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Theorem pntlemg 26166
Description: Lemma for pnt 26182. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝑀 is j^* and 𝑁 is ĵ. (Contributed by Mario Carneiro, 13-Apr-2016.)
Hypotheses
Ref Expression
pntlem1.r 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
pntlem1.a (𝜑𝐴 ∈ ℝ+)
pntlem1.b (𝜑𝐵 ∈ ℝ+)
pntlem1.l (𝜑𝐿 ∈ (0(,)1))
pntlem1.d 𝐷 = (𝐴 + 1)
pntlem1.f 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))
pntlem1.u (𝜑𝑈 ∈ ℝ+)
pntlem1.u2 (𝜑𝑈𝐴)
pntlem1.e 𝐸 = (𝑈 / 𝐷)
pntlem1.k 𝐾 = (exp‘(𝐵 / 𝐸))
pntlem1.y (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))
pntlem1.x (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))
pntlem1.c (𝜑𝐶 ∈ ℝ+)
pntlem1.w 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
pntlem1.z (𝜑𝑍 ∈ (𝑊[,)+∞))
pntlem1.m 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)
pntlem1.n 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))
Assertion
Ref Expression
pntlemg (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀)))
Distinct variable group:   𝐸,𝑎
Allowed substitution hints:   𝜑(𝑎)   𝐴(𝑎)   𝐵(𝑎)   𝐶(𝑎)   𝐷(𝑎)   𝑅(𝑎)   𝑈(𝑎)   𝐹(𝑎)   𝐾(𝑎)   𝐿(𝑎)   𝑀(𝑎)   𝑁(𝑎)   𝑊(𝑎)   𝑋(𝑎)   𝑌(𝑎)   𝑍(𝑎)

Proof of Theorem pntlemg
StepHypRef Expression
1 pntlem1.m . . 3 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)
2 pntlem1.x . . . . . . . . 9 (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))
32simpld 497 . . . . . . . 8 (𝜑𝑋 ∈ ℝ+)
43rpred 12423 . . . . . . 7 (𝜑𝑋 ∈ ℝ)
5 1red 10634 . . . . . . . 8 (𝜑 → 1 ∈ ℝ)
6 pntlem1.y . . . . . . . . . 10 (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))
76simpld 497 . . . . . . . . 9 (𝜑𝑌 ∈ ℝ+)
87rpred 12423 . . . . . . . 8 (𝜑𝑌 ∈ ℝ)
96simprd 498 . . . . . . . 8 (𝜑 → 1 ≤ 𝑌)
102simprd 498 . . . . . . . 8 (𝜑𝑌 < 𝑋)
115, 8, 4, 9, 10lelttrd 10790 . . . . . . 7 (𝜑 → 1 < 𝑋)
124, 11rplogcld 25204 . . . . . 6 (𝜑 → (log‘𝑋) ∈ ℝ+)
13 pntlem1.r . . . . . . . . . 10 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
14 pntlem1.a . . . . . . . . . 10 (𝜑𝐴 ∈ ℝ+)
15 pntlem1.b . . . . . . . . . 10 (𝜑𝐵 ∈ ℝ+)
16 pntlem1.l . . . . . . . . . 10 (𝜑𝐿 ∈ (0(,)1))
17 pntlem1.d . . . . . . . . . 10 𝐷 = (𝐴 + 1)
18 pntlem1.f . . . . . . . . . 10 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))
19 pntlem1.u . . . . . . . . . 10 (𝜑𝑈 ∈ ℝ+)
20 pntlem1.u2 . . . . . . . . . 10 (𝜑𝑈𝐴)
21 pntlem1.e . . . . . . . . . 10 𝐸 = (𝑈 / 𝐷)
22 pntlem1.k . . . . . . . . . 10 𝐾 = (exp‘(𝐵 / 𝐸))
2313, 14, 15, 16, 17, 18, 19, 20, 21, 22pntlemc 26163 . . . . . . . . 9 (𝜑 → (𝐸 ∈ ℝ+𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+)))
2423simp2d 1137 . . . . . . . 8 (𝜑𝐾 ∈ ℝ+)
2524rpred 12423 . . . . . . 7 (𝜑𝐾 ∈ ℝ)
2623simp3d 1138 . . . . . . . 8 (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+))
2726simp2d 1137 . . . . . . 7 (𝜑 → 1 < 𝐾)
2825, 27rplogcld 25204 . . . . . 6 (𝜑 → (log‘𝐾) ∈ ℝ+)
2912, 28rpdivcld 12440 . . . . 5 (𝜑 → ((log‘𝑋) / (log‘𝐾)) ∈ ℝ+)
3029rprege0d 12430 . . . 4 (𝜑 → (((log‘𝑋) / (log‘𝐾)) ∈ ℝ ∧ 0 ≤ ((log‘𝑋) / (log‘𝐾))))
31 flge0nn0 13182 . . . 4 ((((log‘𝑋) / (log‘𝐾)) ∈ ℝ ∧ 0 ≤ ((log‘𝑋) / (log‘𝐾))) → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℕ0)
32 nn0p1nn 11928 . . . 4 ((⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℕ0 → ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) ∈ ℕ)
3330, 31, 323syl 18 . . 3 (𝜑 → ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) ∈ ℕ)
341, 33eqeltrid 2915 . 2 (𝜑𝑀 ∈ ℕ)
3534nnzd 12078 . . 3 (𝜑𝑀 ∈ ℤ)
36 pntlem1.n . . . 4 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))
37 pntlem1.c . . . . . . . . . 10 (𝜑𝐶 ∈ ℝ+)
38 pntlem1.w . . . . . . . . . 10 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
39 pntlem1.z . . . . . . . . . 10 (𝜑𝑍 ∈ (𝑊[,)+∞))
4013, 14, 15, 16, 17, 18, 19, 20, 21, 22, 6, 2, 37, 38, 39pntlemb 26165 . . . . . . . . 9 (𝜑 → (𝑍 ∈ ℝ+ ∧ (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)) ∧ ((4 / (𝐿 · 𝐸)) ≤ (√‘𝑍) ∧ (((log‘𝑋) / (log‘𝐾)) + 2) ≤ (((log‘𝑍) / (log‘𝐾)) / 4) ∧ ((𝑈 · 3) + 𝐶) ≤ (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))))
4140simp1d 1136 . . . . . . . 8 (𝜑𝑍 ∈ ℝ+)
4241relogcld 25198 . . . . . . 7 (𝜑 → (log‘𝑍) ∈ ℝ)
4342, 28rerpdivcld 12454 . . . . . 6 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ)
4443rehalfcld 11876 . . . . 5 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ∈ ℝ)
4544flcld 13160 . . . 4 (𝜑 → (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) ∈ ℤ)
4636, 45eqeltrid 2915 . . 3 (𝜑𝑁 ∈ ℤ)
47 0red 10636 . . . . 5 (𝜑 → 0 ∈ ℝ)
48 4nn 11712 . . . . . 6 4 ∈ ℕ
49 nndivre 11670 . . . . . 6 ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ ∧ 4 ∈ ℕ) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
5043, 48, 49sylancl 588 . . . . 5 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
5146zred 12079 . . . . . 6 (𝜑𝑁 ∈ ℝ)
5234nnred 11645 . . . . . 6 (𝜑𝑀 ∈ ℝ)
5351, 52resubcld 11060 . . . . 5 (𝜑 → (𝑁𝑀) ∈ ℝ)
5441rpred 12423 . . . . . . . . 9 (𝜑𝑍 ∈ ℝ)
5540simp2d 1137 . . . . . . . . . 10 (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
5655simp1d 1136 . . . . . . . . 9 (𝜑 → 1 < 𝑍)
5754, 56rplogcld 25204 . . . . . . . 8 (𝜑 → (log‘𝑍) ∈ ℝ+)
5857, 28rpdivcld 12440 . . . . . . 7 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ+)
59 4re 11713 . . . . . . . 8 4 ∈ ℝ
60 4pos 11736 . . . . . . . 8 0 < 4
6159, 60elrpii 12384 . . . . . . 7 4 ∈ ℝ+
62 rpdivcl 12406 . . . . . . 7 ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ+ ∧ 4 ∈ ℝ+) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ+)
6358, 61, 62sylancl 588 . . . . . 6 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ+)
6463rpge0d 12427 . . . . 5 (𝜑 → 0 ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
6550recnd 10661 . . . . . . . . 9 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℂ)
6634nncnd 11646 . . . . . . . . 9 (𝜑𝑀 ∈ ℂ)
67 1cnd 10628 . . . . . . . . 9 (𝜑 → 1 ∈ ℂ)
6865, 66, 67addassd 10655 . . . . . . . 8 (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)))
6952, 5readdcld 10662 . . . . . . . . . 10 (𝜑 → (𝑀 + 1) ∈ ℝ)
7050, 69readdcld 10662 . . . . . . . . 9 (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ∈ ℝ)
71 peano2re 10805 . . . . . . . . . 10 (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
7251, 71syl 17 . . . . . . . . 9 (𝜑 → (𝑁 + 1) ∈ ℝ)
7329rpred 12423 . . . . . . . . . . . . 13 (𝜑 → ((log‘𝑋) / (log‘𝐾)) ∈ ℝ)
74 2re 11703 . . . . . . . . . . . . . 14 2 ∈ ℝ
7574a1i 11 . . . . . . . . . . . . 13 (𝜑 → 2 ∈ ℝ)
7673, 75readdcld 10662 . . . . . . . . . . . 12 (𝜑 → (((log‘𝑋) / (log‘𝐾)) + 2) ∈ ℝ)
77 reflcl 13158 . . . . . . . . . . . . . . . . 17 (((log‘𝑋) / (log‘𝐾)) ∈ ℝ → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℝ)
7873, 77syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℝ)
7978recnd 10661 . . . . . . . . . . . . . . 15 (𝜑 → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℂ)
8079, 67, 67addassd 10655 . . . . . . . . . . . . . 14 (𝜑 → (((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + (1 + 1)))
811oveq1i 7158 . . . . . . . . . . . . . 14 (𝑀 + 1) = (((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) + 1)
82 df-2 11692 . . . . . . . . . . . . . . 15 2 = (1 + 1)
8382oveq2i 7159 . . . . . . . . . . . . . 14 ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + (1 + 1))
8480, 81, 833eqtr4g 2879 . . . . . . . . . . . . 13 (𝜑 → (𝑀 + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2))
85 flle 13161 . . . . . . . . . . . . . . 15 (((log‘𝑋) / (log‘𝐾)) ∈ ℝ → (⌊‘((log‘𝑋) / (log‘𝐾))) ≤ ((log‘𝑋) / (log‘𝐾)))
8673, 85syl 17 . . . . . . . . . . . . . 14 (𝜑 → (⌊‘((log‘𝑋) / (log‘𝐾))) ≤ ((log‘𝑋) / (log‘𝐾)))
8778, 73, 75, 86leadd1dd 11246 . . . . . . . . . . . . 13 (𝜑 → ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2) ≤ (((log‘𝑋) / (log‘𝐾)) + 2))
8884, 87eqbrtrd 5079 . . . . . . . . . . . 12 (𝜑 → (𝑀 + 1) ≤ (((log‘𝑋) / (log‘𝐾)) + 2))
8940simp3d 1138 . . . . . . . . . . . . 13 (𝜑 → ((4 / (𝐿 · 𝐸)) ≤ (√‘𝑍) ∧ (((log‘𝑋) / (log‘𝐾)) + 2) ≤ (((log‘𝑍) / (log‘𝐾)) / 4) ∧ ((𝑈 · 3) + 𝐶) ≤ (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))))
9089simp2d 1137 . . . . . . . . . . . 12 (𝜑 → (((log‘𝑋) / (log‘𝐾)) + 2) ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
9169, 76, 50, 88, 90letrd 10789 . . . . . . . . . . 11 (𝜑 → (𝑀 + 1) ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
9269, 50, 50, 91leadd2dd 11247 . . . . . . . . . 10 (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
9343recnd 10661 . . . . . . . . . . . . . 14 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℂ)
94 2cnd 11707 . . . . . . . . . . . . . 14 (𝜑 → 2 ∈ ℂ)
95 2ne0 11733 . . . . . . . . . . . . . . 15 2 ≠ 0
9695a1i 11 . . . . . . . . . . . . . 14 (𝜑 → 2 ≠ 0)
9793, 94, 94, 96, 96divdiv1d 11439 . . . . . . . . . . . . 13 (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 2) / 2) = (((log‘𝑍) / (log‘𝐾)) / (2 · 2)))
98 2t2e4 11793 . . . . . . . . . . . . . 14 (2 · 2) = 4
9998oveq2i 7159 . . . . . . . . . . . . 13 (((log‘𝑍) / (log‘𝐾)) / (2 · 2)) = (((log‘𝑍) / (log‘𝐾)) / 4)
10097, 99syl6eq 2870 . . . . . . . . . . . 12 (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 2) / 2) = (((log‘𝑍) / (log‘𝐾)) / 4))
101100oveq2d 7164 . . . . . . . . . . 11 (𝜑 → (2 · ((((log‘𝑍) / (log‘𝐾)) / 2) / 2)) = (2 · (((log‘𝑍) / (log‘𝐾)) / 4)))
10244recnd 10661 . . . . . . . . . . . 12 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ∈ ℂ)
103102, 94, 96divcan2d 11410 . . . . . . . . . . 11 (𝜑 → (2 · ((((log‘𝑍) / (log‘𝐾)) / 2) / 2)) = (((log‘𝑍) / (log‘𝐾)) / 2))
104652timesd 11872 . . . . . . . . . . 11 (𝜑 → (2 · (((log‘𝑍) / (log‘𝐾)) / 4)) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
105101, 103, 1043eqtr3d 2862 . . . . . . . . . 10 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
10692, 105breqtrrd 5085 . . . . . . . . 9 (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ (((log‘𝑍) / (log‘𝐾)) / 2))
107 fllep1 13163 . . . . . . . . . . 11 ((((log‘𝑍) / (log‘𝐾)) / 2) ∈ ℝ → (((log‘𝑍) / (log‘𝐾)) / 2) ≤ ((⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) + 1))
10844, 107syl 17 . . . . . . . . . 10 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ≤ ((⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) + 1))
10936oveq1i 7158 . . . . . . . . . 10 (𝑁 + 1) = ((⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) + 1)
110108, 109breqtrrdi 5099 . . . . . . . . 9 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ≤ (𝑁 + 1))
11170, 44, 72, 106, 110letrd 10789 . . . . . . . 8 (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ (𝑁 + 1))
11268, 111eqbrtrd 5079 . . . . . . 7 (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) ≤ (𝑁 + 1))
11350, 52readdcld 10662 . . . . . . . 8 (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ∈ ℝ)
114113, 51, 5leadd1d 11226 . . . . . . 7 (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ≤ 𝑁 ↔ (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) ≤ (𝑁 + 1)))
115112, 114mpbird 259 . . . . . 6 (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ≤ 𝑁)
116 leaddsub 11108 . . . . . . 7 (((((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ≤ 𝑁 ↔ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀)))
11750, 52, 51, 116syl3anc 1365 . . . . . 6 (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ≤ 𝑁 ↔ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀)))
118115, 117mpbid 234 . . . . 5 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀))
11947, 50, 53, 64, 118letrd 10789 . . . 4 (𝜑 → 0 ≤ (𝑁𝑀))
12051, 52subge0d 11222 . . . 4 (𝜑 → (0 ≤ (𝑁𝑀) ↔ 𝑀𝑁))
121119, 120mpbid 234 . . 3 (𝜑𝑀𝑁)
122 eluz2 12241 . . 3 (𝑁 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁))
12335, 46, 121, 122syl3anbrc 1337 . 2 (𝜑𝑁 ∈ (ℤ𝑀))
12434, 123, 1183jca 1122 1 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  wne 3014   class class class wbr 5057  cmpt 5137  cfv 6348  (class class class)co 7148  cr 10528  0cc0 10529  1c1 10530   + caddc 10532   · cmul 10534  +∞cpnf 10664   < clt 10667  cle 10668  cmin 10862   / cdiv 11289  cn 11630  2c2 11684  3c3 11685  4c4 11686  0cn0 11889  cz 11973  cdc 12090  cuz 12235  +crp 12381  (,)cioo 12730  [,)cico 12732  cfl 13152  cexp 13421  csqrt 14584  expce 15407  eceu 15408  logclog 25130  ψcchp 25662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608  ax-mulf 10609
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-om 7573  df-1st 7681  df-2nd 7682  df-supp 7823  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-fi 8867  df-sup 8898  df-inf 8899  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ioc 12735  df-ico 12736  df-icc 12737  df-fz 12885  df-fzo 13026  df-fl 13154  df-mod 13230  df-seq 13362  df-exp 13422  df-fac 13626  df-bc 13655  df-hash 13683  df-shft 14418  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-limsup 14820  df-clim 14837  df-rlim 14838  df-sum 15035  df-ef 15413  df-e 15414  df-sin 15415  df-cos 15416  df-pi 15418  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-starv 16572  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-hom 16581  df-cco 16582  df-rest 16688  df-topn 16689  df-0g 16707  df-gsum 16708  df-topgen 16709  df-pt 16710  df-prds 16713  df-xrs 16767  df-qtop 16772  df-imas 16773  df-xps 16775  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-mulg 18217  df-cntz 18439  df-cmn 18900  df-psmet 20529  df-xmet 20530  df-met 20531  df-bl 20532  df-mopn 20533  df-fbas 20534  df-fg 20535  df-cnfld 20538  df-top 21494  df-topon 21511  df-topsp 21533  df-bases 21546  df-cld 21619  df-ntr 21620  df-cls 21621  df-nei 21698  df-lp 21736  df-perf 21737  df-cn 21827  df-cnp 21828  df-haus 21915  df-tx 22162  df-hmeo 22355  df-fil 22446  df-fm 22538  df-flim 22539  df-flf 22540  df-xms 22922  df-ms 22923  df-tms 22924  df-cncf 23478  df-limc 24456  df-dv 24457  df-log 25132
This theorem is referenced by:  pntlemh  26167  pntlemq  26169  pntlemr  26170  pntlemj  26171  pntlemf  26173
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