥) < (tan‘𝑦))
80	79	anassrs 473	. . . . . . . 8 ⊢ ((((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) ∧ 0 < 𝑦) → -(tan‘ -𝑥) < (tan‘𝑦))
81		simpl3 1212	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 < 𝑦)
82	15	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 ∈ ℝ)
83	36	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ∈ ℝ)
84	82, 83	ltnegd 11895	. . . . . . . . . . . . 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 11744	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 ∈ ℝ)
87		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ≤ 0)
88	83	le0neg1d 11888	. . . . . . . . . . . . . . 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 1211	. . . . . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(π / 2) < 𝑦)
93		ltnegcon1 11818	. . . . . . . . . . . . . . . 16 ⊢ (((π / 2) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ( -(π / 2) < 𝑦 ↔ -𝑦 < (π / 2)))
94	55, 83, 93	sylancr 599	. . . . . . . . . . . . . . 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 11311	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
97		elico2 13542	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → ( -𝑦 ∈ (0[,)(π / 2)) ↔ ( -𝑦 ∈ ℝ ∧ 0 ≤ -𝑦 ∧ -𝑦 < (π / 2))))
98	96, 60, 97	mp2an 705	. . . . . . . . . . . . . 14 ⊢ ( -𝑦 ∈ (0[,)(π / 2)) ↔ ( -𝑦 ∈ ℝ ∧ 0 ≤ -𝑦 ∧ -𝑦 < (π / 2)))
99	86, 89, 95, 98	syl3anbrc 1362	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 ∈ (0[,)(π / 2)))
100	82	renegcld 11744	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑥 ∈ ℝ)
101		simp3 1156	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
102		0red 11312	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 0 ∈ ℝ)
103		ltletr 11403	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → ((𝑥 < 𝑦 ∧ 𝑦 ≤ 0) → 𝑥 < 0))
104	15, 36, 102, 103	syl3anc 1398	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → ((𝑥 < 𝑦 ∧ 𝑦 ≤ 0) → 𝑥 < 0))
105	101, 104	mpand 708	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → (𝑦 ≤ 0 → 𝑥 < 0))
106	105	imp 412	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 < 0)
107		ltle 11399	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑥 < 0 → 𝑥 ≤ 0))
108	82, 96, 107	sylancl 598	. . . . . . . . . . . . . . . 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 11888	. . . . . . . . . . . . . . 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 1210	. . . . . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(π / 2) < 𝑥)
115	55, 82, 56	sylancr 599	. . . . . . . . . . . . . . 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 13542	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → ( -𝑥 ∈ (0[,)(π / 2)) ↔ ( -𝑥 ∈ ℝ ∧ 0 ≤ -𝑥 ∧ -𝑥 < (π / 2))))
118	96, 60, 117	mp2an 705	. . . . . . . . . . . . . 14 ⊢ ( -𝑥 ∈ (0[,)(π / 2)) ↔ ( -𝑥 ∈ ℝ ∧ 0 ≤ -𝑥 ∧ -𝑥 < (π / 2)))
119	100, 111, 116, 118	syl3anbrc 1362	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑥 ∈ (0[,)(π / 2)))
120		tanord1 26866	. . . . . . . . . . . . 13 ⊢ (( -𝑦 ∈ (0[,)(π / 2)) ∧ -𝑥 ∈ (0[,)(π / 2))) → ( -𝑦 < -𝑥 ↔ (tan‘ -𝑦) < (tan‘ -𝑥)))
121	99, 119, 120	syl2anc 596	. . . . . . . . . . . 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 11338	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ∈ ℂ)
124		cosneg 16316	. . . . . . . . . . . . . . 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 3023	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘ -𝑦) ≠ 0)
128	86, 127	retancld 16314	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘ -𝑦) ∈ ℝ)
129	106, 25	syldan 603	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘ -𝑥) ≠ 0)
130	100, 129	retancld 16314	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘ -𝑥) ∈ ℝ)
131	128, 130	ltnegd 11895	. . . . . . . . . . 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 11661	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → - -𝑦 = 𝑦)
134	133	fveq2d 6890	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘ - -𝑦) = (tan‘𝑦))
135	123	negcld 11657	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 ∈ ℂ)
136		tanneg 16317	. . . . . . . . . . . 12 ⊢ (( -𝑦 ∈ ℂ ∧ (cos‘ -𝑦) ≠ 0) → (tan‘ - -𝑦) = -(tan‘ -𝑦))
137	135, 127, 136	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘ - -𝑦) = -(tan‘ -𝑦))
138	134, 137	eqtr3d 2798	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘𝑦) = -(tan‘ -𝑦))
139	132, 138	breqtrrd 5133	. . . . . . . . 9 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(tan‘ -𝑥) < (tan‘𝑦))
140	139	adantlr 728	. . . . . . . 8 ⊢ ((((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) ∧ 𝑦 ≤ 0) → -(tan‘ -𝑥) < (tan‘𝑦))
141		0red 11312	. . . . . . . 8 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 0 ∈ ℝ)
142		simpl2 1211	. . . . . . . . 9 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑦 ∈ ( -(π / 2)(,)(π / 2)))
143	5, 142	sselid 3927	. . . . . . . 8 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑦 ∈ ℝ)
144	80, 140, 141, 143	ltlecasei 11419	. . . . . . 7 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → -(tan‘ -𝑥) < (tan‘𝑦))
145	28, 144	eqbrtrd 5127	. . . . . 6 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘𝑥) < (tan‘𝑦))
146		simpl3 1212	. . . . . . 7 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 < 𝑦)
147	15	adantr 486	. . . . . . . . 9 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ∈ ℝ)
148		simpr 490	. . . . . . . . 9 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 0 ≤ 𝑥)
149		simpl1 1210	. . . . . . . . . . 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 501	. . . . . . . . 9 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 < (π / 2))
152		elico2 13542	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (𝑥 ∈ (0[,)(π / 2)) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < (π / 2))))
153	96, 60, 152	mp2an 705	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,)(π / 2)) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < (π / 2)))
154	147, 148, 151, 153	syl3anbrc 1362	. . . . . . . 8 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ∈ (0[,)(π / 2)))
155		simpl2 1211	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ ( -(π / 2)(,)(π / 2)))
156	5, 155	sselid 3927	. . . . . . . . 9 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ ℝ)
157		0red 11312	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 0 ∈ ℝ)
158	147, 156, 146	ltled 11459	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ≤ 𝑦)
159	157, 147, 156, 148, 158	letrd 11468	. . . . . . . . 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 501	. . . . . . . . 9 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 < (π / 2))
162		elico2 13542	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ^*) → (𝑦 ∈ (0[,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ∧ 𝑦 < (π / 2))))
163	96, 60, 162	mp2an 705	. . . . . . . . 9 ⊢ (𝑦 ∈ (0[,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦 ∧ 𝑦 < (π / 2)))
164	156, 159, 161, 163	syl3anbrc 1362	. . . . . . . 8 ⊢ (((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ (0[,)(π / 2)))
165		tanord1 26866	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,)(π / 2)) ∧ 𝑦 ∈ (0[,)(π / 2))) → (𝑥 < 𝑦 ↔ (tan‘𝑥) < (tan‘𝑦)))
166	154, 164, 165	syl2anc 596	. . . . . . 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 11419	. . . . 5 ⊢ ((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → (tan‘𝑥) < (tan‘𝑦))
169	168	3expia 1139	. . . 4 ⊢ ((𝑥 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ ( -(π / 2)(,)(π / 2))) → (𝑥 < 𝑦 → (tan‘𝑥) < (tan‘𝑦)))
170	169	adantl 487	. . 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 11843	. 2 ⊢ ((⊤ ∧ (𝐴 ∈ ( -(π / 2)(,)(π / 2)) ∧ 𝐵 ∈ ( -(π / 2)(,)(π / 2)))) → (𝐴 < 𝐵 ↔ (tan‘𝐴) < (tan‘𝐵)))
172	1, 171	mpan 703	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 401 ∧ w3a 1103 = wceq 1570 ⊤wtru 1571 ∈ wcel 2145 ≠ wne 2956 class class class wbr 5103 ‘cfv 6539 (class class class)co 7421 ℂcc 11199 ℝcr 11200 0cc0 11201 ℝ^*cxr 11343 < clt 11344 ≤ cle 11345 -cneg 11543 / cdiv 11974 2c2 12398 ℝ⁺crp 13121 (,)cioo 13477 [,)cico 13479 ℜcre 15265 cosccos 16231 tanctan 16232 πcpi 16233

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7752 ax-inf2 9643 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278 ax-pre-sup 11279 ax-addf 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3900 df-un 3902 df-in 3904 df-ss 3915 df-pss 4035 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6305 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6495 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-supp 8179 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-1o 8477 df-2o 8478 df-er 8718 df-map 8850 df-pm 8851 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9355 df-fi 9404 df-sup 9435 df-inf 9436 df-oi 9505 df-card 10021 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-div 11975 df-nn 12337 df-2 12406 df-3 12407 df-4 12408 df-5 12409 df-6 12410 df-7 12411 df-8 12412 df-9 12413 df-n0 12608 df-z 12695 df-dec 12816 df-uz 12967 df-q 13077 df-rp 13122 df-xneg 13242 df-xadd 13243 df-xmul 13244 df-ioo 13481 df-ioc 13482 df-ico 13483 df-icc 13484 df-fz 13641 df-fzo 13790 df-fl 13933 df-mod 14011 df-seq 14146 df-exp 14206 df-fac 14419 df-bc 14448 df-hash 14476 df-shft 15221 df-cj 15267 df-re 15268 df-im 15269 df-sqrt 15403 df-abs 15404 df-limsup 15639 df-clim 15656 df-rlim 15657 df-sum 15855 df-ef 16234 df-sin 16236 df-cos 16237 df-tan 16238 df-pi 16239 df-struct 17326 df-sets 17343 df-slot 17361 df-ndx 17373 df-base 17389 df-ress 17410 df-plusg 17442 df-mulr 17443 df-starv 17444 df-sca 17445 df-vsca 17446 df-ip 17447 df-tset 17448 df-ple 17449 df-ds 17451 df-unif 17452 df-hom 17453 df-cco 17454 df-rest 17594 df-topn 17595 df-0g 17613 df-gsum 17614 df-topgen 17615 df-pt 17616 df-prds 17619 df-xrs 17675 df-qtop 17680 df-imas 17681 df-xps 17683 df-mre 17757 df-mrc 17758 df-acs 17760 df-mgm 18817 df-sgrp 18909 df-mnd 18925 df-submnd 18980 df-mulg 19279 df-cntz 19532 df-cmn 19997 df-psmet 21671 df-xmet 21672 df-met 21673 df-bl 21674 df-mopn 21675 df-fbas 21676 df-fg 21677 df-cnfld 21680 df-top 23213 df-topon 23230 df-topsp 23252 df-bases 23265 df-cld 23338 df-ntr 23339 df-cls 23340 df-nei 23417 df-lp 23455 df-perf 23456 df-cn 23546 df-cnp 23547 df-haus 23634 df-tx 23882 df-hmeo 24075 df-fil 24166 df-fm 24258 df-flim 24259 df-flf 24260 df-xms 24640 df-ms 24641 df-tms 24642 df-cncf 25200 df-limc 26187 df-dv 26188

This theorem is used by: atanlogsublem 27244 atanord 27256 basellem4 27412

W3C validator