Theorem chpchtsum 27285

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‘(𝐴↑𝑐(1 / 𝑘))))

Distinct variable group:   𝐴,𝑘

Proof of Theorem chpchtsum

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 13988	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin)
2		simpr 488	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
3	2	elin2d 4159	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
4		prmnn 16710	. . . . . . . . 9 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
5	3, 4	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
6	5	nnrpd 13037	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ+)
7	6	relogcld 26690	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
8	7	recnd 11212	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
9		fsumconst 15819	. . . . 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)))
10	1, 8, 9	syl2anc 593	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝) = ((♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝)))
11		simpl 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ)
12		1red 11184	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℝ)
13	5	nnred 12227	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
14		prmuz2 16732	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ≥‘2))
15	3, 14	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (ℤ≥‘2))
16		eluz2gt1 12923	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ℤ≥‘2) → 1 < 𝑝)
17	15, 16	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝑝)
18	2	elin1d 4158	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (0[,]𝐴))
19		0re 11185	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
20		elicc2 13417	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
21	19, 11, 20	sylancr 596	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
22	18, 21	mpbid 234	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴))
23	22	simp3d 1158	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ 𝐴)
24	12, 13, 11, 17, 23	ltletrd 11345	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝐴)
25	11, 24	rplogcld 26696	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ+)
26	13, 17	rplogcld 26696	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ+)
27	25, 26	rpdivcld 13056	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ+)
28	27	rpred 13039	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
29	27	rpge0d 13043	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ≤ ((log‘𝐴) / (log‘𝑝)))
30		flge0nn0 13832	. . . . . . 7 ⊢ ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 0 ≤ ((log‘𝐴) / (log‘𝑝))) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0)
31	28, 29, 30	syl2anc 593	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0)
32		hashfz1 14361	. . . . . 6 ⊢ ((⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0 → (♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) = (⌊‘((log‘𝐴) / (log‘𝑝))))
33	31, 32	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) = (⌊‘((log‘𝐴) / (log‘𝑝))))
34	33	oveq1d 7413	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝)) = ((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝)))
35	28	flcld 13810	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
36	35	zcnd 12680	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℂ)
37	36, 8	mulcomd 11205	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝)) = ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
38	10, 34, 37	3eqtrrd 2804	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) = Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝))
39	38	sumeq2dv 15731	. 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 27284	. 2 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
41		simpl 486	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
42		0red 11186	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ∈ ℝ)
43		1red 11184	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℝ)
44		0lt1 11711	. . . . . . . . 9 ⊢ 0 < 1
45	44	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 < 1)
46		elfzuz2 13536	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...(⌊‘𝐴)) → (⌊‘𝐴) ∈ (ℤ≥‘1))
47		eluzle 12854	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ (ℤ≥‘1) → 1 ≤ (⌊‘𝐴))
48	47	adantl 485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ≥‘1)) → 1 ≤ (⌊‘𝐴))
49		simpl 486	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ≥‘1)) → 𝐴 ∈ ℝ)
50		1z 12603	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
51		flge 13817	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℤ) → (1 ≤ 𝐴 ↔ 1 ≤ (⌊‘𝐴)))
52	49, 50, 51	sylancl 595	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ≥‘1)) → (1 ≤ 𝐴 ↔ 1 ≤ (⌊‘𝐴)))
53	48, 52	mpbird 259	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ≥‘1)) → 1 ≤ 𝐴)
54	46, 53	sylan2 602	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 1 ≤ 𝐴)
55	42, 43, 41, 45, 54	ltletrd 11345	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 < 𝐴)
56	42, 41, 55	ltled 11333	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ≤ 𝐴)
57		elfznn 13560	. . . . . . . 8 ⊢ (𝑘 ∈ (1...(⌊‘𝐴)) → 𝑘 ∈ ℕ)
58	57	adantl 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝑘 ∈ ℕ)
59	58	nnrecred 12266	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (1 / 𝑘) ∈ ℝ)
60	41, 56, 59	recxpcld 26790	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (𝐴↑𝑐(1 / 𝑘)) ∈ ℝ)
61		chtval 27176	. . . . 5 ⊢ ((𝐴↑𝑐(1 / 𝑘)) ∈ ℝ → (θ‘(𝐴↑𝑐(1 / 𝑘))) = Σ𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)(log‘𝑝))
62	60, 61	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (θ‘(𝐴↑𝑐(1 / 𝑘))) = Σ𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)(log‘𝑝))
63	62	sumeq2dv 15731	. . 3 ⊢ (𝐴 ∈ ℝ → Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴↑𝑐(1 / 𝑘))) = Σ𝑘 ∈ (1...(⌊‘𝐴))Σ𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)(log‘𝑝))
64		ppifi 27172	. . . 4 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
65		fzfid 13988	. . . 4 ⊢ (𝐴 ∈ ℝ → (1...(⌊‘𝐴)) ∈ Fin)
66		elinel2 4156	. . . . . . . 8 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ ℙ)
67		elfznn 13560	. . . . . . . 8 ⊢ (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
68	66, 67	anim12i 622	. . . . . . 7 ⊢ ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))
69	68	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
70		0red 11186	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ∈ ℝ)
71		inss2 4191	. . . . . . . . . . . . 13 ⊢ ((0[,]𝐴) ∩ ℙ) ⊆ ℙ
72	71	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ⊆ ℙ)
73	72	sselda 3938	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
74	73, 4	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
75	74	nnred 12227	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
76	74	nngt0d 12264	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝑝)
77	70, 75, 11, 76, 23	ltletrd 11345	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝐴)
78	77	ex 416	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 0 < 𝐴))
79	78	adantrd 495	. . . . . 6 ⊢ (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 0 < 𝐴))
80	69, 79	jcad 520	. . . . 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[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ) → 𝑝 ∈ ℙ)
82	57, 81	anim12ci 623	. . . . . . 7 ⊢ ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))
83	82	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
84	55	ex 416	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝑘 ∈ (1...(⌊‘𝐴)) → 0 < 𝐴))
85	84	adantrd 495	. . . . . 6 ⊢ (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)) → 0 < 𝐴))
86	83, 85	jcad 520	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)))
87		elin 3922	. . . . . . . . 9 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ))
88		simprll 788	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℙ)
89	88	biantrud 539	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ)))
90		0red 11186	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ∈ ℝ)
91		simpl 486	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℝ)
92	88, 4	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℕ)
93	92	nnred 12227	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℝ)
94	92	nnnn0d 12544	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℕ0)
95	94	nn0ge0d 12547	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ 𝑝)
96		df-3an 1101	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ 𝐴))
97	20, 96	bitrdi 289	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ 𝐴)))
98	97	baibd 547	. . . . . . . . . . 11 ⊢ (((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝 ≤ 𝐴))
99	90, 91, 93, 95, 98	syl22anc 849	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝 ≤ 𝐴))
100	89, 99	bitr3d 283	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ) ↔ 𝑝 ≤ 𝐴))
101	87, 100	bitrid 285	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ 𝑝 ≤ 𝐴))
102		simprr 782	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 < 𝐴)
103	91, 102	elrpd 13036	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℝ+)
104	103	relogcld 26690	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘𝐴) ∈ ℝ)
105	88, 14	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ (ℤ≥‘2))
106	105, 16	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 1 < 𝑝)
107	93, 106	rplogcld 26696	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘𝑝) ∈ ℝ+)
108	104, 107	rerpdivcld 13070	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
109		simprlr 789	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℕ)
110	109	nnzd 12596	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℤ)
111		flge 13817	. . . . . . . . . 10 ⊢ ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
112	108, 110, 111	syl2anc 593	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
113	109	nnnn0d 12544	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℕ0)
114	92, 113	nnexpcld 14260	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑𝑘) ∈ ℕ)
115	114	nnrpd 13037	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑𝑘) ∈ ℝ+)
116	115, 103	logled 26694	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (log‘(𝑝↑𝑘)) ≤ (log‘𝐴)))
117	92	nnrpd 13037	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℝ+)
118		relogexp 26663	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℝ+ ∧ 𝑘 ∈ ℤ) → (log‘(𝑝↑𝑘)) = (𝑘 · (log‘𝑝)))
119	117, 110, 118	syl2anc 593	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘(𝑝↑𝑘)) = (𝑘 · (log‘𝑝)))
120	119	breq1d 5112	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((log‘(𝑝↑𝑘)) ≤ (log‘𝐴) ↔ (𝑘 · (log‘𝑝)) ≤ (log‘𝐴)))
121	109	nnred 12227	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℝ)
122	121, 104, 107	lemuldivd 13088	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
123	116, 120, 122	3bitrd 307	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
124		nnuz 12880	. . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1)
125	109, 124	eleqtrdi 2874	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ (ℤ≥‘1))
126	108	flcld 13810	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
127		elfz5 13523	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘1) ∧ (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
128	125, 126, 127	syl2anc 593	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
129	112, 123, 128	3bitr4rd 314	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ (𝑝↑𝑘) ≤ 𝐴))
130	101, 129	anbi12d 641	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑝 ≤ 𝐴 ∧ (𝑝↑𝑘) ≤ 𝐴)))
131	91	flcld 13810	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (⌊‘𝐴) ∈ ℤ)
132		elfz5 13523	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘1) ∧ (⌊‘𝐴) ∈ ℤ) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ (⌊‘𝐴)))
133	125, 131, 132	syl2anc 593	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ (⌊‘𝐴)))
134		flge 13817	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘 ≤ 𝐴 ↔ 𝑘 ≤ (⌊‘𝐴)))
135	91, 110, 134	syl2anc 593	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ≤ 𝐴 ↔ 𝑘 ≤ (⌊‘𝐴)))
136	133, 135	bitr4d 284	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ 𝐴))
137		elin 3922	. . . . . . . . . 10 ⊢ (𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ))
138	88	biantrud 539	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ↔ (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ)))
139	103	rpge0d 13043	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ 𝐴)
140	109	nnrecred 12266	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (1 / 𝑘) ∈ ℝ)
141	91, 139, 140	recxpcld 26790	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴↑𝑐(1 / 𝑘)) ∈ ℝ)
142		elicc2 13417	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ (𝐴↑𝑐(1 / 𝑘)) ∈ ℝ) → (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘)))))
143		df-3an 1101	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘))) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘))))
144	142, 143	bitrdi 289	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ (𝐴↑𝑐(1 / 𝑘)) ∈ ℝ) → (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘)))))
145	144	baibd 547	. . . . . . . . . . . . 13 ⊢ (((0 ∈ ℝ ∧ (𝐴↑𝑐(1 / 𝑘)) ∈ ℝ) ∧ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝)) → (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ↔ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘))))
146	90, 141, 93, 95, 145	syl22anc 849	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ↔ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘))))
147	138, 146	bitr3d 283	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ) ↔ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘))))
148	91, 139, 140	cxpge0d 26791	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ (𝐴↑𝑐(1 / 𝑘)))
149	109	nnrpd 13037	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℝ+)
150	93, 95, 141, 148, 149	cxple2d 26794	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ≤ (𝐴↑𝑐(1 / 𝑘)) ↔ (𝑝↑𝑐𝑘) ≤ ((𝐴↑𝑐(1 / 𝑘))↑𝑐𝑘)))
151	92	nncnd 12228	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℂ)
152		cxpexp 26735	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑝↑𝑐𝑘) = (𝑝↑𝑘))
153	151, 113, 152	syl2anc 593	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑𝑐𝑘) = (𝑝↑𝑘))
154	109	nncnd 12228	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℂ)
155	109	nnne0d 12265	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ≠ 0)
156	154, 155	recid2d 11965	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((1 / 𝑘) · 𝑘) = 1)
157	156	oveq2d 7414	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴↑𝑐((1 / 𝑘) · 𝑘)) = (𝐴↑𝑐1))
158	103, 140, 154	cxpmuld 26804	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴↑𝑐((1 / 𝑘) · 𝑘)) = ((𝐴↑𝑐(1 / 𝑘))↑𝑐𝑘))
159	91	recnd 11212	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℂ)
160	159	cxp1d 26773	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴↑𝑐1) = 𝐴)
161	157, 158, 160	3eqtr3d 2807	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝐴↑𝑐(1 / 𝑘))↑𝑐𝑘) = 𝐴)
162	153, 161	breq12d 5115	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑐𝑘) ≤ ((𝐴↑𝑐(1 / 𝑘))↑𝑐𝑘) ↔ (𝑝↑𝑘) ≤ 𝐴))
163	147, 150, 162	3bitrd 307	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ) ↔ (𝑝↑𝑘) ≤ 𝐴))
164	137, 163	bitrid 285	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝↑𝑘) ≤ 𝐴))
165	136, 164	anbi12d 641	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)) ↔ (𝑘 ≤ 𝐴 ∧ (𝑝↑𝑘) ≤ 𝐴)))
166	114	nnred 12227	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑𝑘) ∈ ℝ)
167		bernneq3 14246	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ (ℤ≥‘2) ∧ 𝑘 ∈ ℕ0) → 𝑘 < (𝑝↑𝑘))
168	105, 113, 167	syl2anc 593	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 < (𝑝↑𝑘))
169	121, 166, 168	ltled 11333	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ≤ (𝑝↑𝑘))
170		letr 11279	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℝ ∧ (𝑝↑𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑘 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑘 ≤ 𝐴))
171	121, 166, 91, 170	syl3anc 1392	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑘 ≤ 𝐴))
172	169, 171	mpand 705	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 → 𝑘 ≤ 𝐴))
173	172	pm4.71rd 570	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑘 ≤ 𝐴 ∧ (𝑝↑𝑘) ≤ 𝐴)))
174	151	exp1d 14156	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑1) = 𝑝)
175	92	nnge1d 12263	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 1 ≤ 𝑝)
176	93, 175, 125	leexp2ad 14269	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑1) ≤ (𝑝↑𝑘))
177	174, 176	eqbrtrrd 5126	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ≤ (𝑝↑𝑘))
178		letr 11279	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℝ ∧ (𝑝↑𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑝 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑝 ≤ 𝐴))
179	93, 166, 91, 178	syl3anc 1392	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑝 ≤ 𝐴))
180	177, 179	mpand 705	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 → 𝑝 ≤ 𝐴))
181	180	pm4.71rd 570	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑝 ≤ 𝐴 ∧ (𝑝↑𝑘) ≤ 𝐴)))
182	165, 173, 181	3bitr2rd 310	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ≤ 𝐴 ∧ (𝑝↑𝑘) ≤ 𝐴) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ))))
183	130, 182	bitrd 281	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ))))
184	183	ex 416	. . . . 5 ⊢ (𝐴 ∈ ℝ → (((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)))))
185	80, 86, 184	pm5.21ndd 381	. . . 4 ⊢ (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ))))
186	8	adantrr 727	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘𝑝) ∈ ℂ)
187	64, 65, 1, 185, 186	fsumcom2 15803	. . 3 ⊢ (𝐴 ∈ ℝ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝) = Σ𝑘 ∈ (1...(⌊‘𝐴))Σ𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)(log‘𝑝))
188	63, 187	eqtr4d 2802	. 2 ⊢ (𝐴 ∈ ℝ → Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴↑𝑐(1 / 𝑘))) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝))
189	39, 40, 188	3eqtr4d 2809	1 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴↑𝑐(1 / 𝑘))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144   ∩ cin 3905   ⊆ wss 3906   class class class wbr 5102  ‘cfv 6523  (class class class)co 7398  Fincfn 8929  ℂcc 11073  ℝcr 11074  0cc0 11075  1c1 11076   · cmul 11080   < clt 11218   ≤ cle 11219   / cdiv 11846  ℕcn 12212  2c2 12274  ℕ₀cn0 12483  ℤcz 12570  ℤ_≥cuz 12841  ℝ⁺crp 12995  [,]cicc 13354  ...cfz 13514  ⌊cfl 13802  ↑cexp 14076  ♯chash 14345  Σcsu 15715  ℙcprime 16707  logclog 26621  ↑_𝑐ccxp 26622  θccht 27157  ψcchp 27159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-inf2 9598  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153  ax-addf 11154

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-isom 6532  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-of 7662  df-om 7849  df-1st 7972  df-2nd 7973  df-supp 8143  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-2o 8440  df-oadd 8443  df-er 8680  df-map 8812  df-pm 8813  df-ixp 8882  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-fsupp 9310  df-fi 9359  df-sup 9390  df-inf 9391  df-oi 9460  df-dju 9861  df-card 9899  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-div 11847  df-nn 12213  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12484  df-z 12571  df-dec 12691  df-uz 12842  df-q 12952  df-rp 12996  df-xneg 13116  df-xadd 13117  df-xmul 13118  df-ioo 13355  df-ioc 13356  df-ico 13357  df-icc 13358  df-fz 13515  df-fzo 13662  df-fl 13804  df-mod 13882  df-seq 14017  df-exp 14077  df-fac 14289  df-bc 14318  df-hash 14346  df-shft 15082  df-cj 15128  df-re 15129  df-im 15130  df-sqrt 15264  df-abs 15265  df-limsup 15500  df-clim 15517  df-rlim 15518  df-sum 15716  df-ef 16099  df-sin 16101  df-cos 16102  df-pi 16104  df-dvds 16289  df-gcd 16531  df-prm 16708  df-pc 16875  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17248  df-ress 17269  df-plusg 17301  df-mulr 17302  df-starv 17303  df-sca 17304  df-vsca 17305  df-ip 17306  df-tset 17307  df-ple 17308  df-ds 17310  df-unif 17311  df-hom 17312  df-cco 17313  df-rest 17453  df-topn 17454  df-0g 17472  df-gsum 17473  df-topgen 17474  df-pt 17475  df-prds 17478  df-xrs 17534  df-qtop 17539  df-imas 17540  df-xps 17542  df-mre 17616  df-mrc 17617  df-acs 17619  df-mgm 18676  df-sgrp 18755  df-mnd 18771  df-submnd 18820  df-mulg 19112  df-cntz 19359  df-cmn 19824  df-psmet 21418  df-xmet 21419  df-met 21420  df-bl 21421  df-mopn 21422  df-fbas 21423  df-fg 21424  df-cnfld 21427  df-top 22956  df-topon 22973  df-topsp 22995  df-bases 23008  df-cld 23081  df-ntr 23082  df-cls 23083  df-nei 23160  df-lp 23198  df-perf 23199  df-cn 23289  df-cnp 23290  df-haus 23377  df-tx 23624  df-hmeo 23817  df-fil 23908  df-fm 24000  df-flim 24001  df-flf 24002  df-xms 24382  df-ms 24383  df-tms 24384  df-cncf 24942  df-limc 25930  df-dv 25931  df-log 26623  df-cxp 26624  df-cht 27163  df-vma 27164  df-chp 27165

This theorem is referenced by: (None)