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Theorem dvtan 33119
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 14734 . . . 4 tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
2 cnvimass 5449 . . . . . . . . 9 (cos “ (ℂ ∖ {0})) ⊆ dom cos
3 cosf 14787 . . . . . . . . . 10 cos:ℂ⟶ℂ
43fdmi 6014 . . . . . . . . 9 dom cos = ℂ
52, 4sseqtri 3621 . . . . . . . 8 (cos “ (ℂ ∖ {0})) ⊆ ℂ
65sseli 3583 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
76sincld 14792 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
86coscld 14793 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ ℂ)
9 ffn 6007 . . . . . . . 8 (cos:ℂ⟶ℂ → cos Fn ℂ)
10 elpreima 6298 . . . . . . . 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 4294 . . . . . . . 8 ((cos‘𝑥) ∈ (ℂ ∖ {0}) → (cos‘𝑥) ≠ 0)
1312adantl 482 . . . . . . 7 ((𝑥 ∈ ℂ ∧ (cos‘𝑥) ∈ (ℂ ∖ {0})) → (cos‘𝑥) ≠ 0)
1411, 13sylbi 207 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (cos‘𝑥) ≠ 0)
157, 8, 14divrecd 10755 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / (cos‘𝑥)) = ((sin‘𝑥) · (1 / (cos‘𝑥))))
1615mpteq2ia 4705 . . . 4 (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
171, 16eqtri 2643 . . 3 tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
1817oveq2i 6621 . 2 (ℂ D tan) = (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥)))))
19 cnelprrecn 9980 . . . . 5 ℂ ∈ {ℝ, ℂ}
2019a1i 11 . . . 4 (⊤ → ℂ ∈ {ℝ, ℂ})
21 difss 3720 . . . . . . . . 9 (ℂ ∖ {0}) ⊆ ℂ
22 imass2 5465 . . . . . . . . 9 ((ℂ ∖ {0}) ⊆ ℂ → (cos “ (ℂ ∖ {0})) ⊆ (cos “ ℂ))
2321, 22ax-mp 5 . . . . . . . 8 (cos “ (ℂ ∖ {0})) ⊆ (cos “ ℂ)
24 fimacnv 6308 . . . . . . . . 9 (cos:ℂ⟶ℂ → (cos “ ℂ) = ℂ)
253, 24ax-mp 5 . . . . . . . 8 (cos “ ℂ) = ℂ
2623, 25sseqtri 3621 . . . . . . 7 (cos “ (ℂ ∖ {0})) ⊆ ℂ
2726sseli 3583 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
2827sincld 14792 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
2928adantl 482 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (sin‘𝑥) ∈ ℂ)
308adantl 482 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (cos‘𝑥) ∈ ℂ)
31 sincl 14788 . . . . . 6 (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
3231adantl 482 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
33 coscl 14789 . . . . . 6 (𝑥 ∈ ℂ → (cos‘𝑥) ∈ ℂ)
3433adantl 482 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ ℂ)
35 dvsin 23662 . . . . . 6 (ℂ D sin) = cos
36 sinf 14786 . . . . . . . . 9 sin:ℂ⟶ℂ
3736a1i 11 . . . . . . . 8 (⊤ → sin:ℂ⟶ℂ)
3837feqmptd 6211 . . . . . . 7 (⊤ → sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
3938oveq2d 6626 . . . . . 6 (⊤ → (ℂ D sin) = (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))))
403a1i 11 . . . . . . 7 (⊤ → cos:ℂ⟶ℂ)
4140feqmptd 6211 . . . . . 6 (⊤ → cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
4235, 39, 413eqtr3a 2679 . . . . 5 (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
4326a1i 11 . . . . 5 (⊤ → (cos “ (ℂ ∖ {0})) ⊆ ℂ)
44 eqid 2621 . . . . . . . 8 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
4544cnfldtopon 22505 . . . . . . 7 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
4645toponunii 20652 . . . . . . . 8 ℂ = (TopOpen‘ℂfld)
4746restid 16022 . . . . . . 7 ((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
4845, 47ax-mp 5 . . . . . 6 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
4948eqcomi 2630 . . . . 5 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
50 dvtanlem 33118 . . . . . 6 (cos “ (ℂ ∖ {0})) ∈ (TopOpen‘ℂfld)
5150a1i 11 . . . . 5 (⊤ → (cos “ (ℂ ∖ {0})) ∈ (TopOpen‘ℂfld))
5220, 32, 34, 42, 43, 49, 44, 51dvmptres 23645 . . . 4 (⊤ → (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (sin‘𝑥))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥)))
538, 14reccld 10745 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (1 / (cos‘𝑥)) ∈ ℂ)
5453adantl 482 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (1 / (cos‘𝑥)) ∈ ℂ)
55 ovex 6638 . . . . 5 (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) ∈ V
5655a1i 11 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) ∈ V)
5711simprbi 480 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
5857adantl 482 . . . . 5 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
5929negcld 10330 . . . . 5 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → -(sin‘𝑥) ∈ ℂ)
60 eldifi 3715 . . . . . . 7 (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ)
61 eldifsni 4294 . . . . . . 7 (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0)
6260, 61reccld 10745 . . . . . 6 (𝑦 ∈ (ℂ ∖ {0}) → (1 / 𝑦) ∈ ℂ)
6362adantl 482 . . . . 5 ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (1 / 𝑦) ∈ ℂ)
64 negex 10230 . . . . . 6 -(1 / (𝑦↑2)) ∈ V
6564a1i 11 . . . . 5 ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(1 / (𝑦↑2)) ∈ V)
6632negcld 10330 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℂ) → -(sin‘𝑥) ∈ ℂ)
67 dvcos 23663 . . . . . . 7 (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥))
6841oveq2d 6626 . . . . . . 7 (⊤ → (ℂ D cos) = (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))))
6967, 68syl5reqr 2670 . . . . . 6 (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)))
7020, 34, 66, 69, 43, 49, 44, 51dvmptres 23645 . . . . 5 (⊤ → (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ -(sin‘𝑥)))
71 ax-1cn 9945 . . . . . 6 1 ∈ ℂ
72 dvrec 23637 . . . . . 6 (1 ∈ ℂ → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(1 / (𝑦↑2))))
7371, 72mp1i 13 . . . . 5 (⊤ → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(1 / (𝑦↑2))))
74 oveq2 6618 . . . . 5 (𝑦 = (cos‘𝑥) → (1 / 𝑦) = (1 / (cos‘𝑥)))
75 oveq1 6617 . . . . . . 7 (𝑦 = (cos‘𝑥) → (𝑦↑2) = ((cos‘𝑥)↑2))
7675oveq2d 6626 . . . . . 6 (𝑦 = (cos‘𝑥) → (1 / (𝑦↑2)) = (1 / ((cos‘𝑥)↑2)))
7776negeqd 10226 . . . . 5 (𝑦 = (cos‘𝑥) → -(1 / (𝑦↑2)) = -(1 / ((cos‘𝑥)↑2)))
7820, 20, 58, 59, 63, 65, 70, 73, 74, 77dvmptco 23654 . . . 4 (⊤ → (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (1 / (cos‘𝑥)))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥))))
7920, 29, 30, 52, 54, 56, 78dvmptmul 23643 . . 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)))))
8079trud 1490 . 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))))
81 ovex 6638 . . . . 5 ((sin‘𝑥) / (cos‘𝑥)) ∈ V
8281, 1dmmpti 5985 . . . 4 dom tan = (cos “ (ℂ ∖ {0}))
8382eqcomi 2630 . . 3 (cos “ (ℂ ∖ {0})) = dom tan
848sqcld 12953 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ∈ ℂ)
857sqcld 12953 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) ∈ ℂ)
86 sqne0 12877 . . . . . . . . 9 ((cos‘𝑥) ∈ ℂ → (((cos‘𝑥)↑2) ≠ 0 ↔ (cos‘𝑥) ≠ 0))
878, 86syl 17 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) ≠ 0 ↔ (cos‘𝑥) ≠ 0))
8814, 87mpbird 247 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ≠ 0)
8984, 85, 84, 88divdird 10790 . . . . . 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))))
9084, 85addcomd 10189 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)))
91 sincossq 14838 . . . . . . . . 9 (𝑥 ∈ ℂ → (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1)
926, 91syl 17 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1)
9390, 92eqtrd 2655 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = 1)
9493oveq1d 6625 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) / ((cos‘𝑥)↑2)) = (1 / ((cos‘𝑥)↑2)))
9589, 94eqtr3d 2657 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) + (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2))) = (1 / ((cos‘𝑥)↑2)))
968, 14recidd 10747 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = 1)
9784, 88dividd 10750 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) = 1)
9896, 97eqtr4d 2658 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)))
997, 7, 84, 88div23d 10789 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
1007sqvald 12952 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) = ((sin‘𝑥) · (sin‘𝑥)))
101100oveq1d 6625 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)))
10284, 88reccld 10745 . . . . . . . . . 10 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (1 / ((cos‘𝑥)↑2)) ∈ ℂ)
103102, 7mul2negd 10436 . . . . . . . . 9 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
1047, 84, 88divrec2d 10756 . . . . . . . . 9 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / ((cos‘𝑥)↑2)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
105103, 104eqtr4d 2658 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((sin‘𝑥) / ((cos‘𝑥)↑2)))
106105oveq1d 6625 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)) = (((sin‘𝑥) / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
10799, 101, 1063eqtr4rd 2666 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)) = (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)))
10898, 107oveq12d 6628 . . . . 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))))
109 2nn0 11260 . . . . . 6 2 ∈ ℕ0
110 expneg 12815 . . . . . 6 (((cos‘𝑥) ∈ ℂ ∧ 2 ∈ ℕ0) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
1118, 109, 110sylancl 693 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
11295, 108, 1113eqtr4d 2665 . . . 4 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2))
113112rgen 2917 . . 3 𝑥 ∈ (cos “ (ℂ ∖ {0}))(((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2)
114 mpteq12 4701 . . 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)))
11583, 113, 114mp2an 707 . 2 (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)))) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2))
11618, 80, 1153eqtri 2647 1 (ℂ D tan) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wtru 1481  wcel 1987  wne 2790  wral 2907  Vcvv 3189  cdif 3556  wss 3559  {csn 4153  {cpr 4155  cmpt 4678  ccnv 5078  dom cdm 5079  cima 5082   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  cc 9885  cr 9886  0cc0 9887  1c1 9888   + caddc 9890   · cmul 9892  -cneg 10218   / cdiv 10635  2c2 11021  0cn0 11243  cexp 12807  sincsin 14726  cosccos 14727  tanctan 14728  t crest 16009  TopOpenctopn 16010  fldccnfld 19674  TopOnctopon 20643   D cdv 23546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965  ax-addf 9966  ax-mulf 9967
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-ixp 7860  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-fsupp 8227  df-fi 8268  df-sup 8299  df-inf 8300  df-oi 8366  df-card 8716  df-cda 8941  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-nn 10972  df-2 11030  df-3 11031  df-4 11032  df-5 11033  df-6 11034  df-7 11035  df-8 11036  df-9 11037  df-n0 11244  df-z 11329  df-dec 11445  df-uz 11639  df-q 11740  df-rp 11784  df-xneg 11897  df-xadd 11898  df-xmul 11899  df-ico 12130  df-icc 12131  df-fz 12276  df-fzo 12414  df-fl 12540  df-seq 12749  df-exp 12808  df-fac 13008  df-bc 13037  df-hash 13065  df-shft 13748  df-cj 13780  df-re 13781  df-im 13782  df-sqrt 13916  df-abs 13917  df-limsup 14143  df-clim 14160  df-rlim 14161  df-sum 14358  df-ef 14730  df-sin 14732  df-cos 14733  df-tan 14734  df-struct 15790  df-ndx 15791  df-slot 15792  df-base 15793  df-sets 15794  df-ress 15795  df-plusg 15882  df-mulr 15883  df-starv 15884  df-sca 15885  df-vsca 15886  df-ip 15887  df-tset 15888  df-ple 15889  df-ds 15892  df-unif 15893  df-hom 15894  df-cco 15895  df-rest 16011  df-topn 16012  df-0g 16030  df-gsum 16031  df-topgen 16032  df-pt 16033  df-prds 16036  df-xrs 16090  df-qtop 16095  df-imas 16096  df-xps 16098  df-mre 16174  df-mrc 16175  df-acs 16177  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-submnd 17264  df-mulg 17469  df-cntz 17678  df-cmn 18123  df-psmet 19666  df-xmet 19667  df-met 19668  df-bl 19669  df-mopn 19670  df-fbas 19671  df-fg 19672  df-cnfld 19675  df-top 20627  df-topon 20644  df-topsp 20657  df-bases 20670  df-cld 20742  df-ntr 20743  df-cls 20744  df-nei 20821  df-lp 20859  df-perf 20860  df-cn 20950  df-cnp 20951  df-t1 21037  df-haus 21038  df-tx 21284  df-hmeo 21477  df-fil 21569  df-fm 21661  df-flim 21662  df-flf 21663  df-xms 22044  df-ms 22045  df-tms 22046  df-cncf 22600  df-limc 23549  df-dv 23550
This theorem is referenced by: (None)
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