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Theorem chpchtsum 26567
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 13878 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin)
2 simpr 485 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
32elin2d 4159 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
4 prmnn 16550 . . . . . . . . 9 (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
53, 4syl 17 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
65nnrpd 12955 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ+)
76relogcld 25978 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
87recnd 11183 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
9 fsumconst 15675 . . . . 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 483 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ)
12 1red 11156 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℝ)
135nnred 12168 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
14 prmuz2 16572 . . . . . . . . . . . . 13 (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ‘2))
153, 14syl 17 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (ℤ‘2))
16 eluz2gt1 12845 . . . . . . . . . . . 12 (𝑝 ∈ (ℤ‘2) → 1 < 𝑝)
1715, 16syl 17 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝑝)
182elin1d 4158 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (0[,]𝐴))
19 0re 11157 . . . . . . . . . . . . . 14 0 ∈ ℝ
20 elicc2 13329 . . . . . . . . . . . . . 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 231 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴))
2322simp3d 1144 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝𝐴)
2412, 13, 11, 17, 23ltletrd 11315 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝐴)
2511, 24rplogcld 25984 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ+)
2613, 17rplogcld 25984 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ+)
2725, 26rpdivcld 12974 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ+)
2827rpred 12957 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
2927rpge0d 12961 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ≤ ((log‘𝐴) / (log‘𝑝)))
30 flge0nn0 13725 . . . . . . 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 14246 . . . . . 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 7372 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝)) = ((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝)))
3528flcld 13703 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
3635zcnd 12608 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℂ)
3736, 8mulcomd 11176 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝)) = ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
3810, 34, 373eqtrrd 2781 . . 3 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) = Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝))
3938sumeq2dv 15588 . 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 26566 . 2 (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
41 simpl 483 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
42 0red 11158 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ∈ ℝ)
43 1red 11156 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℝ)
44 0lt1 11677 . . . . . . . . 9 0 < 1
4544a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 < 1)
46 elfzuz2 13446 . . . . . . . . 9 (𝑘 ∈ (1...(⌊‘𝐴)) → (⌊‘𝐴) ∈ (ℤ‘1))
47 eluzle 12776 . . . . . . . . . . 11 ((⌊‘𝐴) ∈ (ℤ‘1) → 1 ≤ (⌊‘𝐴))
4847adantl 482 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ‘1)) → 1 ≤ (⌊‘𝐴))
49 simpl 483 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ‘1)) → 𝐴 ∈ ℝ)
50 1z 12533 . . . . . . . . . . 11 1 ∈ ℤ
51 flge 13710 . . . . . . . . . . 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 256 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ‘1)) → 1 ≤ 𝐴)
5446, 53sylan2 593 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 1 ≤ 𝐴)
5542, 43, 41, 45, 54ltletrd 11315 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 < 𝐴)
5642, 41, 55ltled 11303 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ≤ 𝐴)
57 elfznn 13470 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘𝐴)) → 𝑘 ∈ ℕ)
5857adantl 482 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝑘 ∈ ℕ)
5958nnrecred 12204 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (1 / 𝑘) ∈ ℝ)
6041, 56, 59recxpcld 26078 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (𝐴𝑐(1 / 𝑘)) ∈ ℝ)
61 chtval 26459 . . . . 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 15588 . . 3 (𝐴 ∈ ℝ → Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴𝑐(1 / 𝑘))) = Σ𝑘 ∈ (1...(⌊‘𝐴))Σ𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)(log‘𝑝))
64 ppifi 26455 . . . 4 (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
65 fzfid 13878 . . . 4 (𝐴 ∈ ℝ → (1...(⌊‘𝐴)) ∈ Fin)
66 elinel2 4156 . . . . . . . 8 (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ ℙ)
67 elfznn 13470 . . . . . . . 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 11158 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ∈ ℝ)
71 inss2 4189 . . . . . . . . . . . . 13 ((0[,]𝐴) ∩ ℙ) ⊆ ℙ
7271a1i 11 . . . . . . . . . . . 12 (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ⊆ ℙ)
7372sselda 3944 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
7473, 4syl 17 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
7574nnred 12168 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
7674nngt0d 12202 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝑝)
7770, 75, 11, 76, 23ltletrd 11315 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝐴)
7877ex 413 . . . . . . 7 (𝐴 ∈ ℝ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 0 < 𝐴))
7978adantrd 492 . . . . . 6 (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 0 < 𝐴))
8069, 79jcad 513 . . . . 5 (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)))
81 elinel2 4156 . . . . . . . 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 413 . . . . . . 7 (𝐴 ∈ ℝ → (𝑘 ∈ (1...(⌊‘𝐴)) → 0 < 𝐴))
8584adantrd 492 . . . . . 6 (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)) → 0 < 𝐴))
8683, 85jcad 513 . . . . 5 (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)))
87 elin 3926 . . . . . . . . 9 (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ))
88 simprll 777 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℙ)
8988biantrud 532 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ)))
90 0red 11158 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ∈ ℝ)
91 simpl 483 . . . . . . . . . . 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 12168 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℝ)
9492nnnn0d 12473 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℕ0)
9594nn0ge0d 12476 . . . . . . . . . . 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 286 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝𝐴)))
9897baibd 540 . . . . . . . . . . 11 (((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝𝐴))
9990, 91, 93, 95, 98syl22anc 837 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝𝐴))
10089, 99bitr3d 280 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ) ↔ 𝑝𝐴))
10187, 100bitrid 282 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ 𝑝𝐴))
102 simprr 771 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 < 𝐴)
10391, 102elrpd 12954 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℝ+)
104103relogcld 25978 . . . . . . . . . . 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 25984 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘𝑝) ∈ ℝ+)
108104, 107rerpdivcld 12988 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
109 simprlr 778 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℕ)
110109nnzd 12526 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℤ)
111 flge 13710 . . . . . . . . . 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 12473 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℕ0)
11492, 113nnexpcld 14148 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝𝑘) ∈ ℕ)
115114nnrpd 12955 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝𝑘) ∈ ℝ+)
116115, 103logled 25982 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴 ↔ (log‘(𝑝𝑘)) ≤ (log‘𝐴)))
11792nnrpd 12955 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℝ+)
118 relogexp 25951 . . . . . . . . . . . 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 5115 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((log‘(𝑝𝑘)) ≤ (log‘𝐴) ↔ (𝑘 · (log‘𝑝)) ≤ (log‘𝐴)))
121109nnred 12168 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℝ)
122121, 104, 107lemuldivd 13006 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
123116, 120, 1223bitrd 304 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
124 nnuz 12806 . . . . . . . . . . 11 ℕ = (ℤ‘1)
125109, 124eleqtrdi 2848 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ (ℤ‘1))
126108flcld 13703 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
127 elfz5 13433 . . . . . . . . . 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 311 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ (𝑝𝑘) ≤ 𝐴))
130101, 129anbi12d 631 . . . . . . 7 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑝𝐴 ∧ (𝑝𝑘) ≤ 𝐴)))
13191flcld 13703 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (⌊‘𝐴) ∈ ℤ)
132 elfz5 13433 . . . . . . . . . . 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 13710 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘𝐴𝑘 ≤ (⌊‘𝐴)))
13591, 110, 134syl2anc 584 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘𝐴𝑘 ≤ (⌊‘𝐴)))
136133, 135bitr4d 281 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘𝐴))
137 elin 3926 . . . . . . . . . 10 (𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ))
13888biantrud 532 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ↔ (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ)))
139103rpge0d 12961 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ 𝐴)
140109nnrecred 12204 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (1 / 𝑘) ∈ ℝ)
14191, 139, 140recxpcld 26078 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴𝑐(1 / 𝑘)) ∈ ℝ)
142 elicc2 13329 . . . . . . . . . . . . . . 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 286 . . . . . . . . . . . . . 14 ((0 ∈ ℝ ∧ (𝐴𝑐(1 / 𝑘)) ∈ ℝ) → (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (𝐴𝑐(1 / 𝑘)))))
145144baibd 540 . . . . . . . . . . . . 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 837 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ↔ 𝑝 ≤ (𝐴𝑐(1 / 𝑘))))
147138, 146bitr3d 280 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ) ↔ 𝑝 ≤ (𝐴𝑐(1 / 𝑘))))
14891, 139, 140cxpge0d 26079 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ (𝐴𝑐(1 / 𝑘)))
149109nnrpd 12955 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℝ+)
15093, 95, 141, 148, 149cxple2d 26082 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ≤ (𝐴𝑐(1 / 𝑘)) ↔ (𝑝𝑐𝑘) ≤ ((𝐴𝑐(1 / 𝑘))↑𝑐𝑘)))
15192nncnd 12169 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℂ)
152 cxpexp 26023 . . . . . . . . . . . . 13 ((𝑝 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑝𝑐𝑘) = (𝑝𝑘))
153151, 113, 152syl2anc 584 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝𝑐𝑘) = (𝑝𝑘))
154109nncnd 12169 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℂ)
155109nnne0d 12203 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ≠ 0)
156154, 155recid2d 11927 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((1 / 𝑘) · 𝑘) = 1)
157156oveq2d 7373 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴𝑐((1 / 𝑘) · 𝑘)) = (𝐴𝑐1))
158103, 140, 154cxpmuld 26091 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴𝑐((1 / 𝑘) · 𝑘)) = ((𝐴𝑐(1 / 𝑘))↑𝑐𝑘))
15991recnd 11183 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℂ)
160159cxp1d 26061 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴𝑐1) = 𝐴)
161157, 158, 1603eqtr3d 2784 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝐴𝑐(1 / 𝑘))↑𝑐𝑘) = 𝐴)
162153, 161breq12d 5118 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑐𝑘) ≤ ((𝐴𝑐(1 / 𝑘))↑𝑐𝑘) ↔ (𝑝𝑘) ≤ 𝐴))
163147, 150, 1623bitrd 304 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ) ↔ (𝑝𝑘) ≤ 𝐴))
164137, 163bitrid 282 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝𝑘) ≤ 𝐴))
165136, 164anbi12d 631 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)) ↔ (𝑘𝐴 ∧ (𝑝𝑘) ≤ 𝐴)))
166114nnred 12168 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝𝑘) ∈ ℝ)
167 bernneq3 14134 . . . . . . . . . . . 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 11303 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ≤ (𝑝𝑘))
170 letr 11249 . . . . . . . . . . 11 ((𝑘 ∈ ℝ ∧ (𝑝𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑘 ≤ (𝑝𝑘) ∧ (𝑝𝑘) ≤ 𝐴) → 𝑘𝐴))
171121, 166, 91, 170syl3anc 1371 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 ≤ (𝑝𝑘) ∧ (𝑝𝑘) ≤ 𝐴) → 𝑘𝐴))
172169, 171mpand 693 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴𝑘𝐴))
173172pm4.71rd 563 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴 ↔ (𝑘𝐴 ∧ (𝑝𝑘) ≤ 𝐴)))
174151exp1d 14046 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑1) = 𝑝)
17592nnge1d 12201 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 1 ≤ 𝑝)
17693, 175, 125leexp2ad 14157 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑1) ≤ (𝑝𝑘))
177174, 176eqbrtrrd 5129 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ≤ (𝑝𝑘))
178 letr 11249 . . . . . . . . . . 11 ((𝑝 ∈ ℝ ∧ (𝑝𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑝 ≤ (𝑝𝑘) ∧ (𝑝𝑘) ≤ 𝐴) → 𝑝𝐴))
17993, 166, 91, 178syl3anc 1371 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ≤ (𝑝𝑘) ∧ (𝑝𝑘) ≤ 𝐴) → 𝑝𝐴))
180177, 179mpand 693 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴𝑝𝐴))
181180pm4.71rd 563 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴 ↔ (𝑝𝐴 ∧ (𝑝𝑘) ≤ 𝐴)))
182165, 173, 1813bitr2rd 307 . . . . . . 7 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝐴 ∧ (𝑝𝑘) ≤ 𝐴) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ))))
183130, 182bitrd 278 . . . . . 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 413 . . . . 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 380 . . . 4 (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ))))
1868adantrr 715 . . . 4 ((𝐴 ∈ ℝ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘𝑝) ∈ ℂ)
18764, 65, 1, 185, 186fsumcom2 15659 . . 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 2779 . 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 2786 1 (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴𝑐(1 / 𝑘))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  cin 3909  wss 3910   class class class wbr 5105  cfv 6496  (class class class)co 7357  Fincfn 8883  cc 11049  cr 11050  0cc0 11051  1c1 11052   · cmul 11056   < clt 11189  cle 11190   / cdiv 11812  cn 12153  2c2 12208  0cn0 12413  cz 12499  cuz 12763  +crp 12915  [,]cicc 13267  ...cfz 13424  cfl 13695  cexp 13967  chash 14230  Σcsu 15570  cprime 16547  logclog 25910  𝑐ccxp 25911  θccht 26440  ψcchp 26442
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-oadd 8416  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-dju 9837  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-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  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-pi 15955  df-dvds 16137  df-gcd 16375  df-prm 16548  df-pc 16709  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-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  df-log 25912  df-cxp 25913  df-cht 26446  df-vma 26447  df-chp 26448
This theorem is referenced by: (None)
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