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Theorem tanord 25109
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.
StepHypRef Expression
1 tru 1541 . 2
2 fveq2 6646 . . 3 (𝑥 = 𝑦 → (tan‘𝑥) = (tan‘𝑦))
3 fveq2 6646 . . 3 (𝑥 = 𝐴 → (tan‘𝑥) = (tan‘𝐴))
4 fveq2 6646 . . 3 (𝑥 = 𝐵 → (tan‘𝑥) = (tan‘𝐵))
5 ioossre 12777 . . 3 (-(π / 2)(,)(π / 2)) ⊆ ℝ
6 elioore 12747 . . . . 5 (𝑥 ∈ (-(π / 2)(,)(π / 2)) → 𝑥 ∈ ℝ)
76recnd 10647 . . . . . 6 (𝑥 ∈ (-(π / 2)(,)(π / 2)) → 𝑥 ∈ ℂ)
86rered 14563 . . . . . . 7 (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝑥) = 𝑥)
9 id 22 . . . . . . 7 (𝑥 ∈ (-(π / 2)(,)(π / 2)) → 𝑥 ∈ (-(π / 2)(,)(π / 2)))
108, 9eqeltrd 2911 . . . . . 6 (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝑥) ∈ (-(π / 2)(,)(π / 2)))
11 cosne0 25100 . . . . . 6 ((𝑥 ∈ ℂ ∧ (ℜ‘𝑥) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝑥) ≠ 0)
127, 10, 11syl2anc 586 . . . . 5 (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (cos‘𝑥) ≠ 0)
136, 12retancld 15478 . . . 4 (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (tan‘𝑥) ∈ ℝ)
1413adantl 484 . . 3 ((⊤ ∧ 𝑥 ∈ (-(π / 2)(,)(π / 2))) → (tan‘𝑥) ∈ ℝ)
1563ad2ant1 1129 . . . . . . . . . . . 12 ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ)
1615adantr 483 . . . . . . . . . . 11 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑥 ∈ ℝ)
1716recnd 10647 . . . . . . . . . 10 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑥 ∈ ℂ)
1817negnegd 10966 . . . . . . . . 9 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → --𝑥 = 𝑥)
1918fveq2d 6650 . . . . . . . 8 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘--𝑥) = (tan‘𝑥))
2017negcld 10962 . . . . . . . . 9 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → -𝑥 ∈ ℂ)
21 cosneg 15480 . . . . . . . . . . 11 (𝑥 ∈ ℂ → (cos‘-𝑥) = (cos‘𝑥))
2217, 21syl 17 . . . . . . . . . 10 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (cos‘-𝑥) = (cos‘𝑥))
23 simpl1 1187 . . . . . . . . . . 11 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑥 ∈ (-(π / 2)(,)(π / 2)))
2423, 12syl 17 . . . . . . . . . 10 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (cos‘𝑥) ≠ 0)
2522, 24eqnetrd 3073 . . . . . . . . 9 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (cos‘-𝑥) ≠ 0)
26 tanneg 15481 . . . . . . . . 9 ((-𝑥 ∈ ℂ ∧ (cos‘-𝑥) ≠ 0) → (tan‘--𝑥) = -(tan‘-𝑥))
2720, 25, 26syl2anc 586 . . . . . . . 8 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘--𝑥) = -(tan‘-𝑥))
2819, 27eqtr3d 2857 . . . . . . 7 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘𝑥) = -(tan‘-𝑥))
2915adantr 483 . . . . . . . . . . . . 13 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑥 ∈ ℝ)
3029renegcld 11045 . . . . . . . . . . . 12 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -𝑥 ∈ ℝ)
3125adantrr 715 . . . . . . . . . . . 12 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (cos‘-𝑥) ≠ 0)
3230, 31retancld 15478 . . . . . . . . . . 11 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘-𝑥) ∈ ℝ)
3332renegcld 11045 . . . . . . . . . 10 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(tan‘-𝑥) ∈ ℝ)
34 0red 10622 . . . . . . . . . 10 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 ∈ ℝ)
35 simp2 1133 . . . . . . . . . . . . 13 ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
365, 35sseldi 3944 . . . . . . . . . . . 12 ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ)
3736adantr 483 . . . . . . . . . . 11 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 ∈ ℝ)
38 simpl2 1188 . . . . . . . . . . . 12 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
39 elioore 12747 . . . . . . . . . . . . . 14 (𝑦 ∈ (-(π / 2)(,)(π / 2)) → 𝑦 ∈ ℝ)
4039recnd 10647 . . . . . . . . . . . . 13 (𝑦 ∈ (-(π / 2)(,)(π / 2)) → 𝑦 ∈ ℂ)
4139rered 14563 . . . . . . . . . . . . . 14 (𝑦 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝑦) = 𝑦)
42 id 22 . . . . . . . . . . . . . 14 (𝑦 ∈ (-(π / 2)(,)(π / 2)) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
4341, 42eqeltrd 2911 . . . . . . . . . . . . 13 (𝑦 ∈ (-(π / 2)(,)(π / 2)) → (ℜ‘𝑦) ∈ (-(π / 2)(,)(π / 2)))
44 cosne0 25100 . . . . . . . . . . . . 13 ((𝑦 ∈ ℂ ∧ (ℜ‘𝑦) ∈ (-(π / 2)(,)(π / 2))) → (cos‘𝑦) ≠ 0)
4540, 43, 44syl2anc 586 . . . . . . . . . . . 12 (𝑦 ∈ (-(π / 2)(,)(π / 2)) → (cos‘𝑦) ≠ 0)
4638, 45syl 17 . . . . . . . . . . 11 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (cos‘𝑦) ≠ 0)
4737, 46retancld 15478 . . . . . . . . . 10 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘𝑦) ∈ ℝ)
48 simprl 769 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑥 < 0)
4929lt0neg1d 11187 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (𝑥 < 0 ↔ 0 < -𝑥))
5048, 49mpbid 234 . . . . . . . . . . . . . 14 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < -𝑥)
51 simpl1 1187 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑥 ∈ (-(π / 2)(,)(π / 2)))
52 eliooord 12775 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (-(π / 2)(,)(π / 2)) → (-(π / 2) < 𝑥𝑥 < (π / 2)))
5351, 52syl 17 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (-(π / 2) < 𝑥𝑥 < (π / 2)))
5453simpld 497 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(π / 2) < 𝑥)
55 halfpire 25036 . . . . . . . . . . . . . . . 16 (π / 2) ∈ ℝ
56 ltnegcon1 11119 . . . . . . . . . . . . . . . 16 (((π / 2) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (-(π / 2) < 𝑥 ↔ -𝑥 < (π / 2)))
5755, 29, 56sylancr 589 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (-(π / 2) < 𝑥 ↔ -𝑥 < (π / 2)))
5854, 57mpbid 234 . . . . . . . . . . . . . 14 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -𝑥 < (π / 2))
59 0xr 10666 . . . . . . . . . . . . . . 15 0 ∈ ℝ*
6055rexri 10677 . . . . . . . . . . . . . . 15 (π / 2) ∈ ℝ*
61 elioo2 12758 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ* ∧ (π / 2) ∈ ℝ*) → (-𝑥 ∈ (0(,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 < -𝑥 ∧ -𝑥 < (π / 2))))
6259, 60, 61mp2an 690 . . . . . . . . . . . . . 14 (-𝑥 ∈ (0(,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 < -𝑥 ∧ -𝑥 < (π / 2)))
6330, 50, 58, 62syl3anbrc 1339 . . . . . . . . . . . . 13 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -𝑥 ∈ (0(,)(π / 2)))
64 tanrpcl 25076 . . . . . . . . . . . . 13 (-𝑥 ∈ (0(,)(π / 2)) → (tan‘-𝑥) ∈ ℝ+)
6563, 64syl 17 . . . . . . . . . . . 12 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘-𝑥) ∈ ℝ+)
6665rpgt0d 12413 . . . . . . . . . . 11 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < (tan‘-𝑥))
6732lt0neg2d 11188 . . . . . . . . . . 11 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (0 < (tan‘-𝑥) ↔ -(tan‘-𝑥) < 0))
6866, 67mpbid 234 . . . . . . . . . 10 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(tan‘-𝑥) < 0)
69 simprr 771 . . . . . . . . . . . . 13 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < 𝑦)
70 eliooord 12775 . . . . . . . . . . . . . . 15 (𝑦 ∈ (-(π / 2)(,)(π / 2)) → (-(π / 2) < 𝑦𝑦 < (π / 2)))
7138, 70syl 17 . . . . . . . . . . . . . 14 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (-(π / 2) < 𝑦𝑦 < (π / 2)))
7271simprd 498 . . . . . . . . . . . . 13 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 < (π / 2))
73 elioo2 12758 . . . . . . . . . . . . . 14 ((0 ∈ ℝ* ∧ (π / 2) ∈ ℝ*) → (𝑦 ∈ (0(,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦𝑦 < (π / 2))))
7459, 60, 73mp2an 690 . . . . . . . . . . . . 13 (𝑦 ∈ (0(,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦𝑦 < (π / 2)))
7537, 69, 72, 74syl3anbrc 1339 . . . . . . . . . . . 12 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 𝑦 ∈ (0(,)(π / 2)))
76 tanrpcl 25076 . . . . . . . . . . . 12 (𝑦 ∈ (0(,)(π / 2)) → (tan‘𝑦) ∈ ℝ+)
7775, 76syl 17 . . . . . . . . . . 11 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → (tan‘𝑦) ∈ ℝ+)
7877rpgt0d 12413 . . . . . . . . . 10 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → 0 < (tan‘𝑦))
7933, 34, 47, 68, 78lttrd 10779 . . . . . . . . 9 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ (𝑥 < 0 ∧ 0 < 𝑦)) → -(tan‘-𝑥) < (tan‘𝑦))
8079anassrs 470 . . . . . . . 8 ((((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) ∧ 0 < 𝑦) → -(tan‘-𝑥) < (tan‘𝑦))
81 simpl3 1189 . . . . . . . . . . . . 13 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 < 𝑦)
8215adantr 483 . . . . . . . . . . . . . 14 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 ∈ ℝ)
8336adantr 483 . . . . . . . . . . . . . 14 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ∈ ℝ)
8482, 83ltnegd 11196 . . . . . . . . . . . . 13 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (𝑥 < 𝑦 ↔ -𝑦 < -𝑥))
8581, 84mpbid 234 . . . . . . . . . . . 12 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 < -𝑥)
8683renegcld 11045 . . . . . . . . . . . . . 14 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 ∈ ℝ)
87 simpr 487 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ≤ 0)
8883le0neg1d 11189 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (𝑦 ≤ 0 ↔ 0 ≤ -𝑦))
8987, 88mpbid 234 . . . . . . . . . . . . . 14 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 0 ≤ -𝑦)
90 simpl2 1188 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
9190, 70syl 17 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (-(π / 2) < 𝑦𝑦 < (π / 2)))
9291simpld 497 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(π / 2) < 𝑦)
93 ltnegcon1 11119 . . . . . . . . . . . . . . . 16 (((π / 2) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (-(π / 2) < 𝑦 ↔ -𝑦 < (π / 2)))
9455, 83, 93sylancr 589 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (-(π / 2) < 𝑦 ↔ -𝑦 < (π / 2)))
9592, 94mpbid 234 . . . . . . . . . . . . . 14 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 < (π / 2))
96 0re 10621 . . . . . . . . . . . . . . 15 0 ∈ ℝ
97 elico2 12780 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ*) → (-𝑦 ∈ (0[,)(π / 2)) ↔ (-𝑦 ∈ ℝ ∧ 0 ≤ -𝑦 ∧ -𝑦 < (π / 2))))
9896, 60, 97mp2an 690 . . . . . . . . . . . . . 14 (-𝑦 ∈ (0[,)(π / 2)) ↔ (-𝑦 ∈ ℝ ∧ 0 ≤ -𝑦 ∧ -𝑦 < (π / 2)))
9986, 89, 95, 98syl3anbrc 1339 . . . . . . . . . . . . 13 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 ∈ (0[,)(π / 2)))
10082renegcld 11045 . . . . . . . . . . . . . 14 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑥 ∈ ℝ)
101 simp3 1134 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
102 0red 10622 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → 0 ∈ ℝ)
103 ltletr 10710 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → ((𝑥 < 𝑦𝑦 ≤ 0) → 𝑥 < 0))
10415, 36, 102, 103syl3anc 1367 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → ((𝑥 < 𝑦𝑦 ≤ 0) → 𝑥 < 0))
105101, 104mpand 693 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → (𝑦 ≤ 0 → 𝑥 < 0))
106105imp 409 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 < 0)
107 ltle 10707 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑥 < 0 → 𝑥 ≤ 0))
10882, 96, 107sylancl 588 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (𝑥 < 0 → 𝑥 ≤ 0))
109106, 108mpd 15 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 ≤ 0)
11082le0neg1d 11189 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (𝑥 ≤ 0 ↔ 0 ≤ -𝑥))
111109, 110mpbid 234 . . . . . . . . . . . . . 14 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 0 ≤ -𝑥)
112 simpl1 1187 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑥 ∈ (-(π / 2)(,)(π / 2)))
113112, 52syl 17 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (-(π / 2) < 𝑥𝑥 < (π / 2)))
114113simpld 497 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(π / 2) < 𝑥)
11555, 82, 56sylancr 589 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (-(π / 2) < 𝑥 ↔ -𝑥 < (π / 2)))
116114, 115mpbid 234 . . . . . . . . . . . . . 14 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑥 < (π / 2))
117 elico2 12780 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ*) → (-𝑥 ∈ (0[,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 ≤ -𝑥 ∧ -𝑥 < (π / 2))))
11896, 60, 117mp2an 690 . . . . . . . . . . . . . 14 (-𝑥 ∈ (0[,)(π / 2)) ↔ (-𝑥 ∈ ℝ ∧ 0 ≤ -𝑥 ∧ -𝑥 < (π / 2)))
119100, 111, 116, 118syl3anbrc 1339 . . . . . . . . . . . . 13 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑥 ∈ (0[,)(π / 2)))
120 tanord1 25108 . . . . . . . . . . . . 13 ((-𝑦 ∈ (0[,)(π / 2)) ∧ -𝑥 ∈ (0[,)(π / 2))) → (-𝑦 < -𝑥 ↔ (tan‘-𝑦) < (tan‘-𝑥)))
12199, 119, 120syl2anc 586 . . . . . . . . . . . 12 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (-𝑦 < -𝑥 ↔ (tan‘-𝑦) < (tan‘-𝑥)))
12285, 121mpbid 234 . . . . . . . . . . 11 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘-𝑦) < (tan‘-𝑥))
12383recnd 10647 . . . . . . . . . . . . . . 15 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → 𝑦 ∈ ℂ)
124 cosneg 15480 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℂ → (cos‘-𝑦) = (cos‘𝑦))
125123, 124syl 17 . . . . . . . . . . . . . 14 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘-𝑦) = (cos‘𝑦))
12690, 45syl 17 . . . . . . . . . . . . . 14 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘𝑦) ≠ 0)
127125, 126eqnetrd 3073 . . . . . . . . . . . . 13 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘-𝑦) ≠ 0)
12886, 127retancld 15478 . . . . . . . . . . . 12 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘-𝑦) ∈ ℝ)
129106, 25syldan 593 . . . . . . . . . . . . 13 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (cos‘-𝑥) ≠ 0)
130100, 129retancld 15478 . . . . . . . . . . . 12 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘-𝑥) ∈ ℝ)
131128, 130ltnegd 11196 . . . . . . . . . . 11 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → ((tan‘-𝑦) < (tan‘-𝑥) ↔ -(tan‘-𝑥) < -(tan‘-𝑦)))
132122, 131mpbid 234 . . . . . . . . . 10 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(tan‘-𝑥) < -(tan‘-𝑦))
133123negnegd 10966 . . . . . . . . . . . 12 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → --𝑦 = 𝑦)
134133fveq2d 6650 . . . . . . . . . . 11 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘--𝑦) = (tan‘𝑦))
135123negcld 10962 . . . . . . . . . . . 12 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -𝑦 ∈ ℂ)
136 tanneg 15481 . . . . . . . . . . . 12 ((-𝑦 ∈ ℂ ∧ (cos‘-𝑦) ≠ 0) → (tan‘--𝑦) = -(tan‘-𝑦))
137135, 127, 136syl2anc 586 . . . . . . . . . . 11 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘--𝑦) = -(tan‘-𝑦))
138134, 137eqtr3d 2857 . . . . . . . . . 10 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → (tan‘𝑦) = -(tan‘-𝑦))
139132, 138breqtrrd 5070 . . . . . . . . 9 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑦 ≤ 0) → -(tan‘-𝑥) < (tan‘𝑦))
140139adantlr 713 . . . . . . . 8 ((((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) ∧ 𝑦 ≤ 0) → -(tan‘-𝑥) < (tan‘𝑦))
141 0red 10622 . . . . . . . 8 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 0 ∈ ℝ)
142 simpl2 1188 . . . . . . . . 9 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
1435, 142sseldi 3944 . . . . . . . 8 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → 𝑦 ∈ ℝ)
14480, 140, 141, 143ltlecasei 10726 . . . . . . 7 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → -(tan‘-𝑥) < (tan‘𝑦))
14528, 144eqbrtrd 5064 . . . . . 6 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 𝑥 < 0) → (tan‘𝑥) < (tan‘𝑦))
146 simpl3 1189 . . . . . . 7 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 < 𝑦)
14715adantr 483 . . . . . . . . 9 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ∈ ℝ)
148 simpr 487 . . . . . . . . 9 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 0 ≤ 𝑥)
149 simpl1 1187 . . . . . . . . . . 11 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ∈ (-(π / 2)(,)(π / 2)))
150149, 52syl 17 . . . . . . . . . 10 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → (-(π / 2) < 𝑥𝑥 < (π / 2)))
151150simprd 498 . . . . . . . . 9 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 < (π / 2))
152 elico2 12780 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ*) → (𝑥 ∈ (0[,)(π / 2)) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥𝑥 < (π / 2))))
15396, 60, 152mp2an 690 . . . . . . . . 9 (𝑥 ∈ (0[,)(π / 2)) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥𝑥 < (π / 2)))
154147, 148, 151, 153syl3anbrc 1339 . . . . . . . 8 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥 ∈ (0[,)(π / 2)))
155 simpl2 1188 . . . . . . . . . 10 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ (-(π / 2)(,)(π / 2)))
1565, 155sseldi 3944 . . . . . . . . 9 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ ℝ)
157 0red 10622 . . . . . . . . . 10 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 0 ∈ ℝ)
158147, 156, 146ltled 10766 . . . . . . . . . 10 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑥𝑦)
159157, 147, 156, 148, 158letrd 10775 . . . . . . . . 9 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 0 ≤ 𝑦)
160155, 70syl 17 . . . . . . . . . 10 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → (-(π / 2) < 𝑦𝑦 < (π / 2)))
161160simprd 498 . . . . . . . . 9 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 < (π / 2))
162 elico2 12780 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ*) → (𝑦 ∈ (0[,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦𝑦 < (π / 2))))
16396, 60, 162mp2an 690 . . . . . . . . 9 (𝑦 ∈ (0[,)(π / 2)) ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦𝑦 < (π / 2)))
164156, 159, 161, 163syl3anbrc 1339 . . . . . . . 8 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → 𝑦 ∈ (0[,)(π / 2)))
165 tanord1 25108 . . . . . . . 8 ((𝑥 ∈ (0[,)(π / 2)) ∧ 𝑦 ∈ (0[,)(π / 2))) → (𝑥 < 𝑦 ↔ (tan‘𝑥) < (tan‘𝑦)))
166154, 164, 165syl2anc 586 . . . . . . 7 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → (𝑥 < 𝑦 ↔ (tan‘𝑥) < (tan‘𝑦)))
167146, 166mpbid 234 . . . . . 6 (((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) ∧ 0 ≤ 𝑥) → (tan‘𝑥) < (tan‘𝑦))
168145, 167, 15, 102ltlecasei 10726 . . . . 5 ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑥 < 𝑦) → (tan‘𝑥) < (tan‘𝑦))
1691683expia 1117 . . . 4 ((𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2))) → (𝑥 < 𝑦 → (tan‘𝑥) < (tan‘𝑦)))
170169adantl 484 . . 3 ((⊤ ∧ (𝑥 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝑦 ∈ (-(π / 2)(,)(π / 2)))) → (𝑥 < 𝑦 → (tan‘𝑥) < (tan‘𝑦)))
1712, 3, 4, 5, 14, 170ltord1 11144 . 2 ((⊤ ∧ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ∧ 𝐵 ∈ (-(π / 2)(,)(π / 2)))) → (𝐴 < 𝐵 ↔ (tan‘𝐴) < (tan‘𝐵)))
1721, 171mpan 688 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 208  wa 398  w3a 1083   = wceq 1537  wtru 1538  wcel 2114  wne 3006   class class class wbr 5042  cfv 6331  (class class class)co 7133  cc 10513  cr 10514  0cc0 10515  *cxr 10652   < clt 10653  cle 10654  -cneg 10849   / cdiv 11275  2c2 11671  +crp 12368  (,)cioo 12717  [,)cico 12719  cre 14436  cosccos 15398  tanctan 15399  πcpi 15400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-inf2 9082  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592  ax-pre-sup 10593  ax-addf 10594  ax-mulf 10595
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-iin 4898  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-se 5491  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-isom 6340  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-of 7387  df-om 7559  df-1st 7667  df-2nd 7668  df-supp 7809  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-2o 8081  df-oadd 8084  df-er 8267  df-map 8386  df-pm 8387  df-ixp 8440  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-fsupp 8812  df-fi 8853  df-sup 8884  df-inf 8885  df-oi 8952  df-card 9346  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-div 11276  df-nn 11617  df-2 11679  df-3 11680  df-4 11681  df-5 11682  df-6 11683  df-7 11684  df-8 11685  df-9 11686  df-n0 11877  df-z 11961  df-dec 12078  df-uz 12223  df-q 12328  df-rp 12369  df-xneg 12486  df-xadd 12487  df-xmul 12488  df-ioo 12721  df-ioc 12722  df-ico 12723  df-icc 12724  df-fz 12877  df-fzo 13018  df-fl 13146  df-mod 13222  df-seq 13354  df-exp 13415  df-fac 13619  df-bc 13648  df-hash 13676  df-shft 14406  df-cj 14438  df-re 14439  df-im 14440  df-sqrt 14574  df-abs 14575  df-limsup 14808  df-clim 14825  df-rlim 14826  df-sum 15023  df-ef 15401  df-sin 15403  df-cos 15404  df-tan 15405  df-pi 15406  df-struct 16464  df-ndx 16465  df-slot 16466  df-base 16468  df-sets 16469  df-ress 16470  df-plusg 16557  df-mulr 16558  df-starv 16559  df-sca 16560  df-vsca 16561  df-ip 16562  df-tset 16563  df-ple 16564  df-ds 16566  df-unif 16567  df-hom 16568  df-cco 16569  df-rest 16675  df-topn 16676  df-0g 16694  df-gsum 16695  df-topgen 16696  df-pt 16697  df-prds 16700  df-xrs 16754  df-qtop 16759  df-imas 16760  df-xps 16762  df-mre 16836  df-mrc 16837  df-acs 16839  df-mgm 17831  df-sgrp 17880  df-mnd 17891  df-submnd 17936  df-mulg 18204  df-cntz 18426  df-cmn 18887  df-psmet 20513  df-xmet 20514  df-met 20515  df-bl 20516  df-mopn 20517  df-fbas 20518  df-fg 20519  df-cnfld 20522  df-top 21478  df-topon 21495  df-topsp 21517  df-bases 21530  df-cld 21603  df-ntr 21604  df-cls 21605  df-nei 21682  df-lp 21720  df-perf 21721  df-cn 21811  df-cnp 21812  df-haus 21899  df-tx 22146  df-hmeo 22339  df-fil 22430  df-fm 22522  df-flim 22523  df-flf 22524  df-xms 22906  df-ms 22907  df-tms 22908  df-cncf 23462  df-limc 24448  df-dv 24449
This theorem is referenced by:  atanlogsublem  25480  atanord  25492  basellem4  25648
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