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Theorem pntlemg 24999
Description: Lemma for pnt 25015. 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 473 . . . . . . . 8 (𝜑𝑋 ∈ ℝ+)
43rpred 11699 . . . . . . 7 (𝜑𝑋 ∈ ℝ)
5 1red 9906 . . . . . . . 8 (𝜑 → 1 ∈ ℝ)
6 pntlem1.y . . . . . . . . . 10 (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))
76simpld 473 . . . . . . . . 9 (𝜑𝑌 ∈ ℝ+)
87rpred 11699 . . . . . . . 8 (𝜑𝑌 ∈ ℝ)
96simprd 477 . . . . . . . 8 (𝜑 → 1 ≤ 𝑌)
102simprd 477 . . . . . . . 8 (𝜑𝑌 < 𝑋)
115, 8, 4, 9, 10lelttrd 10041 . . . . . . 7 (𝜑 → 1 < 𝑋)
124, 11rplogcld 24091 . . . . . 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 24996 . . . . . . . . 9 (𝜑 → (𝐸 ∈ ℝ+𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+)))
2423simp2d 1066 . . . . . . . 8 (𝜑𝐾 ∈ ℝ+)
2524rpred 11699 . . . . . . 7 (𝜑𝐾 ∈ ℝ)
2623simp3d 1067 . . . . . . . 8 (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+))
2726simp2d 1066 . . . . . . 7 (𝜑 → 1 < 𝐾)
2825, 27rplogcld 24091 . . . . . 6 (𝜑 → (log‘𝐾) ∈ ℝ+)
2912, 28rpdivcld 11716 . . . . 5 (𝜑 → ((log‘𝑋) / (log‘𝐾)) ∈ ℝ+)
3029rprege0d 11706 . . . 4 (𝜑 → (((log‘𝑋) / (log‘𝐾)) ∈ ℝ ∧ 0 ≤ ((log‘𝑋) / (log‘𝐾))))
31 flge0nn0 12433 . . . 4 ((((log‘𝑋) / (log‘𝐾)) ∈ ℝ ∧ 0 ≤ ((log‘𝑋) / (log‘𝐾))) → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℕ0)
32 nn0p1nn 11174 . . . 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, 33syl5eqel 2686 . 2 (𝜑𝑀 ∈ ℕ)
3534nnzd 11308 . . 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 24998 . . . . . . . . 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 1065 . . . . . . . 8 (𝜑𝑍 ∈ ℝ+)
4241relogcld 24085 . . . . . . 7 (𝜑 → (log‘𝑍) ∈ ℝ)
4342, 28rerpdivcld 11730 . . . . . 6 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ)
4443rehalfcld 11121 . . . . 5 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ∈ ℝ)
4544flcld 12411 . . . 4 (𝜑 → (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) ∈ ℤ)
4636, 45syl5eqel 2686 . . 3 (𝜑𝑁 ∈ ℤ)
47 0red 9892 . . . . 5 (𝜑 → 0 ∈ ℝ)
48 4nn 11029 . . . . . 6 4 ∈ ℕ
49 nndivre 10898 . . . . . 6 ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ ∧ 4 ∈ ℕ) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
5043, 48, 49sylancl 692 . . . . 5 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
5146zred 11309 . . . . . 6 (𝜑𝑁 ∈ ℝ)
5234nnred 10877 . . . . . 6 (𝜑𝑀 ∈ ℝ)
5351, 52resubcld 10304 . . . . 5 (𝜑 → (𝑁𝑀) ∈ ℝ)
5441rpred 11699 . . . . . . . . 9 (𝜑𝑍 ∈ ℝ)
5540simp2d 1066 . . . . . . . . . 10 (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
5655simp1d 1065 . . . . . . . . 9 (𝜑 → 1 < 𝑍)
5754, 56rplogcld 24091 . . . . . . . 8 (𝜑 → (log‘𝑍) ∈ ℝ+)
5857, 28rpdivcld 11716 . . . . . . 7 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ+)
59 4re 10939 . . . . . . . 8 4 ∈ ℝ
60 4pos 10958 . . . . . . . 8 0 < 4
6159, 60elrpii 11662 . . . . . . 7 4 ∈ ℝ+
62 rpdivcl 11683 . . . . . . 7 ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ+ ∧ 4 ∈ ℝ+) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ+)
6358, 61, 62sylancl 692 . . . . . 6 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ+)
6463rpge0d 11703 . . . . 5 (𝜑 → 0 ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
6550recnd 9919 . . . . . . . . 9 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℂ)
6634nncnd 10878 . . . . . . . . 9 (𝜑𝑀 ∈ ℂ)
67 1cnd 9907 . . . . . . . . 9 (𝜑 → 1 ∈ ℂ)
6865, 66, 67addassd 9913 . . . . . . . 8 (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)))
6952, 5readdcld 9920 . . . . . . . . . 10 (𝜑 → (𝑀 + 1) ∈ ℝ)
7050, 69readdcld 9920 . . . . . . . . 9 (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ∈ ℝ)
71 peano2re 10055 . . . . . . . . . 10 (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
7251, 71syl 17 . . . . . . . . 9 (𝜑 → (𝑁 + 1) ∈ ℝ)
7329rpred 11699 . . . . . . . . . . . . 13 (𝜑 → ((log‘𝑋) / (log‘𝐾)) ∈ ℝ)
74 2re 10932 . . . . . . . . . . . . . 14 2 ∈ ℝ
7574a1i 11 . . . . . . . . . . . . 13 (𝜑 → 2 ∈ ℝ)
7673, 75readdcld 9920 . . . . . . . . . . . 12 (𝜑 → (((log‘𝑋) / (log‘𝐾)) + 2) ∈ ℝ)
77 reflcl 12409 . . . . . . . . . . . . . . . . 17 (((log‘𝑋) / (log‘𝐾)) ∈ ℝ → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℝ)
7873, 77syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℝ)
7978recnd 9919 . . . . . . . . . . . . . . 15 (𝜑 → (⌊‘((log‘𝑋) / (log‘𝐾))) ∈ ℂ)
8079, 67, 67addassd 9913 . . . . . . . . . . . . . 14 (𝜑 → (((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + (1 + 1)))
811oveq1i 6532 . . . . . . . . . . . . . 14 (𝑀 + 1) = (((⌊‘((log‘𝑋) / (log‘𝐾))) + 1) + 1)
82 df-2 10921 . . . . . . . . . . . . . . 15 2 = (1 + 1)
8382oveq2i 6533 . . . . . . . . . . . . . 14 ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + (1 + 1))
8480, 81, 833eqtr4g 2663 . . . . . . . . . . . . 13 (𝜑 → (𝑀 + 1) = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2))
85 flle 12412 . . . . . . . . . . . . . . 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 10485 . . . . . . . . . . . . 13 (𝜑 → ((⌊‘((log‘𝑋) / (log‘𝐾))) + 2) ≤ (((log‘𝑋) / (log‘𝐾)) + 2))
8884, 87eqbrtrd 4594 . . . . . . . . . . . 12 (𝜑 → (𝑀 + 1) ≤ (((log‘𝑋) / (log‘𝐾)) + 2))
8940simp3d 1067 . . . . . . . . . . . . 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 1066 . . . . . . . . . . . 12 (𝜑 → (((log‘𝑋) / (log‘𝐾)) + 2) ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
9169, 76, 50, 88, 90letrd 10040 . . . . . . . . . . 11 (𝜑 → (𝑀 + 1) ≤ (((log‘𝑍) / (log‘𝐾)) / 4))
9269, 50, 50, 91leadd2dd 10486 . . . . . . . . . 10 (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
9343recnd 9919 . . . . . . . . . . . . . 14 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℂ)
94 2cnd 10935 . . . . . . . . . . . . . 14 (𝜑 → 2 ∈ ℂ)
95 2ne0 10955 . . . . . . . . . . . . . . 15 2 ≠ 0
9695a1i 11 . . . . . . . . . . . . . 14 (𝜑 → 2 ≠ 0)
9793, 94, 94, 96, 96divdiv1d 10676 . . . . . . . . . . . . 13 (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 2) / 2) = (((log‘𝑍) / (log‘𝐾)) / (2 · 2)))
98 2t2e4 11019 . . . . . . . . . . . . . 14 (2 · 2) = 4
9998oveq2i 6533 . . . . . . . . . . . . 13 (((log‘𝑍) / (log‘𝐾)) / (2 · 2)) = (((log‘𝑍) / (log‘𝐾)) / 4)
10097, 99syl6eq 2654 . . . . . . . . . . . 12 (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 2) / 2) = (((log‘𝑍) / (log‘𝐾)) / 4))
101100oveq2d 6538 . . . . . . . . . . 11 (𝜑 → (2 · ((((log‘𝑍) / (log‘𝐾)) / 2) / 2)) = (2 · (((log‘𝑍) / (log‘𝐾)) / 4)))
10244recnd 9919 . . . . . . . . . . . 12 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ∈ ℂ)
103102, 94, 96divcan2d 10647 . . . . . . . . . . 11 (𝜑 → (2 · ((((log‘𝑍) / (log‘𝐾)) / 2) / 2)) = (((log‘𝑍) / (log‘𝐾)) / 2))
104652timesd 11117 . . . . . . . . . . 11 (𝜑 → (2 · (((log‘𝑍) / (log‘𝐾)) / 4)) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
105101, 103, 1043eqtr3d 2646 . . . . . . . . . 10 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) = ((((log‘𝑍) / (log‘𝐾)) / 4) + (((log‘𝑍) / (log‘𝐾)) / 4)))
10692, 105breqtrrd 4600 . . . . . . . . 9 (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ (((log‘𝑍) / (log‘𝐾)) / 2))
107 fllep1 12414 . . . . . . . . . . 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 6532 . . . . . . . . . 10 (𝑁 + 1) = ((⌊‘(((log‘𝑍) / (log‘𝐾)) / 2)) + 1)
110108, 109syl6breqr 4614 . . . . . . . . 9 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 2) ≤ (𝑁 + 1))
11170, 44, 72, 106, 110letrd 10040 . . . . . . . 8 (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + (𝑀 + 1)) ≤ (𝑁 + 1))
11268, 111eqbrtrd 4594 . . . . . . 7 (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) ≤ (𝑁 + 1))
11350, 52readdcld 9920 . . . . . . . 8 (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ∈ ℝ)
114113, 51, 5leadd1d 10465 . . . . . . 7 (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ≤ 𝑁 ↔ (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) + 1) ≤ (𝑁 + 1)))
115112, 114mpbird 245 . . . . . 6 (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ≤ 𝑁)
116 leaddsub 10348 . . . . . . 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 1317 . . . . . 6 (𝜑 → (((((log‘𝑍) / (log‘𝐾)) / 4) + 𝑀) ≤ 𝑁 ↔ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀)))
118115, 117mpbid 220 . . . . 5 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀))
11947, 50, 53, 64, 118letrd 10040 . . . 4 (𝜑 → 0 ≤ (𝑁𝑀))
12051, 52subge0d 10461 . . . 4 (𝜑 → (0 ≤ (𝑁𝑀) ↔ 𝑀𝑁))
121119, 120mpbid 220 . . 3 (𝜑𝑀𝑁)
122 eluz2 11520 . . 3 (𝑁 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁))
12335, 46, 121, 122syl3anbrc 1238 . 2 (𝜑𝑁 ∈ (ℤ𝑀))
12434, 123, 1183jca 1234 1 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2774   class class class wbr 4572  cmpt 4632  cfv 5785  (class class class)co 6522  cr 9786  0cc0 9787  1c1 9788   + caddc 9790   · cmul 9792  +∞cpnf 9922   < clt 9925  cle 9926  cmin 10112   / cdiv 10528  cn 10862  2c2 10912  3c3 10913  4c4 10914  0cn0 11134  cz 11205  cdc 11320  cuz 11514  +crp 11659  (,)cioo 11997  [,)cico 11999  cfl 12403  cexp 12672  csqrt 13762  expce 14572  eceu 14573  logclog 24017  ψcchp 24531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-inf2 8393  ax-cnex 9843  ax-resscn 9844  ax-1cn 9845  ax-icn 9846  ax-addcl 9847  ax-addrcl 9848  ax-mulcl 9849  ax-mulrcl 9850  ax-mulcom 9851  ax-addass 9852  ax-mulass 9853  ax-distr 9854  ax-i2m1 9855  ax-1ne0 9856  ax-1rid 9857  ax-rnegex 9858  ax-rrecex 9859  ax-cnre 9860  ax-pre-lttri 9861  ax-pre-lttrn 9862  ax-pre-ltadd 9863  ax-pre-mulgt0 9864  ax-pre-sup 9865  ax-addf 9866  ax-mulf 9867
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-nel 2777  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-iin 4447  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-se 4983  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-isom 5794  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-of 6767  df-om 6930  df-1st 7031  df-2nd 7032  df-supp 7155  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-1o 7419  df-2o 7420  df-oadd 7423  df-er 7601  df-map 7718  df-pm 7719  df-ixp 7767  df-en 7814  df-dom 7815  df-sdom 7816  df-fin 7817  df-fsupp 8131  df-fi 8172  df-sup 8203  df-inf 8204  df-oi 8270  df-card 8620  df-cda 8845  df-pnf 9927  df-mnf 9928  df-xr 9929  df-ltxr 9930  df-le 9931  df-sub 10114  df-neg 10115  df-div 10529  df-nn 10863  df-2 10921  df-3 10922  df-4 10923  df-5 10924  df-6 10925  df-7 10926  df-8 10927  df-9 10928  df-n0 11135  df-z 11206  df-dec 11321  df-uz 11515  df-q 11616  df-rp 11660  df-xneg 11773  df-xadd 11774  df-xmul 11775  df-ioo 12001  df-ioc 12002  df-ico 12003  df-icc 12004  df-fz 12148  df-fzo 12285  df-fl 12405  df-mod 12481  df-seq 12614  df-exp 12673  df-fac 12873  df-bc 12902  df-hash 12930  df-shft 13596  df-cj 13628  df-re 13629  df-im 13630  df-sqrt 13764  df-abs 13765  df-limsup 13991  df-clim 14008  df-rlim 14009  df-sum 14206  df-ef 14578  df-e 14579  df-sin 14580  df-cos 14581  df-pi 14583  df-struct 15638  df-ndx 15639  df-slot 15640  df-base 15641  df-sets 15642  df-ress 15643  df-plusg 15722  df-mulr 15723  df-starv 15724  df-sca 15725  df-vsca 15726  df-ip 15727  df-tset 15728  df-ple 15729  df-ds 15732  df-unif 15733  df-hom 15734  df-cco 15735  df-rest 15847  df-topn 15848  df-0g 15866  df-gsum 15867  df-topgen 15868  df-pt 15869  df-prds 15872  df-xrs 15926  df-qtop 15931  df-imas 15932  df-xps 15934  df-mre 16010  df-mrc 16011  df-acs 16013  df-mgm 17006  df-sgrp 17048  df-mnd 17059  df-submnd 17100  df-mulg 17305  df-cntz 17514  df-cmn 17959  df-psmet 19500  df-xmet 19501  df-met 19502  df-bl 19503  df-mopn 19504  df-fbas 19505  df-fg 19506  df-cnfld 19509  df-top 20458  df-bases 20459  df-topon 20460  df-topsp 20461  df-cld 20570  df-ntr 20571  df-cls 20572  df-nei 20649  df-lp 20687  df-perf 20688  df-cn 20778  df-cnp 20779  df-haus 20866  df-tx 21112  df-hmeo 21305  df-fil 21397  df-fm 21489  df-flim 21490  df-flf 21491  df-xms 21871  df-ms 21872  df-tms 21873  df-cncf 22415  df-limc 23348  df-dv 23349  df-log 24019
This theorem is referenced by:  pntlemh  25000  pntlemq  25002  pntlemr  25003  pntlemj  25004  pntlemf  25006
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