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Theorem dvtan 33590
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 14846 . . . 4 tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
2 cnvimass 5520 . . . . . . . . 9 (cos “ (ℂ ∖ {0})) ⊆ dom cos
3 cosf 14899 . . . . . . . . . 10 cos:ℂ⟶ℂ
43fdmi 6090 . . . . . . . . 9 dom cos = ℂ
52, 4sseqtri 3670 . . . . . . . 8 (cos “ (ℂ ∖ {0})) ⊆ ℂ
65sseli 3632 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
76sincld 14904 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
86coscld 14905 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ ℂ)
9 ffn 6083 . . . . . . . 8 (cos:ℂ⟶ℂ → cos Fn ℂ)
10 elpreima 6377 . . . . . . . 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 4353 . . . . . . . 8 ((cos‘𝑥) ∈ (ℂ ∖ {0}) → (cos‘𝑥) ≠ 0)
1312adantl 481 . . . . . . 7 ((𝑥 ∈ ℂ ∧ (cos‘𝑥) ∈ (ℂ ∖ {0})) → (cos‘𝑥) ≠ 0)
1411, 13sylbi 207 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (cos‘𝑥) ≠ 0)
157, 8, 14divrecd 10842 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / (cos‘𝑥)) = ((sin‘𝑥) · (1 / (cos‘𝑥))))
1615mpteq2ia 4773 . . . 4 (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
171, 16eqtri 2673 . . 3 tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
1817oveq2i 6701 . 2 (ℂ D tan) = (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥)))))
19 cnelprrecn 10067 . . . . 5 ℂ ∈ {ℝ, ℂ}
2019a1i 11 . . . 4 (⊤ → ℂ ∈ {ℝ, ℂ})
21 difss 3770 . . . . . . . . 9 (ℂ ∖ {0}) ⊆ ℂ
22 imass2 5536 . . . . . . . . 9 ((ℂ ∖ {0}) ⊆ ℂ → (cos “ (ℂ ∖ {0})) ⊆ (cos “ ℂ))
2321, 22ax-mp 5 . . . . . . . 8 (cos “ (ℂ ∖ {0})) ⊆ (cos “ ℂ)
24 fimacnv 6387 . . . . . . . . 9 (cos:ℂ⟶ℂ → (cos “ ℂ) = ℂ)
253, 24ax-mp 5 . . . . . . . 8 (cos “ ℂ) = ℂ
2623, 25sseqtri 3670 . . . . . . 7 (cos “ (ℂ ∖ {0})) ⊆ ℂ
2726sseli 3632 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
2827sincld 14904 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
2928adantl 481 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (sin‘𝑥) ∈ ℂ)
308adantl 481 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (cos‘𝑥) ∈ ℂ)
31 sincl 14900 . . . . . 6 (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
3231adantl 481 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
33 coscl 14901 . . . . . 6 (𝑥 ∈ ℂ → (cos‘𝑥) ∈ ℂ)
3433adantl 481 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ ℂ)
35 dvsin 23790 . . . . . 6 (ℂ D sin) = cos
36 sinf 14898 . . . . . . . . 9 sin:ℂ⟶ℂ
3736a1i 11 . . . . . . . 8 (⊤ → sin:ℂ⟶ℂ)
3837feqmptd 6288 . . . . . . 7 (⊤ → sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
3938oveq2d 6706 . . . . . 6 (⊤ → (ℂ D sin) = (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))))
403a1i 11 . . . . . . 7 (⊤ → cos:ℂ⟶ℂ)
4140feqmptd 6288 . . . . . 6 (⊤ → cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
4235, 39, 413eqtr3a 2709 . . . . 5 (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
4326a1i 11 . . . . 5 (⊤ → (cos “ (ℂ ∖ {0})) ⊆ ℂ)
44 eqid 2651 . . . . . . . 8 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
4544cnfldtopon 22633 . . . . . . 7 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
4645toponunii 20769 . . . . . . . 8 ℂ = (TopOpen‘ℂfld)
4746restid 16141 . . . . . . 7 ((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
4845, 47ax-mp 5 . . . . . 6 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
4948eqcomi 2660 . . . . 5 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
50 dvtanlem 33589 . . . . . 6 (cos “ (ℂ ∖ {0})) ∈ (TopOpen‘ℂfld)
5150a1i 11 . . . . 5 (⊤ → (cos “ (ℂ ∖ {0})) ∈ (TopOpen‘ℂfld))
5220, 32, 34, 42, 43, 49, 44, 51dvmptres 23771 . . . 4 (⊤ → (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (sin‘𝑥))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥)))
538, 14reccld 10832 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (1 / (cos‘𝑥)) ∈ ℂ)
5453adantl 481 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (1 / (cos‘𝑥)) ∈ ℂ)
55 ovexd 6720 . . . 4 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) ∈ V)
5611simprbi 479 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
5756adantl 481 . . . . 5 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
5829negcld 10417 . . . . 5 ((⊤ ∧ 𝑥 ∈ (cos “ (ℂ ∖ {0}))) → -(sin‘𝑥) ∈ ℂ)
59 eldifi 3765 . . . . . . 7 (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ)
60 eldifsni 4353 . . . . . . 7 (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0)
6159, 60reccld 10832 . . . . . 6 (𝑦 ∈ (ℂ ∖ {0}) → (1 / 𝑦) ∈ ℂ)
6261adantl 481 . . . . 5 ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (1 / 𝑦) ∈ ℂ)
63 negex 10317 . . . . . 6 -(1 / (𝑦↑2)) ∈ V
6463a1i 11 . . . . 5 ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(1 / (𝑦↑2)) ∈ V)
6532negcld 10417 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℂ) → -(sin‘𝑥) ∈ ℂ)
66 dvcos 23791 . . . . . . 7 (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥))
6741oveq2d 6706 . . . . . . 7 (⊤ → (ℂ D cos) = (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))))
6866, 67syl5reqr 2700 . . . . . 6 (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)))
6920, 34, 65, 68, 43, 49, 44, 51dvmptres 23771 . . . . 5 (⊤ → (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ -(sin‘𝑥)))
70 ax-1cn 10032 . . . . . 6 1 ∈ ℂ
71 dvrec 23763 . . . . . 6 (1 ∈ ℂ → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(1 / (𝑦↑2))))
7270, 71mp1i 13 . . . . 5 (⊤ → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(1 / (𝑦↑2))))
73 oveq2 6698 . . . . 5 (𝑦 = (cos‘𝑥) → (1 / 𝑦) = (1 / (cos‘𝑥)))
74 oveq1 6697 . . . . . . 7 (𝑦 = (cos‘𝑥) → (𝑦↑2) = ((cos‘𝑥)↑2))
7574oveq2d 6706 . . . . . 6 (𝑦 = (cos‘𝑥) → (1 / (𝑦↑2)) = (1 / ((cos‘𝑥)↑2)))
7675negeqd 10313 . . . . 5 (𝑦 = (cos‘𝑥) → -(1 / (𝑦↑2)) = -(1 / ((cos‘𝑥)↑2)))
7720, 20, 57, 58, 62, 64, 69, 72, 73, 76dvmptco 23780 . . . 4 (⊤ → (ℂ D (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (1 / (cos‘𝑥)))) = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥))))
7820, 29, 30, 52, 54, 55, 77dvmptmul 23769 . . 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)))))
7978trud 1533 . 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))))
80 ovex 6718 . . . . 5 ((sin‘𝑥) / (cos‘𝑥)) ∈ V
8180, 1dmmpti 6061 . . . 4 dom tan = (cos “ (ℂ ∖ {0}))
8281eqcomi 2660 . . 3 (cos “ (ℂ ∖ {0})) = dom tan
838sqcld 13046 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ∈ ℂ)
847sqcld 13046 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) ∈ ℂ)
85 sqne0 12970 . . . . . . . . 9 ((cos‘𝑥) ∈ ℂ → (((cos‘𝑥)↑2) ≠ 0 ↔ (cos‘𝑥) ≠ 0))
868, 85syl 17 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) ≠ 0 ↔ (cos‘𝑥) ≠ 0))
8714, 86mpbird 247 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ≠ 0)
8883, 84, 83, 87divdird 10877 . . . . . 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))))
8983, 84addcomd 10276 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)))
90 sincossq 14950 . . . . . . . . 9 (𝑥 ∈ ℂ → (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1)
916, 90syl 17 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1)
9289, 91eqtrd 2685 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = 1)
9392oveq1d 6705 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) / ((cos‘𝑥)↑2)) = (1 / ((cos‘𝑥)↑2)))
9488, 93eqtr3d 2687 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) + (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2))) = (1 / ((cos‘𝑥)↑2)))
958, 14recidd 10834 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = 1)
9683, 87dividd 10837 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) = 1)
9795, 96eqtr4d 2688 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)))
987, 7, 83, 87div23d 10876 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
997sqvald 13045 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) = ((sin‘𝑥) · (sin‘𝑥)))
10099oveq1d 6705 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)))
10183, 87reccld 10832 . . . . . . . . . 10 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (1 / ((cos‘𝑥)↑2)) ∈ ℂ)
102101, 7mul2negd 10523 . . . . . . . . 9 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
1037, 83, 87divrec2d 10843 . . . . . . . . 9 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / ((cos‘𝑥)↑2)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
104102, 103eqtr4d 2688 . . . . . . . 8 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((sin‘𝑥) / ((cos‘𝑥)↑2)))
105104oveq1d 6705 . . . . . . 7 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)) = (((sin‘𝑥) / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
10698, 100, 1053eqtr4rd 2696 . . . . . 6 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)) = (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)))
10797, 106oveq12d 6708 . . . . 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))))
108 2nn0 11347 . . . . . 6 2 ∈ ℕ0
109 expneg 12908 . . . . . 6 (((cos‘𝑥) ∈ ℂ ∧ 2 ∈ ℕ0) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
1108, 108, 109sylancl 695 . . . . 5 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
11194, 107, 1103eqtr4d 2695 . . . 4 (𝑥 ∈ (cos “ (ℂ ∖ {0})) → (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2))
112111rgen 2951 . . 3 𝑥 ∈ (cos “ (ℂ ∖ {0}))(((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2)
113 mpteq12 4769 . . 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)))
11482, 112, 113mp2an 708 . 2 (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)))) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2))
11518, 79, 1143eqtri 2677 1 (ℂ D tan) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wtru 1524  wcel 2030  wne 2823  wral 2941  Vcvv 3231  cdif 3604  wss 3607  {csn 4210  {cpr 4212  cmpt 4762  ccnv 5142  dom cdm 5143  cima 5146   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  -cneg 10305   / cdiv 10722  2c2 11108  0cn0 11330  cexp 12900  sincsin 14838  cosccos 14839  tanctan 14840  t crest 16128  TopOpenctopn 16129  fldccnfld 19794  TopOnctopon 20763   D cdv 23672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-sum 14461  df-ef 14842  df-sin 14844  df-cos 14845  df-tan 14846  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-t1 21166  df-haus 21167  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676
This theorem is referenced by: (None)
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