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Theorem chpchtsum 27263
Description: The second Chebyshev function is the sum of the theta function at arguments quickly approaching zero. (This is usually stated as an infinite sum, but after a certain point, the terms are all zero, and it is easier for us to use an explicit finite sum.) (Contributed by Mario Carneiro, 7-Apr-2016.)
Assertion
Ref Expression
chpchtsum (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴𝑐(1 / 𝑘))))
Distinct variable group:   𝐴,𝑘

Proof of Theorem chpchtsum
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 fzfid 14014 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin)
2 simpr 484 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
32elin2d 4205 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
4 prmnn 16711 . . . . . . . . 9 (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
53, 4syl 17 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
65nnrpd 13075 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ+)
76relogcld 26665 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
87recnd 11289 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
9 fsumconst 15826 . . . . 5 (((1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin ∧ (log‘𝑝) ∈ ℂ) → Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝) = ((♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝)))
101, 8, 9syl2anc 584 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝) = ((♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝)))
11 simpl 482 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ)
12 1red 11262 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℝ)
135nnred 12281 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
14 prmuz2 16733 . . . . . . . . . . . . 13 (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ‘2))
153, 14syl 17 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (ℤ‘2))
16 eluz2gt1 12962 . . . . . . . . . . . 12 (𝑝 ∈ (ℤ‘2) → 1 < 𝑝)
1715, 16syl 17 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝑝)
182elin1d 4204 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (0[,]𝐴))
19 0re 11263 . . . . . . . . . . . . . 14 0 ∈ ℝ
20 elicc2 13452 . . . . . . . . . . . . . 14 ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴)))
2119, 11, 20sylancr 587 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴)))
2218, 21mpbid 232 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴))
2322simp3d 1145 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝𝐴)
2412, 13, 11, 17, 23ltletrd 11421 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝐴)
2511, 24rplogcld 26671 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ+)
2613, 17rplogcld 26671 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ+)
2725, 26rpdivcld 13094 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ+)
2827rpred 13077 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
2927rpge0d 13081 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ≤ ((log‘𝐴) / (log‘𝑝)))
30 flge0nn0 13860 . . . . . . 7 ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 0 ≤ ((log‘𝐴) / (log‘𝑝))) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0)
3128, 29, 30syl2anc 584 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0)
32 hashfz1 14385 . . . . . 6 ((⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0 → (♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) = (⌊‘((log‘𝐴) / (log‘𝑝))))
3331, 32syl 17 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) = (⌊‘((log‘𝐴) / (log‘𝑝))))
3433oveq1d 7446 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝)) = ((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝)))
3528flcld 13838 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
3635zcnd 12723 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℂ)
3736, 8mulcomd 11282 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝)) = ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
3810, 34, 373eqtrrd 2782 . . 3 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) = Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝))
3938sumeq2dv 15738 . 2 (𝐴 ∈ ℝ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝))
40 chpval2 27262 . 2 (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
41 simpl 482 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
42 0red 11264 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ∈ ℝ)
43 1red 11262 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℝ)
44 0lt1 11785 . . . . . . . . 9 0 < 1
4544a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 < 1)
46 elfzuz2 13569 . . . . . . . . 9 (𝑘 ∈ (1...(⌊‘𝐴)) → (⌊‘𝐴) ∈ (ℤ‘1))
47 eluzle 12891 . . . . . . . . . . 11 ((⌊‘𝐴) ∈ (ℤ‘1) → 1 ≤ (⌊‘𝐴))
4847adantl 481 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ‘1)) → 1 ≤ (⌊‘𝐴))
49 simpl 482 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ‘1)) → 𝐴 ∈ ℝ)
50 1z 12647 . . . . . . . . . . 11 1 ∈ ℤ
51 flge 13845 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 1 ∈ ℤ) → (1 ≤ 𝐴 ↔ 1 ≤ (⌊‘𝐴)))
5249, 50, 51sylancl 586 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ‘1)) → (1 ≤ 𝐴 ↔ 1 ≤ (⌊‘𝐴)))
5348, 52mpbird 257 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ‘1)) → 1 ≤ 𝐴)
5446, 53sylan2 593 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 1 ≤ 𝐴)
5542, 43, 41, 45, 54ltletrd 11421 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 < 𝐴)
5642, 41, 55ltled 11409 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ≤ 𝐴)
57 elfznn 13593 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘𝐴)) → 𝑘 ∈ ℕ)
5857adantl 481 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝑘 ∈ ℕ)
5958nnrecred 12317 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (1 / 𝑘) ∈ ℝ)
6041, 56, 59recxpcld 26765 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (𝐴𝑐(1 / 𝑘)) ∈ ℝ)
61 chtval 27153 . . . . 5 ((𝐴𝑐(1 / 𝑘)) ∈ ℝ → (θ‘(𝐴𝑐(1 / 𝑘))) = Σ𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)(log‘𝑝))
6260, 61syl 17 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (θ‘(𝐴𝑐(1 / 𝑘))) = Σ𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)(log‘𝑝))
6362sumeq2dv 15738 . . 3 (𝐴 ∈ ℝ → Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴𝑐(1 / 𝑘))) = Σ𝑘 ∈ (1...(⌊‘𝐴))Σ𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)(log‘𝑝))
64 ppifi 27149 . . . 4 (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
65 fzfid 14014 . . . 4 (𝐴 ∈ ℝ → (1...(⌊‘𝐴)) ∈ Fin)
66 elinel2 4202 . . . . . . . 8 (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ ℙ)
67 elfznn 13593 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
6866, 67anim12i 613 . . . . . . 7 ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))
6968a1i 11 . . . . . 6 (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
70 0red 11264 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ∈ ℝ)
71 inss2 4238 . . . . . . . . . . . . 13 ((0[,]𝐴) ∩ ℙ) ⊆ ℙ
7271a1i 11 . . . . . . . . . . . 12 (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ⊆ ℙ)
7372sselda 3983 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
7473, 4syl 17 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
7574nnred 12281 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
7674nngt0d 12315 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝑝)
7770, 75, 11, 76, 23ltletrd 11421 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝐴)
7877ex 412 . . . . . . 7 (𝐴 ∈ ℝ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 0 < 𝐴))
7978adantrd 491 . . . . . 6 (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 0 < 𝐴))
8069, 79jcad 512 . . . . 5 (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)))
81 elinel2 4202 . . . . . . . 8 (𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ) → 𝑝 ∈ ℙ)
8257, 81anim12ci 614 . . . . . . 7 ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))
8382a1i 11 . . . . . 6 (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
8455ex 412 . . . . . . 7 (𝐴 ∈ ℝ → (𝑘 ∈ (1...(⌊‘𝐴)) → 0 < 𝐴))
8584adantrd 491 . . . . . 6 (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)) → 0 < 𝐴))
8683, 85jcad 512 . . . . 5 (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)))
87 elin 3967 . . . . . . . . 9 (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ))
88 simprll 779 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℙ)
8988biantrud 531 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ)))
90 0red 11264 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ∈ ℝ)
91 simpl 482 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℝ)
9288, 4syl 17 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℕ)
9392nnred 12281 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℝ)
9492nnnn0d 12587 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℕ0)
9594nn0ge0d 12590 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ 𝑝)
96 df-3an 1089 . . . . . . . . . . . . 13 ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝𝐴))
9720, 96bitrdi 287 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝𝐴)))
9897baibd 539 . . . . . . . . . . 11 (((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝𝐴))
9990, 91, 93, 95, 98syl22anc 839 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝𝐴))
10089, 99bitr3d 281 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ) ↔ 𝑝𝐴))
10187, 100bitrid 283 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ 𝑝𝐴))
102 simprr 773 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 < 𝐴)
10391, 102elrpd 13074 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℝ+)
104103relogcld 26665 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘𝐴) ∈ ℝ)
10588, 14syl 17 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ (ℤ‘2))
106105, 16syl 17 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 1 < 𝑝)
10793, 106rplogcld 26671 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘𝑝) ∈ ℝ+)
108104, 107rerpdivcld 13108 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
109 simprlr 780 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℕ)
110109nnzd 12640 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℤ)
111 flge 13845 . . . . . . . . . 10 ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
112108, 110, 111syl2anc 584 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
113109nnnn0d 12587 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℕ0)
11492, 113nnexpcld 14284 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝𝑘) ∈ ℕ)
115114nnrpd 13075 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝𝑘) ∈ ℝ+)
116115, 103logled 26669 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴 ↔ (log‘(𝑝𝑘)) ≤ (log‘𝐴)))
11792nnrpd 13075 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℝ+)
118 relogexp 26638 . . . . . . . . . . . 12 ((𝑝 ∈ ℝ+𝑘 ∈ ℤ) → (log‘(𝑝𝑘)) = (𝑘 · (log‘𝑝)))
119117, 110, 118syl2anc 584 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘(𝑝𝑘)) = (𝑘 · (log‘𝑝)))
120119breq1d 5153 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((log‘(𝑝𝑘)) ≤ (log‘𝐴) ↔ (𝑘 · (log‘𝑝)) ≤ (log‘𝐴)))
121109nnred 12281 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℝ)
122121, 104, 107lemuldivd 13126 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
123116, 120, 1223bitrd 305 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
124 nnuz 12921 . . . . . . . . . . 11 ℕ = (ℤ‘1)
125109, 124eleqtrdi 2851 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ (ℤ‘1))
126108flcld 13838 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
127 elfz5 13556 . . . . . . . . . 10 ((𝑘 ∈ (ℤ‘1) ∧ (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
128125, 126, 127syl2anc 584 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
129112, 123, 1283bitr4rd 312 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ (𝑝𝑘) ≤ 𝐴))
130101, 129anbi12d 632 . . . . . . 7 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑝𝐴 ∧ (𝑝𝑘) ≤ 𝐴)))
13191flcld 13838 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (⌊‘𝐴) ∈ ℤ)
132 elfz5 13556 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ‘1) ∧ (⌊‘𝐴) ∈ ℤ) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ (⌊‘𝐴)))
133125, 131, 132syl2anc 584 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ (⌊‘𝐴)))
134 flge 13845 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘𝐴𝑘 ≤ (⌊‘𝐴)))
13591, 110, 134syl2anc 584 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘𝐴𝑘 ≤ (⌊‘𝐴)))
136133, 135bitr4d 282 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘𝐴))
137 elin 3967 . . . . . . . . . 10 (𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ))
13888biantrud 531 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ↔ (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ)))
139103rpge0d 13081 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ 𝐴)
140109nnrecred 12317 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (1 / 𝑘) ∈ ℝ)
14191, 139, 140recxpcld 26765 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴𝑐(1 / 𝑘)) ∈ ℝ)
142 elicc2 13452 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ ∧ (𝐴𝑐(1 / 𝑘)) ∈ ℝ) → (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝 ≤ (𝐴𝑐(1 / 𝑘)))))
143 df-3an 1089 . . . . . . . . . . . . . . 15 ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝 ≤ (𝐴𝑐(1 / 𝑘))) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (𝐴𝑐(1 / 𝑘))))
144142, 143bitrdi 287 . . . . . . . . . . . . . 14 ((0 ∈ ℝ ∧ (𝐴𝑐(1 / 𝑘)) ∈ ℝ) → (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (𝐴𝑐(1 / 𝑘)))))
145144baibd 539 . . . . . . . . . . . . 13 (((0 ∈ ℝ ∧ (𝐴𝑐(1 / 𝑘)) ∈ ℝ) ∧ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝)) → (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ↔ 𝑝 ≤ (𝐴𝑐(1 / 𝑘))))
14690, 141, 93, 95, 145syl22anc 839 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ↔ 𝑝 ≤ (𝐴𝑐(1 / 𝑘))))
147138, 146bitr3d 281 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ) ↔ 𝑝 ≤ (𝐴𝑐(1 / 𝑘))))
14891, 139, 140cxpge0d 26766 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ (𝐴𝑐(1 / 𝑘)))
149109nnrpd 13075 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℝ+)
15093, 95, 141, 148, 149cxple2d 26769 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ≤ (𝐴𝑐(1 / 𝑘)) ↔ (𝑝𝑐𝑘) ≤ ((𝐴𝑐(1 / 𝑘))↑𝑐𝑘)))
15192nncnd 12282 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℂ)
152 cxpexp 26710 . . . . . . . . . . . . 13 ((𝑝 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑝𝑐𝑘) = (𝑝𝑘))
153151, 113, 152syl2anc 584 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝𝑐𝑘) = (𝑝𝑘))
154109nncnd 12282 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℂ)
155109nnne0d 12316 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ≠ 0)
156154, 155recid2d 12039 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((1 / 𝑘) · 𝑘) = 1)
157156oveq2d 7447 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴𝑐((1 / 𝑘) · 𝑘)) = (𝐴𝑐1))
158103, 140, 154cxpmuld 26779 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴𝑐((1 / 𝑘) · 𝑘)) = ((𝐴𝑐(1 / 𝑘))↑𝑐𝑘))
15991recnd 11289 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℂ)
160159cxp1d 26748 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴𝑐1) = 𝐴)
161157, 158, 1603eqtr3d 2785 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝐴𝑐(1 / 𝑘))↑𝑐𝑘) = 𝐴)
162153, 161breq12d 5156 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑐𝑘) ≤ ((𝐴𝑐(1 / 𝑘))↑𝑐𝑘) ↔ (𝑝𝑘) ≤ 𝐴))
163147, 150, 1623bitrd 305 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ) ↔ (𝑝𝑘) ≤ 𝐴))
164137, 163bitrid 283 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝𝑘) ≤ 𝐴))
165136, 164anbi12d 632 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)) ↔ (𝑘𝐴 ∧ (𝑝𝑘) ≤ 𝐴)))
166114nnred 12281 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝𝑘) ∈ ℝ)
167 bernneq3 14270 . . . . . . . . . . . 12 ((𝑝 ∈ (ℤ‘2) ∧ 𝑘 ∈ ℕ0) → 𝑘 < (𝑝𝑘))
168105, 113, 167syl2anc 584 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 < (𝑝𝑘))
169121, 166, 168ltled 11409 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ≤ (𝑝𝑘))
170 letr 11355 . . . . . . . . . . 11 ((𝑘 ∈ ℝ ∧ (𝑝𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑘 ≤ (𝑝𝑘) ∧ (𝑝𝑘) ≤ 𝐴) → 𝑘𝐴))
171121, 166, 91, 170syl3anc 1373 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 ≤ (𝑝𝑘) ∧ (𝑝𝑘) ≤ 𝐴) → 𝑘𝐴))
172169, 171mpand 695 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴𝑘𝐴))
173172pm4.71rd 562 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴 ↔ (𝑘𝐴 ∧ (𝑝𝑘) ≤ 𝐴)))
174151exp1d 14181 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑1) = 𝑝)
17592nnge1d 12314 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 1 ≤ 𝑝)
17693, 175, 125leexp2ad 14293 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑1) ≤ (𝑝𝑘))
177174, 176eqbrtrrd 5167 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ≤ (𝑝𝑘))
178 letr 11355 . . . . . . . . . . 11 ((𝑝 ∈ ℝ ∧ (𝑝𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑝 ≤ (𝑝𝑘) ∧ (𝑝𝑘) ≤ 𝐴) → 𝑝𝐴))
17993, 166, 91, 178syl3anc 1373 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ≤ (𝑝𝑘) ∧ (𝑝𝑘) ≤ 𝐴) → 𝑝𝐴))
180177, 179mpand 695 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴𝑝𝐴))
181180pm4.71rd 562 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴 ↔ (𝑝𝐴 ∧ (𝑝𝑘) ≤ 𝐴)))
182165, 173, 1813bitr2rd 308 . . . . . . 7 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝐴 ∧ (𝑝𝑘) ≤ 𝐴) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ))))
183130, 182bitrd 279 . . . . . 6 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ))))
184183ex 412 . . . . 5 (𝐴 ∈ ℝ → (((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)))))
18580, 86, 184pm5.21ndd 379 . . . 4 (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ))))
1868adantrr 717 . . . 4 ((𝐴 ∈ ℝ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘𝑝) ∈ ℂ)
18764, 65, 1, 185, 186fsumcom2 15810 . . 3 (𝐴 ∈ ℝ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝) = Σ𝑘 ∈ (1...(⌊‘𝐴))Σ𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)(log‘𝑝))
18863, 187eqtr4d 2780 . 2 (𝐴 ∈ ℝ → Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴𝑐(1 / 𝑘))) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝))
18939, 40, 1883eqtr4d 2787 1 (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴𝑐(1 / 𝑘))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  cin 3950  wss 3951   class class class wbr 5143  cfv 6561  (class class class)co 7431  Fincfn 8985  cc 11153  cr 11154  0cc0 11155  1c1 11156   · cmul 11160   < clt 11295  cle 11296   / cdiv 11920  cn 12266  2c2 12321  0cn0 12526  cz 12613  cuz 12878  +crp 13034  [,]cicc 13390  ...cfz 13547  cfl 13830  cexp 14102  chash 14369  Σcsu 15722  cprime 16708  logclog 26596  𝑐ccxp 26597  θccht 27134  ψcchp 27136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ioc 13392  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-shft 15106  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15507  df-clim 15524  df-rlim 15525  df-sum 15723  df-ef 16103  df-sin 16105  df-cos 16106  df-pi 16108  df-dvds 16291  df-gcd 16532  df-prm 16709  df-pc 16875  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-fbas 21361  df-fg 21362  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-lp 23144  df-perf 23145  df-cn 23235  df-cnp 23236  df-haus 23323  df-tx 23570  df-hmeo 23763  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-xms 24330  df-ms 24331  df-tms 24332  df-cncf 24904  df-limc 25901  df-dv 25902  df-log 26598  df-cxp 26599  df-cht 27140  df-vma 27141  df-chp 27142
This theorem is referenced by: (None)
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