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Theorem chpchtsum 25143
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 12966 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin)
2 inss2 3977 . . . . . . . . . 10 ((0[,]𝐴) ∩ ℙ) ⊆ ℙ
3 simpr 479 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
42, 3sseldi 3742 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
5 prmnn 15590 . . . . . . . . 9 (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
64, 5syl 17 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
76nnrpd 12063 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ+)
87relogcld 24568 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
98recnd 10260 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
10 fsumconst 14721 . . . . 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)))
111, 9, 10syl2anc 696 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝) = ((♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝)))
12 simpl 474 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ)
13 1red 10247 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℝ)
146nnred 11227 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
15 prmuz2 15610 . . . . . . . . . . . . 13 (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ‘2))
164, 15syl 17 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (ℤ‘2))
17 eluz2b2 11954 . . . . . . . . . . . . 13 (𝑝 ∈ (ℤ‘2) ↔ (𝑝 ∈ ℕ ∧ 1 < 𝑝))
1817simprbi 483 . . . . . . . . . . . 12 (𝑝 ∈ (ℤ‘2) → 1 < 𝑝)
1916, 18syl 17 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝑝)
20 inss1 3976 . . . . . . . . . . . . . 14 ((0[,]𝐴) ∩ ℙ) ⊆ (0[,]𝐴)
2120, 3sseldi 3742 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (0[,]𝐴))
22 0re 10232 . . . . . . . . . . . . . 14 0 ∈ ℝ
23 elicc2 12431 . . . . . . . . . . . . . 14 ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴)))
2422, 12, 23sylancr 698 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴)))
2521, 24mpbid 222 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴))
2625simp3d 1139 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝𝐴)
2713, 14, 12, 19, 26ltletrd 10389 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝐴)
2812, 27rplogcld 24574 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ+)
2914, 19rplogcld 24574 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ+)
3028, 29rpdivcld 12082 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ+)
3130rpred 12065 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
3230rpge0d 12069 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ≤ ((log‘𝐴) / (log‘𝑝)))
33 flge0nn0 12815 . . . . . . 7 ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 0 ≤ ((log‘𝐴) / (log‘𝑝))) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0)
3431, 32, 33syl2anc 696 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0)
35 hashfz1 13328 . . . . . 6 ((⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0 → (♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) = (⌊‘((log‘𝐴) / (log‘𝑝))))
3634, 35syl 17 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) = (⌊‘((log‘𝐴) / (log‘𝑝))))
3736oveq1d 6828 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝)) = ((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝)))
3831flcld 12793 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
3938zcnd 11675 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℂ)
4039, 9mulcomd 10253 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝)) = ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
4111, 37, 403eqtrrd 2799 . . 3 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) = Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝))
4241sumeq2dv 14632 . 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))
43 chpval2 25142 . 2 (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
44 simpl 474 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
45 0red 10233 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ∈ ℝ)
46 1red 10247 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℝ)
47 0lt1 10742 . . . . . . . . 9 0 < 1
4847a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 < 1)
49 elfzuz2 12539 . . . . . . . . 9 (𝑘 ∈ (1...(⌊‘𝐴)) → (⌊‘𝐴) ∈ (ℤ‘1))
50 eluzle 11892 . . . . . . . . . . 11 ((⌊‘𝐴) ∈ (ℤ‘1) → 1 ≤ (⌊‘𝐴))
5150adantl 473 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ‘1)) → 1 ≤ (⌊‘𝐴))
52 simpl 474 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ‘1)) → 𝐴 ∈ ℝ)
53 1z 11599 . . . . . . . . . . 11 1 ∈ ℤ
54 flge 12800 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 1 ∈ ℤ) → (1 ≤ 𝐴 ↔ 1 ≤ (⌊‘𝐴)))
5552, 53, 54sylancl 697 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ‘1)) → (1 ≤ 𝐴 ↔ 1 ≤ (⌊‘𝐴)))
5651, 55mpbird 247 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ‘1)) → 1 ≤ 𝐴)
5749, 56sylan2 492 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 1 ≤ 𝐴)
5845, 46, 44, 48, 57ltletrd 10389 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 < 𝐴)
5945, 44, 58ltled 10377 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ≤ 𝐴)
60 elfznn 12563 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘𝐴)) → 𝑘 ∈ ℕ)
6160adantl 473 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝑘 ∈ ℕ)
6261nnrecred 11258 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (1 / 𝑘) ∈ ℝ)
6344, 59, 62recxpcld 24668 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (𝐴𝑐(1 / 𝑘)) ∈ ℝ)
64 chtval 25035 . . . . 5 ((𝐴𝑐(1 / 𝑘)) ∈ ℝ → (θ‘(𝐴𝑐(1 / 𝑘))) = Σ𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)(log‘𝑝))
6563, 64syl 17 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (θ‘(𝐴𝑐(1 / 𝑘))) = Σ𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)(log‘𝑝))
6665sumeq2dv 14632 . . 3 (𝐴 ∈ ℝ → Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴𝑐(1 / 𝑘))) = Σ𝑘 ∈ (1...(⌊‘𝐴))Σ𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)(log‘𝑝))
67 ppifi 25031 . . . 4 (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
68 fzfid 12966 . . . 4 (𝐴 ∈ ℝ → (1...(⌊‘𝐴)) ∈ Fin)
692sseli 3740 . . . . . . . 8 (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ ℙ)
70 elfznn 12563 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
7169, 70anim12i 591 . . . . . . 7 ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))
7271a1i 11 . . . . . 6 (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
73 0red 10233 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ∈ ℝ)
742a1i 11 . . . . . . . . . . . 12 (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ⊆ ℙ)
7574sselda 3744 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
7675, 5syl 17 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
7776nnred 11227 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
7876nngt0d 11256 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝑝)
7973, 77, 12, 78, 26ltletrd 10389 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝐴)
8079ex 449 . . . . . . 7 (𝐴 ∈ ℝ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 0 < 𝐴))
8180adantrd 485 . . . . . 6 (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 0 < 𝐴))
8272, 81jcad 556 . . . . 5 (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)))
83 inss2 3977 . . . . . . . . 9 ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ) ⊆ ℙ
8483sseli 3740 . . . . . . . 8 (𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ) → 𝑝 ∈ ℙ)
8560, 84anim12ci 592 . . . . . . 7 ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))
8685a1i 11 . . . . . 6 (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
8758ex 449 . . . . . . 7 (𝐴 ∈ ℝ → (𝑘 ∈ (1...(⌊‘𝐴)) → 0 < 𝐴))
8887adantrd 485 . . . . . 6 (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)) → 0 < 𝐴))
8986, 88jcad 556 . . . . 5 (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)))
90 elin 3939 . . . . . . . . 9 (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ))
91 simprll 821 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℙ)
9291biantrud 529 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ)))
93 0red 10233 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ∈ ℝ)
94 simpl 474 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℝ)
9591, 5syl 17 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℕ)
9695nnred 11227 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℝ)
9795nnnn0d 11543 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℕ0)
9897nn0ge0d 11546 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ 𝑝)
99 df-3an 1074 . . . . . . . . . . . . 13 ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝𝐴))
10023, 99syl6bb 276 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝𝐴)))
101100baibd 986 . . . . . . . . . . 11 (((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝𝐴))
10293, 94, 96, 98, 101syl22anc 1478 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝𝐴))
10392, 102bitr3d 270 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ) ↔ 𝑝𝐴))
10490, 103syl5bb 272 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ 𝑝𝐴))
105 simprr 813 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 < 𝐴)
10694, 105elrpd 12062 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℝ+)
107106relogcld 24568 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘𝐴) ∈ ℝ)
10891, 15syl 17 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ (ℤ‘2))
109108, 18syl 17 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 1 < 𝑝)
11096, 109rplogcld 24574 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘𝑝) ∈ ℝ+)
111107, 110rerpdivcld 12096 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
112 simprlr 822 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℕ)
113112nnzd 11673 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℤ)
114 flge 12800 . . . . . . . . . 10 ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
115111, 113, 114syl2anc 696 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
116112nnnn0d 11543 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℕ0)
11795, 116nnexpcld 13224 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝𝑘) ∈ ℕ)
118117nnrpd 12063 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝𝑘) ∈ ℝ+)
119118, 106logled 24572 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴 ↔ (log‘(𝑝𝑘)) ≤ (log‘𝐴)))
12095nnrpd 12063 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℝ+)
121 relogexp 24541 . . . . . . . . . . . 12 ((𝑝 ∈ ℝ+𝑘 ∈ ℤ) → (log‘(𝑝𝑘)) = (𝑘 · (log‘𝑝)))
122120, 113, 121syl2anc 696 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘(𝑝𝑘)) = (𝑘 · (log‘𝑝)))
123122breq1d 4814 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((log‘(𝑝𝑘)) ≤ (log‘𝐴) ↔ (𝑘 · (log‘𝑝)) ≤ (log‘𝐴)))
124112nnred 11227 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℝ)
125124, 107, 110lemuldivd 12114 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
126119, 123, 1253bitrd 294 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
127 nnuz 11916 . . . . . . . . . . 11 ℕ = (ℤ‘1)
128112, 127syl6eleq 2849 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ (ℤ‘1))
129111flcld 12793 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
130 elfz5 12527 . . . . . . . . . 10 ((𝑘 ∈ (ℤ‘1) ∧ (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
131128, 129, 130syl2anc 696 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
132115, 126, 1313bitr4rd 301 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ (𝑝𝑘) ≤ 𝐴))
133104, 132anbi12d 749 . . . . . . 7 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑝𝐴 ∧ (𝑝𝑘) ≤ 𝐴)))
13494flcld 12793 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (⌊‘𝐴) ∈ ℤ)
135 elfz5 12527 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ‘1) ∧ (⌊‘𝐴) ∈ ℤ) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ (⌊‘𝐴)))
136128, 134, 135syl2anc 696 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ (⌊‘𝐴)))
137 flge 12800 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘𝐴𝑘 ≤ (⌊‘𝐴)))
13894, 113, 137syl2anc 696 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘𝐴𝑘 ≤ (⌊‘𝐴)))
139136, 138bitr4d 271 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘𝐴))
140 elin 3939 . . . . . . . . . 10 (𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ))
14191biantrud 529 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ↔ (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ)))
142106rpge0d 12069 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ 𝐴)
143112nnrecred 11258 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (1 / 𝑘) ∈ ℝ)
14494, 142, 143recxpcld 24668 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴𝑐(1 / 𝑘)) ∈ ℝ)
145 elicc2 12431 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ ∧ (𝐴𝑐(1 / 𝑘)) ∈ ℝ) → (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝 ≤ (𝐴𝑐(1 / 𝑘)))))
146 df-3an 1074 . . . . . . . . . . . . . . 15 ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝 ≤ (𝐴𝑐(1 / 𝑘))) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (𝐴𝑐(1 / 𝑘))))
147145, 146syl6bb 276 . . . . . . . . . . . . . 14 ((0 ∈ ℝ ∧ (𝐴𝑐(1 / 𝑘)) ∈ ℝ) → (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (𝐴𝑐(1 / 𝑘)))))
148147baibd 986 . . . . . . . . . . . . 13 (((0 ∈ ℝ ∧ (𝐴𝑐(1 / 𝑘)) ∈ ℝ) ∧ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝)) → (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ↔ 𝑝 ≤ (𝐴𝑐(1 / 𝑘))))
14993, 144, 96, 98, 148syl22anc 1478 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ↔ 𝑝 ≤ (𝐴𝑐(1 / 𝑘))))
150141, 149bitr3d 270 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ) ↔ 𝑝 ≤ (𝐴𝑐(1 / 𝑘))))
15194, 142, 143cxpge0d 24669 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ (𝐴𝑐(1 / 𝑘)))
152112nnrpd 12063 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℝ+)
15396, 98, 144, 151, 152cxple2d 24672 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ≤ (𝐴𝑐(1 / 𝑘)) ↔ (𝑝𝑐𝑘) ≤ ((𝐴𝑐(1 / 𝑘))↑𝑐𝑘)))
15495nncnd 11228 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℂ)
155 cxpexp 24613 . . . . . . . . . . . . 13 ((𝑝 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑝𝑐𝑘) = (𝑝𝑘))
156154, 116, 155syl2anc 696 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝𝑐𝑘) = (𝑝𝑘))
157112nncnd 11228 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℂ)
158112nnne0d 11257 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ≠ 0)
159157, 158recid2d 10989 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((1 / 𝑘) · 𝑘) = 1)
160159oveq2d 6829 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴𝑐((1 / 𝑘) · 𝑘)) = (𝐴𝑐1))
161106, 143, 157cxpmuld 24679 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴𝑐((1 / 𝑘) · 𝑘)) = ((𝐴𝑐(1 / 𝑘))↑𝑐𝑘))
16294recnd 10260 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℂ)
163162cxp1d 24651 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴𝑐1) = 𝐴)
164160, 161, 1633eqtr3d 2802 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝐴𝑐(1 / 𝑘))↑𝑐𝑘) = 𝐴)
165156, 164breq12d 4817 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑐𝑘) ≤ ((𝐴𝑐(1 / 𝑘))↑𝑐𝑘) ↔ (𝑝𝑘) ≤ 𝐴))
166150, 153, 1653bitrd 294 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ) ↔ (𝑝𝑘) ≤ 𝐴))
167140, 166syl5bb 272 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝𝑘) ≤ 𝐴))
168139, 167anbi12d 749 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)) ↔ (𝑘𝐴 ∧ (𝑝𝑘) ≤ 𝐴)))
169117nnred 11227 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝𝑘) ∈ ℝ)
170 bernneq3 13186 . . . . . . . . . . . 12 ((𝑝 ∈ (ℤ‘2) ∧ 𝑘 ∈ ℕ0) → 𝑘 < (𝑝𝑘))
171108, 116, 170syl2anc 696 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 < (𝑝𝑘))
172124, 169, 171ltled 10377 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ≤ (𝑝𝑘))
173 letr 10323 . . . . . . . . . . 11 ((𝑘 ∈ ℝ ∧ (𝑝𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑘 ≤ (𝑝𝑘) ∧ (𝑝𝑘) ≤ 𝐴) → 𝑘𝐴))
174124, 169, 94, 173syl3anc 1477 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 ≤ (𝑝𝑘) ∧ (𝑝𝑘) ≤ 𝐴) → 𝑘𝐴))
175172, 174mpand 713 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴𝑘𝐴))
176175pm4.71rd 670 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴 ↔ (𝑘𝐴 ∧ (𝑝𝑘) ≤ 𝐴)))
177154exp1d 13197 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑1) = 𝑝)
17895nnge1d 11255 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 1 ≤ 𝑝)
17996, 178, 128leexp2ad 13235 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑1) ≤ (𝑝𝑘))
180177, 179eqbrtrrd 4828 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ≤ (𝑝𝑘))
181 letr 10323 . . . . . . . . . . 11 ((𝑝 ∈ ℝ ∧ (𝑝𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑝 ≤ (𝑝𝑘) ∧ (𝑝𝑘) ≤ 𝐴) → 𝑝𝐴))
18296, 169, 94, 181syl3anc 1477 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ≤ (𝑝𝑘) ∧ (𝑝𝑘) ≤ 𝐴) → 𝑝𝐴))
183180, 182mpand 713 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴𝑝𝐴))
184183pm4.71rd 670 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴 ↔ (𝑝𝐴 ∧ (𝑝𝑘) ≤ 𝐴)))
185168, 176, 1843bitr2rd 297 . . . . . . 7 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝐴 ∧ (𝑝𝑘) ≤ 𝐴) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ))))
186133, 185bitrd 268 . . . . . 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))))
187186ex 449 . . . . 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)))))
18882, 89, 187pm5.21ndd 368 . . . 4 (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ))))
1899adantrr 755 . . . 4 ((𝐴 ∈ ℝ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘𝑝) ∈ ℂ)
19067, 68, 1, 188, 189fsumcom2 14704 . . 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))
19166, 190eqtr4d 2797 . 2 (𝐴 ∈ ℝ → Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴𝑐(1 / 𝑘))) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝))
19242, 43, 1913eqtr4d 2804 1 (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴𝑐(1 / 𝑘))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  cin 3714  wss 3715   class class class wbr 4804  cfv 6049  (class class class)co 6813  Fincfn 8121  cc 10126  cr 10127  0cc0 10128  1c1 10129   · cmul 10133   < clt 10266  cle 10267   / cdiv 10876  cn 11212  2c2 11262  0cn0 11484  cz 11569  cuz 11879  +crp 12025  [,]cicc 12371  ...cfz 12519  cfl 12785  cexp 13054  chash 13311  Σcsu 14615  cprime 15587  logclog 24500  𝑐ccxp 24501  θccht 25016  ψcchp 25018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206  ax-addf 10207  ax-mulf 10208
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-fi 8482  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-q 11982  df-rp 12026  df-xneg 12139  df-xadd 12140  df-xmul 12141  df-ioo 12372  df-ioc 12373  df-ico 12374  df-icc 12375  df-fz 12520  df-fzo 12660  df-fl 12787  df-mod 12863  df-seq 12996  df-exp 13055  df-fac 13255  df-bc 13284  df-hash 13312  df-shft 14006  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-limsup 14401  df-clim 14418  df-rlim 14419  df-sum 14616  df-ef 14997  df-sin 14999  df-cos 15000  df-pi 15002  df-dvds 15183  df-gcd 15419  df-prm 15588  df-pc 15744  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-starv 16158  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-unif 16167  df-hom 16168  df-cco 16169  df-rest 16285  df-topn 16286  df-0g 16304  df-gsum 16305  df-topgen 16306  df-pt 16307  df-prds 16310  df-xrs 16364  df-qtop 16369  df-imas 16370  df-xps 16372  df-mre 16448  df-mrc 16449  df-acs 16451  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-submnd 17537  df-mulg 17742  df-cntz 17950  df-cmn 18395  df-psmet 19940  df-xmet 19941  df-met 19942  df-bl 19943  df-mopn 19944  df-fbas 19945  df-fg 19946  df-cnfld 19949  df-top 20901  df-topon 20918  df-topsp 20939  df-bases 20952  df-cld 21025  df-ntr 21026  df-cls 21027  df-nei 21104  df-lp 21142  df-perf 21143  df-cn 21233  df-cnp 21234  df-haus 21321  df-tx 21567  df-hmeo 21760  df-fil 21851  df-fm 21943  df-flim 21944  df-flf 21945  df-xms 22326  df-ms 22327  df-tms 22328  df-cncf 22882  df-limc 23829  df-dv 23830  df-log 24502  df-cxp 24503  df-cht 25022  df-vma 25023  df-chp 25024
This theorem is referenced by: (None)
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