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Theorem chpchtsum 24744
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 12634 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin)
2 inss2 3796 . . . . . . . . . 10 ((0[,]𝐴) ∩ ℙ) ⊆ ℙ
3 simpr 476 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
42, 3sseldi 3566 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
5 prmnn 15226 . . . . . . . . 9 (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
64, 5syl 17 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
76nnrpd 11746 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ+)
87relogcld 24173 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
98recnd 9947 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
10 fsumconst 14364 . . . . 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 691 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝) = ((#‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝)))
12 simpl 472 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ)
13 1red 9934 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℝ)
146nnred 10912 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
15 prmuz2 15246 . . . . . . . . . . . . 13 (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ‘2))
164, 15syl 17 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (ℤ‘2))
17 eluz2b2 11637 . . . . . . . . . . . . 13 (𝑝 ∈ (ℤ‘2) ↔ (𝑝 ∈ ℕ ∧ 1 < 𝑝))
1817simprbi 479 . . . . . . . . . . . 12 (𝑝 ∈ (ℤ‘2) → 1 < 𝑝)
1916, 18syl 17 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝑝)
20 inss1 3795 . . . . . . . . . . . . . 14 ((0[,]𝐴) ∩ ℙ) ⊆ (0[,]𝐴)
2120, 3sseldi 3566 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (0[,]𝐴))
22 0re 9919 . . . . . . . . . . . . . 14 0 ∈ ℝ
23 elicc2 12109 . . . . . . . . . . . . . 14 ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴)))
2422, 12, 23sylancr 694 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴)))
2521, 24mpbid 221 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴))
2625simp3d 1068 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝𝐴)
2713, 14, 12, 19, 26ltletrd 10076 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝐴)
2812, 27rplogcld 24179 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ+)
2914, 19rplogcld 24179 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ+)
3028, 29rpdivcld 11765 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ+)
3130rpred 11748 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
3230rpge0d 11752 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ≤ ((log‘𝐴) / (log‘𝑝)))
33 flge0nn0 12483 . . . . . . 7 ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 0 ≤ ((log‘𝐴) / (log‘𝑝))) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0)
3431, 32, 33syl2anc 691 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0)
35 hashfz1 12996 . . . . . 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 6564 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((#‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝)) = ((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝)))
3831flcld 12461 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
3938zcnd 11359 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℂ)
4039, 9mulcomd 9940 . . . 4 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝)) = ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
4111, 37, 403eqtrrd 2649 . . 3 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) = Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝))
4241sumeq2dv 14281 . 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 24743 . 2 (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
44 simpl 472 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
45 0red 9920 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ∈ ℝ)
46 1red 9934 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℝ)
47 0lt1 10429 . . . . . . . . 9 0 < 1
4847a1i 11 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 < 1)
49 elfzuz2 12217 . . . . . . . . 9 (𝑘 ∈ (1...(⌊‘𝐴)) → (⌊‘𝐴) ∈ (ℤ‘1))
50 eluzle 11576 . . . . . . . . . . 11 ((⌊‘𝐴) ∈ (ℤ‘1) → 1 ≤ (⌊‘𝐴))
5150adantl 481 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ‘1)) → 1 ≤ (⌊‘𝐴))
52 simpl 472 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ‘1)) → 𝐴 ∈ ℝ)
53 1z 11284 . . . . . . . . . . 11 1 ∈ ℤ
54 flge 12468 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 1 ∈ ℤ) → (1 ≤ 𝐴 ↔ 1 ≤ (⌊‘𝐴)))
5552, 53, 54sylancl 693 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ‘1)) → (1 ≤ 𝐴 ↔ 1 ≤ (⌊‘𝐴)))
5651, 55mpbird 246 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ‘1)) → 1 ≤ 𝐴)
5749, 56sylan2 490 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 1 ≤ 𝐴)
5845, 46, 44, 48, 57ltletrd 10076 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 < 𝐴)
5945, 44, 58ltled 10064 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ≤ 𝐴)
60 elfznn 12241 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘𝐴)) → 𝑘 ∈ ℕ)
6160adantl 481 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝑘 ∈ ℕ)
6261nnrecred 10943 . . . . . 6 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (1 / 𝑘) ∈ ℝ)
6344, 59, 62recxpcld 24269 . . . . 5 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (𝐴𝑐(1 / 𝑘)) ∈ ℝ)
64 chtval 24636 . . . . 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 14281 . . 3 (𝐴 ∈ ℝ → Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴𝑐(1 / 𝑘))) = Σ𝑘 ∈ (1...(⌊‘𝐴))Σ𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)(log‘𝑝))
67 ppifi 24632 . . . 4 (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
68 fzfid 12634 . . . 4 (𝐴 ∈ ℝ → (1...(⌊‘𝐴)) ∈ Fin)
692sseli 3564 . . . . . . . 8 (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ ℙ)
70 elfznn 12241 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
7169, 70anim12i 588 . . . . . . 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 9920 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ∈ ℝ)
742a1i 11 . . . . . . . . . . . 12 (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ⊆ ℙ)
7574sselda 3568 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
7675, 5syl 17 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
7776nnred 10912 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
7876nngt0d 10941 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝑝)
7973, 77, 12, 78, 26ltletrd 10076 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝐴)
8079ex 449 . . . . . . 7 (𝐴 ∈ ℝ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 0 < 𝐴))
8180adantrd 483 . . . . . 6 (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 0 < 𝐴))
8272, 81jcad 554 . . . . 5 (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)))
83 inss2 3796 . . . . . . . . 9 ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ) ⊆ ℙ
8483sseli 3564 . . . . . . . 8 (𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ) → 𝑝 ∈ ℙ)
8560, 84anim12ci 589 . . . . . . 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 483 . . . . . 6 (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)) → 0 < 𝐴))
8986, 88jcad 554 . . . . 5 (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)))
90 elin 3758 . . . . . . . . 9 (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ))
91 simprll 798 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℙ)
9291biantrud 527 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ)))
93 0red 9920 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ∈ ℝ)
94 simpl 472 . . . . . . . . . . 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 10912 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℝ)
9795nnnn0d 11228 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℕ0)
9897nn0ge0d 11231 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ 𝑝)
99 df-3an 1033 . . . . . . . . . . . . 13 ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝𝐴))
10023, 99syl6bb 275 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝𝐴)))
101100baibd 946 . . . . . . . . . . 11 (((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝𝐴))
10293, 94, 96, 98, 101syl22anc 1319 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝𝐴))
10392, 102bitr3d 269 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ) ↔ 𝑝𝐴))
10490, 103syl5bb 271 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ 𝑝𝐴))
105 simprr 792 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 < 𝐴)
10694, 105elrpd 11745 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℝ+)
107106relogcld 24173 . . . . . . . . . . 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 24179 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘𝑝) ∈ ℝ+)
111107, 110rerpdivcld 11779 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
112 simprlr 799 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℕ)
113112nnzd 11357 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℤ)
114 flge 12468 . . . . . . . . . 10 ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
115111, 113, 114syl2anc 691 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
116112nnnn0d 11228 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℕ0)
11795, 116nnexpcld 12892 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝𝑘) ∈ ℕ)
118117nnrpd 11746 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝𝑘) ∈ ℝ+)
119118, 106logled 24177 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴 ↔ (log‘(𝑝𝑘)) ≤ (log‘𝐴)))
12095nnrpd 11746 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℝ+)
121 relogexp 24146 . . . . . . . . . . . 12 ((𝑝 ∈ ℝ+𝑘 ∈ ℤ) → (log‘(𝑝𝑘)) = (𝑘 · (log‘𝑝)))
122120, 113, 121syl2anc 691 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘(𝑝𝑘)) = (𝑘 · (log‘𝑝)))
123122breq1d 4593 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((log‘(𝑝𝑘)) ≤ (log‘𝐴) ↔ (𝑘 · (log‘𝑝)) ≤ (log‘𝐴)))
124112nnred 10912 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℝ)
125124, 107, 110lemuldivd 11797 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
126119, 123, 1253bitrd 293 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
127 nnuz 11599 . . . . . . . . . . 11 ℕ = (ℤ‘1)
128112, 127syl6eleq 2698 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ (ℤ‘1))
129111flcld 12461 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
130 elfz5 12205 . . . . . . . . . 10 ((𝑘 ∈ (ℤ‘1) ∧ (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
131128, 129, 130syl2anc 691 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
132115, 126, 1313bitr4rd 300 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ (𝑝𝑘) ≤ 𝐴))
133104, 132anbi12d 743 . . . . . . 7 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑝𝐴 ∧ (𝑝𝑘) ≤ 𝐴)))
13494flcld 12461 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (⌊‘𝐴) ∈ ℤ)
135 elfz5 12205 . . . . . . . . . . 11 ((𝑘 ∈ (ℤ‘1) ∧ (⌊‘𝐴) ∈ ℤ) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ (⌊‘𝐴)))
136128, 134, 135syl2anc 691 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ (⌊‘𝐴)))
137 flge 12468 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘𝐴𝑘 ≤ (⌊‘𝐴)))
13894, 113, 137syl2anc 691 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘𝐴𝑘 ≤ (⌊‘𝐴)))
139136, 138bitr4d 270 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘𝐴))
140 elin 3758 . . . . . . . . . 10 (𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ))
14191biantrud 527 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ↔ (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ)))
142106rpge0d 11752 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ 𝐴)
143112nnrecred 10943 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (1 / 𝑘) ∈ ℝ)
14494, 142, 143recxpcld 24269 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴𝑐(1 / 𝑘)) ∈ ℝ)
145 elicc2 12109 . . . . . . . . . . . . . . 15 ((0 ∈ ℝ ∧ (𝐴𝑐(1 / 𝑘)) ∈ ℝ) → (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝 ≤ (𝐴𝑐(1 / 𝑘)))))
146 df-3an 1033 . . . . . . . . . . . . . . 15 ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝 ≤ (𝐴𝑐(1 / 𝑘))) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (𝐴𝑐(1 / 𝑘))))
147145, 146syl6bb 275 . . . . . . . . . . . . . 14 ((0 ∈ ℝ ∧ (𝐴𝑐(1 / 𝑘)) ∈ ℝ) → (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (𝐴𝑐(1 / 𝑘)))))
148147baibd 946 . . . . . . . . . . . . 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 1319 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ↔ 𝑝 ≤ (𝐴𝑐(1 / 𝑘))))
150141, 149bitr3d 269 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ) ↔ 𝑝 ≤ (𝐴𝑐(1 / 𝑘))))
15194, 142, 143cxpge0d 24270 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ (𝐴𝑐(1 / 𝑘)))
152112nnrpd 11746 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℝ+)
15396, 98, 144, 151, 152cxple2d 24273 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ≤ (𝐴𝑐(1 / 𝑘)) ↔ (𝑝𝑐𝑘) ≤ ((𝐴𝑐(1 / 𝑘))↑𝑐𝑘)))
15495nncnd 10913 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℂ)
155 cxpexp 24214 . . . . . . . . . . . . 13 ((𝑝 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑝𝑐𝑘) = (𝑝𝑘))
156154, 116, 155syl2anc 691 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝𝑐𝑘) = (𝑝𝑘))
157112nncnd 10913 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℂ)
158112nnne0d 10942 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ≠ 0)
159157, 158recid2d 10676 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((1 / 𝑘) · 𝑘) = 1)
160159oveq2d 6565 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴𝑐((1 / 𝑘) · 𝑘)) = (𝐴𝑐1))
161106, 143, 157cxpmuld 24280 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴𝑐((1 / 𝑘) · 𝑘)) = ((𝐴𝑐(1 / 𝑘))↑𝑐𝑘))
16294recnd 9947 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℂ)
163162cxp1d 24252 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴𝑐1) = 𝐴)
164160, 161, 1633eqtr3d 2652 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝐴𝑐(1 / 𝑘))↑𝑐𝑘) = 𝐴)
165156, 164breq12d 4596 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑐𝑘) ≤ ((𝐴𝑐(1 / 𝑘))↑𝑐𝑘) ↔ (𝑝𝑘) ≤ 𝐴))
166150, 153, 1653bitrd 293 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,](𝐴𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ) ↔ (𝑝𝑘) ≤ 𝐴))
167140, 166syl5bb 271 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝𝑘) ≤ 𝐴))
168139, 167anbi12d 743 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ)) ↔ (𝑘𝐴 ∧ (𝑝𝑘) ≤ 𝐴)))
169117nnred 10912 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝𝑘) ∈ ℝ)
170 bernneq3 12854 . . . . . . . . . . . 12 ((𝑝 ∈ (ℤ‘2) ∧ 𝑘 ∈ ℕ0) → 𝑘 < (𝑝𝑘))
171108, 116, 170syl2anc 691 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 < (𝑝𝑘))
172124, 169, 171ltled 10064 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ≤ (𝑝𝑘))
173 letr 10010 . . . . . . . . . . 11 ((𝑘 ∈ ℝ ∧ (𝑝𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑘 ≤ (𝑝𝑘) ∧ (𝑝𝑘) ≤ 𝐴) → 𝑘𝐴))
174124, 169, 94, 173syl3anc 1318 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 ≤ (𝑝𝑘) ∧ (𝑝𝑘) ≤ 𝐴) → 𝑘𝐴))
175172, 174mpand 707 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴𝑘𝐴))
176175pm4.71rd 665 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴 ↔ (𝑘𝐴 ∧ (𝑝𝑘) ≤ 𝐴)))
177154exp1d 12865 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑1) = 𝑝)
17895nnge1d 10940 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 1 ≤ 𝑝)
17996, 178, 128leexp2ad 12903 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑1) ≤ (𝑝𝑘))
180177, 179eqbrtrrd 4607 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ≤ (𝑝𝑘))
181 letr 10010 . . . . . . . . . . 11 ((𝑝 ∈ ℝ ∧ (𝑝𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑝 ≤ (𝑝𝑘) ∧ (𝑝𝑘) ≤ 𝐴) → 𝑝𝐴))
18296, 169, 94, 181syl3anc 1318 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ≤ (𝑝𝑘) ∧ (𝑝𝑘) ≤ 𝐴) → 𝑝𝐴))
183180, 182mpand 707 . . . . . . . . 9 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴𝑝𝐴))
184183pm4.71rd 665 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝑘) ≤ 𝐴 ↔ (𝑝𝐴 ∧ (𝑝𝑘) ≤ 𝐴)))
185168, 176, 1843bitr2rd 296 . . . . . . 7 ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝𝐴 ∧ (𝑝𝑘) ≤ 𝐴) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴𝑐(1 / 𝑘))) ∩ ℙ))))
186133, 185bitrd 267 . . . . . 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 749 . . . 4 ((𝐴 ∈ ℝ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘𝑝) ∈ ℂ)
19067, 68, 1, 188, 189fsumcom2 14347 . . 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 2647 . 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 2654 1 (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴𝑐(1 / 𝑘))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  cin 3539  wss 3540   class class class wbr 4583  cfv 5804  (class class class)co 6549  Fincfn 7841  cc 9813  cr 9814  0cc0 9815  1c1 9816   · cmul 9820   < clt 9953  cle 9954   / cdiv 10563  cn 10897  2c2 10947  0cn0 11169  cz 11254  cuz 11563  +crp 11708  [,]cicc 12049  ...cfz 12197  cfl 12453  cexp 12722  #chash 12979  Σcsu 14264  cprime 15223  logclog 24105  𝑐ccxp 24106  θccht 24617  ψcchp 24619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637  df-sin 14639  df-cos 14640  df-pi 14642  df-dvds 14822  df-gcd 15055  df-prm 15224  df-pc 15380  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-haus 20929  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-limc 23436  df-dv 23437  df-log 24107  df-cxp 24108  df-cht 24623  df-vma 24624  df-chp 24625
This theorem is referenced by: (None)
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