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Theorem dvtan 36128
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 15954 . . . 4 tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
2 cnvimass 6033 . . . . . . . . 9 (cos “ (ℂ ∖ {0})) ⊆ dom cos
3 cosf 16007 . . . . . . . . . 10 cos:ℂ⟶ℂ
43fdmi 6680 . . . . . . . . 9 dom cos = ℂ
52, 4sseqtri 3980 . . . . . . . 8 (cos “ (ℂ ∖ {0})) ⊆ ℂ
65sseli 3940 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
76sincld 16012 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
86coscld 16013 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ ℂ)
9 ffn 6668 . . . . . . . 8 (cos:ℂ⟶ℂ → cos Fn ℂ)
10 elpreima 7008 . . . . . . . 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 4750 . . . . . . . 8 ((cos‘𝑥) ∈ (ℂ ∖ {0}) → (cos‘𝑥) ≠ 0)
1312adantl 482 . . . . . . 7 ((𝑥 ∈ ℂ ∧ (cos‘𝑥) ∈ (ℂ ∖ {0})) → (cos‘𝑥) ≠ 0)
1411, 13sylbi 216 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (cos‘𝑥) ≠ 0)
157, 8, 14divrecd 11934 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / (cos‘𝑥)) = ((sin‘𝑥) · (1 / (cos‘𝑥))))
1615mpteq2ia 5208 . . . 4 (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
171, 16eqtri 2764 . . 3 tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
1817oveq2i 7368 . 2 (ℂ D tan) = (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥)))))
19 cnelprrecn 11144 . . . . 5 ℂ ∈ {ℝ, ℂ}
2019a1i 11 . . . 4 (⊤ → ℂ ∈ {ℝ, ℂ})
21 difss 4091 . . . . . . . . 9 (ℂ ∖ {0}) ⊆ ℂ
22 imass2 6054 . . . . . . . . 9 ((ℂ ∖ {0}) ⊆ ℂ → (cos “ (ℂ ∖ {0})) ⊆ (cos “ ℂ))
2321, 22ax-mp 5 . . . . . . . 8 (cos “ (ℂ ∖ {0})) ⊆ (cos “ ℂ)
24 fimacnv 6690 . . . . . . . . 9 (cos:ℂ⟶ℂ → (cos “ ℂ) = ℂ)
253, 24ax-mp 5 . . . . . . . 8 (cos “ ℂ) = ℂ
2623, 25sseqtri 3980 . . . . . . 7 (cos “ (ℂ ∖ {0})) ⊆ ℂ
2726sseli 3940 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
2827sincld 16012 . . . . 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 16008 . . . . . 6 (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
3231adantl 482 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
33 coscl 16009 . . . . . 6 (𝑥 ∈ ℂ → (cos‘𝑥) ∈ ℂ)
3433adantl 482 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ ℂ)
35 dvsin 25346 . . . . . 6 (ℂ D sin) = cos
36 sinf 16006 . . . . . . . . 9 sin:ℂ⟶ℂ
3736a1i 11 . . . . . . . 8 (⊤ → sin:ℂ⟶ℂ)
3837feqmptd 6910 . . . . . . 7 (⊤ → sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
3938oveq2d 7373 . . . . . 6 (⊤ → (ℂ D sin) = (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))))
403a1i 11 . . . . . . 7 (⊤ → cos:ℂ⟶ℂ)
4140feqmptd 6910 . . . . . 6 (⊤ → cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
4235, 39, 413eqtr3a 2800 . . . . 5 (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
4326a1i 11 . . . . 5 (⊤ → (cos “ (ℂ ∖ {0})) ⊆ ℂ)
44 eqid 2736 . . . . . . 7 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
4544cnfldtopon 24146 . . . . . 6 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
4645toponrestid 22270 . . . . 5 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
47 dvtanlem 36127 . . . . . 6 (cos “ (ℂ ∖ {0})) ∈ (TopOpen‘ℂfld)
4847a1i 11 . . . . 5 (⊤ → (cos “ (ℂ ∖ {0})) ∈ (TopOpen‘ℂfld))
4920, 32, 34, 42, 43, 46, 44, 48dvmptres 25327 . . . 4 (⊤ → (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (sin‘𝑥))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥)))
508, 14reccld 11924 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (1 / (cos‘𝑥)) ∈ ℂ)
5150adantl 482 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (1 / (cos‘𝑥)) ∈ ℂ)
52 ovexd 7392 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) ∈ V)
5311simprbi 497 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
5453adantl 482 . . . . 5 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
5529negcld 11499 . . . . 5 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → -(sin‘𝑥) ∈ ℂ)
56 eldifi 4086 . . . . . . 7 (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ)
57 eldifsni 4750 . . . . . . 7 (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0)
5856, 57reccld 11924 . . . . . 6 (𝑦 ∈ (ℂ ∖ {0}) → (1 / 𝑦) ∈ ℂ)
5958adantl 482 . . . . 5 ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (1 / 𝑦) ∈ ℂ)
60 negex 11399 . . . . . 6 -(1 / (𝑦↑2)) ∈ V
6160a1i 11 . . . . 5 ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(1 / (𝑦↑2)) ∈ V)
6232negcld 11499 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℂ) → -(sin‘𝑥) ∈ ℂ)
6341oveq2d 7373 . . . . . . 7 (⊤ → (ℂ D cos) = (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))))
64 dvcos 25347 . . . . . . 7 (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥))
6563, 64eqtr3di 2791 . . . . . 6 (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)))
6620, 34, 62, 65, 43, 46, 44, 48dvmptres 25327 . . . . 5 (⊤ → (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ -(sin‘𝑥)))
67 ax-1cn 11109 . . . . . 6 1 ∈ ℂ
68 dvrec 25319 . . . . . 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 7365 . . . . 5 (𝑦 = (cos‘𝑥) → (1 / 𝑦) = (1 / (cos‘𝑥)))
71 oveq1 7364 . . . . . . 7 (𝑦 = (cos‘𝑥) → (𝑦↑2) = ((cos‘𝑥)↑2))
7271oveq2d 7373 . . . . . 6 (𝑦 = (cos‘𝑥) → (1 / (𝑦↑2)) = (1 / ((cos‘𝑥)↑2)))
7372negeqd 11395 . . . . 5 (𝑦 = (cos‘𝑥) → -(1 / (𝑦↑2)) = -(1 / ((cos‘𝑥)↑2)))
7420, 20, 54, 55, 59, 61, 66, 69, 70, 73dvmptco 25336 . . . 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 25325 . . 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 1548 . 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 7390 . . . . 5 ((sin‘𝑥) / (cos‘𝑥)) ∈ V
7877, 1dmmpti 6645 . . . 4 dom tan = (cos “ (ℂ ∖ {0}))
7978eqcomi 2745 . . 3 (cos “ (ℂ ∖ {0})) = dom tan
808sqcld 14049 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ∈ ℂ)
817sqcld 14049 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) ∈ ℂ)
82 sqne0 14028 . . . . . . . . 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 256 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ≠ 0)
8580, 81, 80, 84divdird 11969 . . . . . 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 11357 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)))
87 sincossq 16058 . . . . . . . . 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 2776 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = 1)
9089oveq1d 7372 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) / ((cos‘𝑥)↑2)) = (1 / ((cos‘𝑥)↑2)))
9185, 90eqtr3d 2778 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) + (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2))) = (1 / ((cos‘𝑥)↑2)))
928, 14recidd 11926 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = 1)
9380, 84dividd 11929 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) = 1)
9492, 93eqtr4d 2779 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)))
957, 7, 80, 84div23d 11968 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
967sqvald 14048 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) = ((sin‘𝑥) · (sin‘𝑥)))
9796oveq1d 7372 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)))
9880, 84reccld 11924 . . . . . . . . . 10 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (1 / ((cos‘𝑥)↑2)) ∈ ℂ)
9998, 7mul2negd 11610 . . . . . . . . 9 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
1007, 80, 84divrec2d 11935 . . . . . . . . 9 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / ((cos‘𝑥)↑2)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
10199, 100eqtr4d 2779 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((sin‘𝑥) / ((cos‘𝑥)↑2)))
102101oveq1d 7372 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)) = (((sin‘𝑥) / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
10395, 97, 1023eqtr4rd 2787 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)) = (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)))
10494, 103oveq12d 7375 . . . . 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 12430 . . . . . 6 2 ∈ ℕ0
106 expneg 13975 . . . . . 6 (((cos‘𝑥) ∈ ℂ ∧ 2 ∈ ℕ0) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
1078, 105, 106sylancl 586 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
10891, 104, 1073eqtr4d 2786 . . . 4 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2))
109108rgen 3066 . . 3 𝑥 ∈ (cos “ (ℂ ∖ {0}))(((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2)
110 mpteq12 5197 . . 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 690 . 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 2768 1 (ℂ D tan) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wtru 1542  wcel 2106  wne 2943  wral 3064  Vcvv 3445  cdif 3907  wss 3910  {csn 4586  {cpr 4588  cmpt 5188  ccnv 5632  dom cdm 5633  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  -cneg 11386   / cdiv 11812  2c2 12208  0cn0 12413  cexp 13967  sincsin 15946  cosccos 15947  tanctan 15948  TopOpenctopn 17303  fldccnfld 20796   D cdv 25227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-ef 15950  df-sin 15952  df-cos 15953  df-tan 15954  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-t1 22665  df-haus 22666  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241  df-limc 25230  df-dv 25231
This theorem is referenced by: (None)
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