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Theorem dvtan 33815
Description: Derivative of tangent. (Contributed by Brendan Leahy, 7-Aug-2018.)
Assertion
Ref Expression
dvtan (ℂ D tan) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2))

Proof of Theorem dvtan
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-tan 15085 . . . 4 tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
2 cnvimass 5666 . . . . . . . . 9 (cos “ (ℂ ∖ {0})) ⊆ dom cos
3 cosf 15138 . . . . . . . . . 10 cos:ℂ⟶ℂ
43fdmi 6232 . . . . . . . . 9 dom cos = ℂ
52, 4sseqtri 3796 . . . . . . . 8 (cos “ (ℂ ∖ {0})) ⊆ ℂ
65sseli 3756 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
76sincld 15143 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
86coscld 15144 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ ℂ)
9 ffn 6222 . . . . . . . 8 (cos:ℂ⟶ℂ → cos Fn ℂ)
10 elpreima 6526 . . . . . . . 8 (cos Fn ℂ → (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↔ (𝑥 ∈ ℂ ∧ (cos‘𝑥) ∈ (ℂ ∖ {0}))))
113, 9, 10mp2b 10 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↔ (𝑥 ∈ ℂ ∧ (cos‘𝑥) ∈ (ℂ ∖ {0})))
12 eldifsni 4475 . . . . . . . 8 ((cos‘𝑥) ∈ (ℂ ∖ {0}) → (cos‘𝑥) ≠ 0)
1312adantl 473 . . . . . . 7 ((𝑥 ∈ ℂ ∧ (cos‘𝑥) ∈ (ℂ ∖ {0})) → (cos‘𝑥) ≠ 0)
1411, 13sylbi 208 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (cos‘𝑥) ≠ 0)
157, 8, 14divrecd 11057 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / (cos‘𝑥)) = ((sin‘𝑥) · (1 / (cos‘𝑥))))
1615mpteq2ia 4898 . . . 4 (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
171, 16eqtri 2786 . . 3 tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
1817oveq2i 6852 . 2 (ℂ D tan) = (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥)))))
19 cnelprrecn 10281 . . . . 5 ℂ ∈ {ℝ, ℂ}
2019a1i 11 . . . 4 (⊤ → ℂ ∈ {ℝ, ℂ})
21 difss 3898 . . . . . . . . 9 (ℂ ∖ {0}) ⊆ ℂ
22 imass2 5682 . . . . . . . . 9 ((ℂ ∖ {0}) ⊆ ℂ → (cos “ (ℂ ∖ {0})) ⊆ (cos “ ℂ))
2321, 22ax-mp 5 . . . . . . . 8 (cos “ (ℂ ∖ {0})) ⊆ (cos “ ℂ)
24 fimacnv 6536 . . . . . . . . 9 (cos:ℂ⟶ℂ → (cos “ ℂ) = ℂ)
253, 24ax-mp 5 . . . . . . . 8 (cos “ ℂ) = ℂ
2623, 25sseqtri 3796 . . . . . . 7 (cos “ (ℂ ∖ {0})) ⊆ ℂ
2726sseli 3756 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
2827sincld 15143 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
2928adantl 473 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (sin‘𝑥) ∈ ℂ)
308adantl 473 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (cos‘𝑥) ∈ ℂ)
31 sincl 15139 . . . . . 6 (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
3231adantl 473 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
33 coscl 15140 . . . . . 6 (𝑥 ∈ ℂ → (cos‘𝑥) ∈ ℂ)
3433adantl 473 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ ℂ)
35 dvsin 24035 . . . . . 6 (ℂ D sin) = cos
36 sinf 15137 . . . . . . . . 9 sin:ℂ⟶ℂ
3736a1i 11 . . . . . . . 8 (⊤ → sin:ℂ⟶ℂ)
3837feqmptd 6437 . . . . . . 7 (⊤ → sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
3938oveq2d 6857 . . . . . 6 (⊤ → (ℂ D sin) = (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))))
403a1i 11 . . . . . . 7 (⊤ → cos:ℂ⟶ℂ)
4140feqmptd 6437 . . . . . 6 (⊤ → cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
4235, 39, 413eqtr3a 2822 . . . . 5 (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
4326a1i 11 . . . . 5 (⊤ → (cos “ (ℂ ∖ {0})) ⊆ ℂ)
44 eqid 2764 . . . . . . 7 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
4544cnfldtopon 22864 . . . . . 6 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
4645toponrestid 21004 . . . . 5 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
47 dvtanlem 33814 . . . . . 6 (cos “ (ℂ ∖ {0})) ∈ (TopOpen‘ℂfld)
4847a1i 11 . . . . 5 (⊤ → (cos “ (ℂ ∖ {0})) ∈ (TopOpen‘ℂfld))
4920, 32, 34, 42, 43, 46, 44, 48dvmptres 24016 . . . 4 (⊤ → (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (sin‘𝑥))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥)))
508, 14reccld 11047 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (1 / (cos‘𝑥)) ∈ ℂ)
5150adantl 473 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (1 / (cos‘𝑥)) ∈ ℂ)
52 ovexd 6875 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) ∈ V)
5311simprbi 490 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
5453adantl 473 . . . . 5 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
5529negcld 10632 . . . . 5 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → -(sin‘𝑥) ∈ ℂ)
56 eldifi 3893 . . . . . . 7 (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ)
57 eldifsni 4475 . . . . . . 7 (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0)
5856, 57reccld 11047 . . . . . 6 (𝑦 ∈ (ℂ ∖ {0}) → (1 / 𝑦) ∈ ℂ)
5958adantl 473 . . . . 5 ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (1 / 𝑦) ∈ ℂ)
60 negex 10532 . . . . . 6 -(1 / (𝑦↑2)) ∈ V
6160a1i 11 . . . . 5 ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(1 / (𝑦↑2)) ∈ V)
6232negcld 10632 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℂ) → -(sin‘𝑥) ∈ ℂ)
63 dvcos 24036 . . . . . . 7 (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥))
6441oveq2d 6857 . . . . . . 7 (⊤ → (ℂ D cos) = (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))))
6563, 64syl5reqr 2813 . . . . . 6 (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)))
6620, 34, 62, 65, 43, 46, 44, 48dvmptres 24016 . . . . 5 (⊤ → (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ -(sin‘𝑥)))
67 ax-1cn 10246 . . . . . 6 1 ∈ ℂ
68 dvrec 24008 . . . . . 6 (1 ∈ ℂ → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(1 / (𝑦↑2))))
6967, 68mp1i 13 . . . . 5 (⊤ → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(1 / (𝑦↑2))))
70 oveq2 6849 . . . . 5 (𝑦 = (cos‘𝑥) → (1 / 𝑦) = (1 / (cos‘𝑥)))
71 oveq1 6848 . . . . . . 7 (𝑦 = (cos‘𝑥) → (𝑦↑2) = ((cos‘𝑥)↑2))
7271oveq2d 6857 . . . . . 6 (𝑦 = (cos‘𝑥) → (1 / (𝑦↑2)) = (1 / ((cos‘𝑥)↑2)))
7372negeqd 10528 . . . . 5 (𝑦 = (cos‘𝑥) → -(1 / (𝑦↑2)) = -(1 / ((cos‘𝑥)↑2)))
7420, 20, 54, 55, 59, 61, 66, 69, 70, 73dvmptco 24025 . . . 4 (⊤ → (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (1 / (cos‘𝑥)))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥))))
7520, 29, 30, 49, 51, 52, 74dvmptmul 24014 . . 3 (⊤ → (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)))))
7675mptru 1660 . 2 (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))))
77 ovex 6873 . . . . 5 ((sin‘𝑥) / (cos‘𝑥)) ∈ V
7877, 1dmmpti 6200 . . . 4 dom tan = (cos “ (ℂ ∖ {0}))
7978eqcomi 2773 . . 3 (cos “ (ℂ ∖ {0})) = dom tan
808sqcld 13212 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ∈ ℂ)
817sqcld 13212 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) ∈ ℂ)
82 sqne0 13136 . . . . . . . . 9 ((cos‘𝑥) ∈ ℂ → (((cos‘𝑥)↑2) ≠ 0 ↔ (cos‘𝑥) ≠ 0))
838, 82syl 17 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) ≠ 0 ↔ (cos‘𝑥) ≠ 0))
8414, 83mpbird 248 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ≠ 0)
8580, 81, 80, 84divdird 11092 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) / ((cos‘𝑥)↑2)) = ((((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) + (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2))))
8680, 81addcomd 10491 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)))
87 sincossq 15189 . . . . . . . . 9 (𝑥 ∈ ℂ → (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1)
886, 87syl 17 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1)
8986, 88eqtrd 2798 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = 1)
9089oveq1d 6856 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) / ((cos‘𝑥)↑2)) = (1 / ((cos‘𝑥)↑2)))
9185, 90eqtr3d 2800 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) + (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2))) = (1 / ((cos‘𝑥)↑2)))
928, 14recidd 11049 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = 1)
9380, 84dividd 11052 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) = 1)
9492, 93eqtr4d 2801 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)))
957, 7, 80, 84div23d 11091 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
967sqvald 13211 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) = ((sin‘𝑥) · (sin‘𝑥)))
9796oveq1d 6856 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)))
9880, 84reccld 11047 . . . . . . . . . 10 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (1 / ((cos‘𝑥)↑2)) ∈ ℂ)
9998, 7mul2negd 10738 . . . . . . . . 9 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
1007, 80, 84divrec2d 11058 . . . . . . . . 9 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / ((cos‘𝑥)↑2)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
10199, 100eqtr4d 2801 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((sin‘𝑥) / ((cos‘𝑥)↑2)))
102101oveq1d 6856 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)) = (((sin‘𝑥) / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
10395, 97, 1023eqtr4rd 2809 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)) = (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)))
10494, 103oveq12d 6859 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) + (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2))))
105 2nn0 11556 . . . . . 6 2 ∈ ℕ0
106 expneg 13074 . . . . . 6 (((cos‘𝑥) ∈ ℂ ∧ 2 ∈ ℕ0) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
1078, 105, 106sylancl 580 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
10891, 104, 1073eqtr4d 2808 . . . 4 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2))
109108rgen 3068 . . 3 𝑥 ∈ (cos “ (ℂ ∖ {0}))(((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2)
110 mpteq12 4894 . . 3 (((cos “ (ℂ ∖ {0})) = dom tan ∧ ∀𝑥 ∈ (cos “ (ℂ ∖ {0}))(((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2)) → (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)))) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2)))
11179, 109, 110mp2an 683 . 2 (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)))) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2))
11218, 76, 1113eqtri 2790 1 (ℂ D tan) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wtru 1653  wcel 2155  wne 2936  wral 3054  Vcvv 3349  cdif 3728  wss 3731  {csn 4333  {cpr 4335  cmpt 4887  ccnv 5275  dom cdm 5276  cima 5279   Fn wfn 6062  wf 6063  cfv 6067  (class class class)co 6841  cc 10186  cr 10187  0cc0 10188  1c1 10189   + caddc 10191   · cmul 10193  -cneg 10520   / cdiv 10937  2c2 11326  0cn0 11537  cexp 13066  sincsin 15077  cosccos 15078  tanctan 15079  TopOpenctopn 16349  fldccnfld 20018   D cdv 23917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-inf2 8752  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265  ax-pre-sup 10266  ax-addf 10267  ax-mulf 10268
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-int 4633  df-iun 4677  df-iin 4678  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-se 5236  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-isom 6076  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-of 7094  df-om 7263  df-1st 7365  df-2nd 7366  df-supp 7497  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-2o 7764  df-oadd 7767  df-er 7946  df-map 8061  df-pm 8062  df-ixp 8113  df-en 8160  df-dom 8161  df-sdom 8162  df-fin 8163  df-fsupp 8482  df-fi 8523  df-sup 8554  df-inf 8555  df-oi 8621  df-card 9015  df-cda 9242  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-div 10938  df-nn 11274  df-2 11334  df-3 11335  df-4 11336  df-5 11337  df-6 11338  df-7 11339  df-8 11340  df-9 11341  df-n0 11538  df-z 11624  df-dec 11740  df-uz 11886  df-q 11989  df-rp 12028  df-xneg 12145  df-xadd 12146  df-xmul 12147  df-ico 12382  df-icc 12383  df-fz 12533  df-fzo 12673  df-fl 12800  df-seq 13008  df-exp 13067  df-fac 13264  df-bc 13293  df-hash 13321  df-shft 14093  df-cj 14125  df-re 14126  df-im 14127  df-sqrt 14261  df-abs 14262  df-limsup 14488  df-clim 14505  df-rlim 14506  df-sum 14703  df-ef 15081  df-sin 15083  df-cos 15084  df-tan 15085  df-struct 16133  df-ndx 16134  df-slot 16135  df-base 16137  df-sets 16138  df-ress 16139  df-plusg 16228  df-mulr 16229  df-starv 16230  df-sca 16231  df-vsca 16232  df-ip 16233  df-tset 16234  df-ple 16235  df-ds 16237  df-unif 16238  df-hom 16239  df-cco 16240  df-rest 16350  df-topn 16351  df-0g 16369  df-gsum 16370  df-topgen 16371  df-pt 16372  df-prds 16375  df-xrs 16429  df-qtop 16434  df-imas 16435  df-xps 16437  df-mre 16513  df-mrc 16514  df-acs 16516  df-mgm 17509  df-sgrp 17551  df-mnd 17562  df-submnd 17603  df-mulg 17809  df-cntz 18014  df-cmn 18460  df-psmet 20010  df-xmet 20011  df-met 20012  df-bl 20013  df-mopn 20014  df-fbas 20015  df-fg 20016  df-cnfld 20019  df-top 20977  df-topon 20994  df-topsp 21016  df-bases 21029  df-cld 21102  df-ntr 21103  df-cls 21104  df-nei 21181  df-lp 21219  df-perf 21220  df-cn 21310  df-cnp 21311  df-t1 21397  df-haus 21398  df-tx 21644  df-hmeo 21837  df-fil 21928  df-fm 22020  df-flim 22021  df-flf 22022  df-xms 22403  df-ms 22404  df-tms 22405  df-cncf 22959  df-limc 23920  df-dv 23921
This theorem is referenced by: (None)
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