Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvtan Structured version   Visualization version   GIF version

Theorem dvtan 35564
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 15633 . . . 4 tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
2 cnvimass 5949 . . . . . . . . 9 (cos “ (ℂ ∖ {0})) ⊆ dom cos
3 cosf 15686 . . . . . . . . . 10 cos:ℂ⟶ℂ
43fdmi 6557 . . . . . . . . 9 dom cos = ℂ
52, 4sseqtri 3937 . . . . . . . 8 (cos “ (ℂ ∖ {0})) ⊆ ℂ
65sseli 3896 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
76sincld 15691 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
86coscld 15692 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ ℂ)
9 ffn 6545 . . . . . . . 8 (cos:ℂ⟶ℂ → cos Fn ℂ)
10 elpreima 6878 . . . . . . . 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 4703 . . . . . . . 8 ((cos‘𝑥) ∈ (ℂ ∖ {0}) → (cos‘𝑥) ≠ 0)
1312adantl 485 . . . . . . 7 ((𝑥 ∈ ℂ ∧ (cos‘𝑥) ∈ (ℂ ∖ {0})) → (cos‘𝑥) ≠ 0)
1411, 13sylbi 220 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (cos‘𝑥) ≠ 0)
157, 8, 14divrecd 11611 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / (cos‘𝑥)) = ((sin‘𝑥) · (1 / (cos‘𝑥))))
1615mpteq2ia 5146 . . . 4 (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
171, 16eqtri 2765 . . 3 tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
1817oveq2i 7224 . 2 (ℂ D tan) = (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥)))))
19 cnelprrecn 10822 . . . . 5 ℂ ∈ {ℝ, ℂ}
2019a1i 11 . . . 4 (⊤ → ℂ ∈ {ℝ, ℂ})
21 difss 4046 . . . . . . . . 9 (ℂ ∖ {0}) ⊆ ℂ
22 imass2 5970 . . . . . . . . 9 ((ℂ ∖ {0}) ⊆ ℂ → (cos “ (ℂ ∖ {0})) ⊆ (cos “ ℂ))
2321, 22ax-mp 5 . . . . . . . 8 (cos “ (ℂ ∖ {0})) ⊆ (cos “ ℂ)
24 fimacnv 6567 . . . . . . . . 9 (cos:ℂ⟶ℂ → (cos “ ℂ) = ℂ)
253, 24ax-mp 5 . . . . . . . 8 (cos “ ℂ) = ℂ
2623, 25sseqtri 3937 . . . . . . 7 (cos “ (ℂ ∖ {0})) ⊆ ℂ
2726sseli 3896 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
2827sincld 15691 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
2928adantl 485 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (sin‘𝑥) ∈ ℂ)
308adantl 485 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (cos‘𝑥) ∈ ℂ)
31 sincl 15687 . . . . . 6 (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
3231adantl 485 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
33 coscl 15688 . . . . . 6 (𝑥 ∈ ℂ → (cos‘𝑥) ∈ ℂ)
3433adantl 485 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ ℂ)
35 dvsin 24879 . . . . . 6 (ℂ D sin) = cos
36 sinf 15685 . . . . . . . . 9 sin:ℂ⟶ℂ
3736a1i 11 . . . . . . . 8 (⊤ → sin:ℂ⟶ℂ)
3837feqmptd 6780 . . . . . . 7 (⊤ → sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
3938oveq2d 7229 . . . . . 6 (⊤ → (ℂ D sin) = (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))))
403a1i 11 . . . . . . 7 (⊤ → cos:ℂ⟶ℂ)
4140feqmptd 6780 . . . . . 6 (⊤ → cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
4235, 39, 413eqtr3a 2802 . . . . 5 (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
4326a1i 11 . . . . 5 (⊤ → (cos “ (ℂ ∖ {0})) ⊆ ℂ)
44 eqid 2737 . . . . . . 7 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
4544cnfldtopon 23680 . . . . . 6 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
4645toponrestid 21818 . . . . 5 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
47 dvtanlem 35563 . . . . . 6 (cos “ (ℂ ∖ {0})) ∈ (TopOpen‘ℂfld)
4847a1i 11 . . . . 5 (⊤ → (cos “ (ℂ ∖ {0})) ∈ (TopOpen‘ℂfld))
4920, 32, 34, 42, 43, 46, 44, 48dvmptres 24860 . . . 4 (⊤ → (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (sin‘𝑥))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥)))
508, 14reccld 11601 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (1 / (cos‘𝑥)) ∈ ℂ)
5150adantl 485 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (1 / (cos‘𝑥)) ∈ ℂ)
52 ovexd 7248 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) ∈ V)
5311simprbi 500 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
5453adantl 485 . . . . 5 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
5529negcld 11176 . . . . 5 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → -(sin‘𝑥) ∈ ℂ)
56 eldifi 4041 . . . . . . 7 (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ)
57 eldifsni 4703 . . . . . . 7 (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0)
5856, 57reccld 11601 . . . . . 6 (𝑦 ∈ (ℂ ∖ {0}) → (1 / 𝑦) ∈ ℂ)
5958adantl 485 . . . . 5 ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (1 / 𝑦) ∈ ℂ)
60 negex 11076 . . . . . 6 -(1 / (𝑦↑2)) ∈ V
6160a1i 11 . . . . 5 ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(1 / (𝑦↑2)) ∈ V)
6232negcld 11176 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℂ) → -(sin‘𝑥) ∈ ℂ)
6341oveq2d 7229 . . . . . . 7 (⊤ → (ℂ D cos) = (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))))
64 dvcos 24880 . . . . . . 7 (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥))
6563, 64eqtr3di 2793 . . . . . 6 (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)))
6620, 34, 62, 65, 43, 46, 44, 48dvmptres 24860 . . . . 5 (⊤ → (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ -(sin‘𝑥)))
67 ax-1cn 10787 . . . . . 6 1 ∈ ℂ
68 dvrec 24852 . . . . . 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 7221 . . . . 5 (𝑦 = (cos‘𝑥) → (1 / 𝑦) = (1 / (cos‘𝑥)))
71 oveq1 7220 . . . . . . 7 (𝑦 = (cos‘𝑥) → (𝑦↑2) = ((cos‘𝑥)↑2))
7271oveq2d 7229 . . . . . 6 (𝑦 = (cos‘𝑥) → (1 / (𝑦↑2)) = (1 / ((cos‘𝑥)↑2)))
7372negeqd 11072 . . . . 5 (𝑦 = (cos‘𝑥) → -(1 / (𝑦↑2)) = -(1 / ((cos‘𝑥)↑2)))
7420, 20, 54, 55, 59, 61, 66, 69, 70, 73dvmptco 24869 . . . 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 24858 . . 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 1550 . 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 7246 . . . . 5 ((sin‘𝑥) / (cos‘𝑥)) ∈ V
7877, 1dmmpti 6522 . . . 4 dom tan = (cos “ (ℂ ∖ {0}))
7978eqcomi 2746 . . 3 (cos “ (ℂ ∖ {0})) = dom tan
808sqcld 13714 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ∈ ℂ)
817sqcld 13714 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) ∈ ℂ)
82 sqne0 13695 . . . . . . . . 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 260 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ≠ 0)
8580, 81, 80, 84divdird 11646 . . . . . 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 11034 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)))
87 sincossq 15737 . . . . . . . . 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 2777 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = 1)
9089oveq1d 7228 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) / ((cos‘𝑥)↑2)) = (1 / ((cos‘𝑥)↑2)))
9185, 90eqtr3d 2779 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) + (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2))) = (1 / ((cos‘𝑥)↑2)))
928, 14recidd 11603 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = 1)
9380, 84dividd 11606 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) = 1)
9492, 93eqtr4d 2780 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)))
957, 7, 80, 84div23d 11645 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
967sqvald 13713 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) = ((sin‘𝑥) · (sin‘𝑥)))
9796oveq1d 7228 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)))
9880, 84reccld 11601 . . . . . . . . . 10 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (1 / ((cos‘𝑥)↑2)) ∈ ℂ)
9998, 7mul2negd 11287 . . . . . . . . 9 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
1007, 80, 84divrec2d 11612 . . . . . . . . 9 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / ((cos‘𝑥)↑2)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
10199, 100eqtr4d 2780 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((sin‘𝑥) / ((cos‘𝑥)↑2)))
102101oveq1d 7228 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)) = (((sin‘𝑥) / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
10395, 97, 1023eqtr4rd 2788 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)) = (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)))
10494, 103oveq12d 7231 . . . . 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 12107 . . . . . 6 2 ∈ ℕ0
106 expneg 13643 . . . . . 6 (((cos‘𝑥) ∈ ℂ ∧ 2 ∈ ℕ0) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
1078, 105, 106sylancl 589 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
10891, 104, 1073eqtr4d 2787 . . . 4 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2))
109108rgen 3071 . . 3 𝑥 ∈ (cos “ (ℂ ∖ {0}))(((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2)
110 mpteq12 5142 . . 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 692 . 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 2769 1 (ℂ D tan) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  wtru 1544  wcel 2110  wne 2940  wral 3061  Vcvv 3408  cdif 3863  wss 3866  {csn 4541  {cpr 4543  cmpt 5135  ccnv 5550  dom cdm 5551  cima 5554   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  cc 10727  cr 10728  0cc0 10729  1c1 10730   + caddc 10732   · cmul 10734  -cneg 11063   / cdiv 11489  2c2 11885  0cn0 12090  cexp 13635  sincsin 15625  cosccos 15626  tanctan 15627  TopOpenctopn 16926  fldccnfld 20363   D cdv 24760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807  ax-addf 10808  ax-mulf 10809
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-supp 7904  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-er 8391  df-map 8510  df-pm 8511  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fsupp 8986  df-fi 9027  df-sup 9058  df-inf 9059  df-oi 9126  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-dec 12294  df-uz 12439  df-q 12545  df-rp 12587  df-xneg 12704  df-xadd 12705  df-xmul 12706  df-ico 12941  df-icc 12942  df-fz 13096  df-fzo 13239  df-fl 13367  df-seq 13575  df-exp 13636  df-fac 13840  df-bc 13869  df-hash 13897  df-shft 14630  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-limsup 15032  df-clim 15049  df-rlim 15050  df-sum 15250  df-ef 15629  df-sin 15631  df-cos 15632  df-tan 15633  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-starv 16817  df-sca 16818  df-vsca 16819  df-ip 16820  df-tset 16821  df-ple 16822  df-ds 16824  df-unif 16825  df-hom 16826  df-cco 16827  df-rest 16927  df-topn 16928  df-0g 16946  df-gsum 16947  df-topgen 16948  df-pt 16949  df-prds 16952  df-xrs 17007  df-qtop 17012  df-imas 17013  df-xps 17015  df-mre 17089  df-mrc 17090  df-acs 17092  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-submnd 18219  df-mulg 18489  df-cntz 18711  df-cmn 19172  df-psmet 20355  df-xmet 20356  df-met 20357  df-bl 20358  df-mopn 20359  df-fbas 20360  df-fg 20361  df-cnfld 20364  df-top 21791  df-topon 21808  df-topsp 21830  df-bases 21843  df-cld 21916  df-ntr 21917  df-cls 21918  df-nei 21995  df-lp 22033  df-perf 22034  df-cn 22124  df-cnp 22125  df-t1 22211  df-haus 22212  df-tx 22459  df-hmeo 22652  df-fil 22743  df-fm 22835  df-flim 22836  df-flf 22837  df-xms 23218  df-ms 23219  df-tms 23220  df-cncf 23775  df-limc 24763  df-dv 24764
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator