Theorem chpchtsum 27163

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 13914	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin)
2		simpr 484	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
3	2	elin2d 4164	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
4		prmnn 16620	. . . . . . . . 9 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
5	3, 4	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
6	5	nnrpd 12969	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ+)
7	6	relogcld 26565	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
8	7	recnd 11178	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
9		fsumconst 15732	. . . . 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 584	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝) = ((♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝)))
11		simpl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ)
12		1red 11151	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℝ)
13	5	nnred 12177	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
14		prmuz2 16642	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ≥‘2))
15	3, 14	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (ℤ≥‘2))
16		eluz2gt1 12855	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ℤ≥‘2) → 1 < 𝑝)
17	15, 16	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝑝)
18	2	elin1d 4163	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (0[,]𝐴))
19		0re 11152	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
20		elicc2 13348	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
21	19, 11, 20	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
22	18, 21	mpbid 232	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴))
23	22	simp3d 1144	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ 𝐴)
24	12, 13, 11, 17, 23	ltletrd 11310	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝐴)
25	11, 24	rplogcld 26571	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ+)
26	13, 17	rplogcld 26571	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ+)
27	25, 26	rpdivcld 12988	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ+)
28	27	rpred 12971	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
29	27	rpge0d 12975	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ≤ ((log‘𝐴) / (log‘𝑝)))
30		flge0nn0 13758	. . . . . . 7 ⊢ ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 0 ≤ ((log‘𝐴) / (log‘𝑝))) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0)
31	28, 29, 30	syl2anc 584	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0)
32		hashfz1 14287	. . . . . 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 7384	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝)) = ((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝)))
35	28	flcld 13736	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
36	35	zcnd 12615	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℂ)
37	36, 8	mulcomd 11171	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝)) = ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
38	10, 34, 37	3eqtrrd 2769	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) = Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝))
39	38	sumeq2dv 15644	. 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 27162	. 2 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
41		simpl 482	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
42		0red 11153	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ∈ ℝ)
43		1red 11151	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℝ)
44		0lt1 11676	. . . . . . . . 9 ⊢ 0 < 1
45	44	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 < 1)
46		elfzuz2 13466	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...(⌊‘𝐴)) → (⌊‘𝐴) ∈ (ℤ≥‘1))
47		eluzle 12782	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ (ℤ≥‘1) → 1 ≤ (⌊‘𝐴))
48	47	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ≥‘1)) → 1 ≤ (⌊‘𝐴))
49		simpl 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ≥‘1)) → 𝐴 ∈ ℝ)
50		1z 12539	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
51		flge 13743	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℤ) → (1 ≤ 𝐴 ↔ 1 ≤ (⌊‘𝐴)))
52	49, 50, 51	sylancl 586	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ≥‘1)) → (1 ≤ 𝐴 ↔ 1 ≤ (⌊‘𝐴)))
53	48, 52	mpbird 257	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ≥‘1)) → 1 ≤ 𝐴)
54	46, 53	sylan2 593	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 1 ≤ 𝐴)
55	42, 43, 41, 45, 54	ltletrd 11310	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 < 𝐴)
56	42, 41, 55	ltled 11298	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ≤ 𝐴)
57		elfznn 13490	. . . . . . . 8 ⊢ (𝑘 ∈ (1...(⌊‘𝐴)) → 𝑘 ∈ ℕ)
58	57	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝑘 ∈ ℕ)
59	58	nnrecred 12213	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (1 / 𝑘) ∈ ℝ)
60	41, 56, 59	recxpcld 26665	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (𝐴↑𝑐(1 / 𝑘)) ∈ ℝ)
61		chtval 27053	. . . . 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 15644	. . 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 27049	. . . 4 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
65		fzfid 13914	. . . 4 ⊢ (𝐴 ∈ ℝ → (1...(⌊‘𝐴)) ∈ Fin)
66		elinel2 4161	. . . . . . . 8 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ ℙ)
67		elfznn 13490	. . . . . . . 8 ⊢ (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
68	66, 67	anim12i 613	. . . . . . 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 11153	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ∈ ℝ)
71		inss2 4197	. . . . . . . . . . . . 13 ⊢ ((0[,]𝐴) ∩ ℙ) ⊆ ℙ
72	71	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ⊆ ℙ)
73	72	sselda 3943	. . . . . . . . . . 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 12177	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
76	74	nngt0d 12211	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝑝)
77	70, 75, 11, 76, 23	ltletrd 11310	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝐴)
78	77	ex 412	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 0 < 𝐴))
79	78	adantrd 491	. . . . . 6 ⊢ (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 0 < 𝐴))
80	69, 79	jcad 512	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)))
81		elinel2 4161	. . . . . . . 8 ⊢ (𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ) → 𝑝 ∈ ℙ)
82	57, 81	anim12ci 614	. . . . . . 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 412	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝑘 ∈ (1...(⌊‘𝐴)) → 0 < 𝐴))
85	84	adantrd 491	. . . . . 6 ⊢ (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)) → 0 < 𝐴))
86	83, 85	jcad 512	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)))
87		elin 3927	. . . . . . . . 9 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ))
88		simprll 778	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℙ)
89	88	biantrud 531	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ)))
90		0red 11153	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ∈ ℝ)
91		simpl 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℝ)
92	88, 4	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℕ)
93	92	nnred 12177	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℝ)
94	92	nnnn0d 12479	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℕ0)
95	94	nn0ge0d 12482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ 𝑝)
96		df-3an 1088	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ 𝐴))
97	20, 96	bitrdi 287	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ 𝐴)))
98	97	baibd 539	. . . . . . . . . . 11 ⊢ (((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝 ≤ 𝐴))
99	90, 91, 93, 95, 98	syl22anc 838	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝 ≤ 𝐴))
100	89, 99	bitr3d 281	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ) ↔ 𝑝 ≤ 𝐴))
101	87, 100	bitrid 283	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ 𝑝 ≤ 𝐴))
102		simprr 772	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 < 𝐴)
103	91, 102	elrpd 12968	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℝ+)
104	103	relogcld 26565	. . . . . . . . . . 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 26571	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘𝑝) ∈ ℝ+)
108	104, 107	rerpdivcld 13002	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
109		simprlr 779	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℕ)
110	109	nnzd 12532	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℤ)
111		flge 13743	. . . . . . . . . 10 ⊢ ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
112	108, 110, 111	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
113	109	nnnn0d 12479	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℕ0)
114	92, 113	nnexpcld 14186	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑𝑘) ∈ ℕ)
115	114	nnrpd 12969	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑𝑘) ∈ ℝ+)
116	115, 103	logled 26569	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (log‘(𝑝↑𝑘)) ≤ (log‘𝐴)))
117	92	nnrpd 12969	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℝ+)
118		relogexp 26538	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℝ+ ∧ 𝑘 ∈ ℤ) → (log‘(𝑝↑𝑘)) = (𝑘 · (log‘𝑝)))
119	117, 110, 118	syl2anc 584	. . . . . . . . . . 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 12177	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℝ)
122	121, 104, 107	lemuldivd 13020	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
123	116, 120, 122	3bitrd 305	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
124		nnuz 12812	. . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1)
125	109, 124	eleqtrdi 2838	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ (ℤ≥‘1))
126	108	flcld 13736	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
127		elfz5 13453	. . . . . . . . . 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 584	. . . . . . . . 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 312	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ (𝑝↑𝑘) ≤ 𝐴))
130	101, 129	anbi12d 632	. . . . . . 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 13736	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (⌊‘𝐴) ∈ ℤ)
132		elfz5 13453	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘1) ∧ (⌊‘𝐴) ∈ ℤ) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ (⌊‘𝐴)))
133	125, 131, 132	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ (⌊‘𝐴)))
134		flge 13743	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘 ≤ 𝐴 ↔ 𝑘 ≤ (⌊‘𝐴)))
135	91, 110, 134	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ≤ 𝐴 ↔ 𝑘 ≤ (⌊‘𝐴)))
136	133, 135	bitr4d 282	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ 𝐴))
137		elin 3927	. . . . . . . . . 10 ⊢ (𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ))
138	88	biantrud 531	. . . . . . . . . . . 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 12975	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ 𝐴)
140	109	nnrecred 12213	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (1 / 𝑘) ∈ ℝ)
141	91, 139, 140	recxpcld 26665	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴↑𝑐(1 / 𝑘)) ∈ ℝ)
142		elicc2 13348	. . . . . . . . . . . . . . 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 1088	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘))) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘))))
144	142, 143	bitrdi 287	. . . . . . . . . . . . . 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 539	. . . . . . . . . . . . 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 838	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ↔ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘))))
147	138, 146	bitr3d 281	. . . . . . . . . . 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 26666	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ (𝐴↑𝑐(1 / 𝑘)))
149	109	nnrpd 12969	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℝ+)
150	93, 95, 141, 148, 149	cxple2d 26669	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ≤ (𝐴↑𝑐(1 / 𝑘)) ↔ (𝑝↑𝑐𝑘) ≤ ((𝐴↑𝑐(1 / 𝑘))↑𝑐𝑘)))
151	92	nncnd 12178	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℂ)
152		cxpexp 26610	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑝↑𝑐𝑘) = (𝑝↑𝑘))
153	151, 113, 152	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑𝑐𝑘) = (𝑝↑𝑘))
154	109	nncnd 12178	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℂ)
155	109	nnne0d 12212	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ≠ 0)
156	154, 155	recid2d 11930	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((1 / 𝑘) · 𝑘) = 1)
157	156	oveq2d 7385	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴↑𝑐((1 / 𝑘) · 𝑘)) = (𝐴↑𝑐1))
158	103, 140, 154	cxpmuld 26679	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴↑𝑐((1 / 𝑘) · 𝑘)) = ((𝐴↑𝑐(1 / 𝑘))↑𝑐𝑘))
159	91	recnd 11178	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℂ)
160	159	cxp1d 26648	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴↑𝑐1) = 𝐴)
161	157, 158, 160	3eqtr3d 2772	. . . . . . . . . . . 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 305	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ) ↔ (𝑝↑𝑘) ≤ 𝐴))
164	137, 163	bitrid 283	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝↑𝑘) ≤ 𝐴))
165	136, 164	anbi12d 632	. . . . . . . 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 12177	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑𝑘) ∈ ℝ)
167		bernneq3 14172	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ (ℤ≥‘2) ∧ 𝑘 ∈ ℕ0) → 𝑘 < (𝑝↑𝑘))
168	105, 113, 167	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 < (𝑝↑𝑘))
169	121, 166, 168	ltled 11298	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ≤ (𝑝↑𝑘))
170		letr 11244	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℝ ∧ (𝑝↑𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑘 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑘 ≤ 𝐴))
171	121, 166, 91, 170	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑘 ≤ 𝐴))
172	169, 171	mpand 695	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 → 𝑘 ≤ 𝐴))
173	172	pm4.71rd 562	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑘 ≤ 𝐴 ∧ (𝑝↑𝑘) ≤ 𝐴)))
174	151	exp1d 14082	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑1) = 𝑝)
175	92	nnge1d 12210	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 1 ≤ 𝑝)
176	93, 175, 125	leexp2ad 14195	. . . . . . . . . . 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 11244	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℝ ∧ (𝑝↑𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑝 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑝 ≤ 𝐴))
179	93, 166, 91, 178	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑝 ≤ 𝐴))
180	177, 179	mpand 695	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 → 𝑝 ≤ 𝐴))
181	180	pm4.71rd 562	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑝 ≤ 𝐴 ∧ (𝑝↑𝑘) ≤ 𝐴)))
182	165, 173, 181	3bitr2rd 308	. . . . . . 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 279	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ))))
184	183	ex 412	. . . . 5 ⊢ (𝐴 ∈ ℝ → (((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)))))
185	80, 86, 184	pm5.21ndd 379	. . . 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 717	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘𝑝) ∈ ℂ)
187	64, 65, 1, 185, 186	fsumcom2 15716	. . 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 2767	. 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 2774	1 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴↑𝑐(1 / 𝑘))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∩ cin 3910   ⊆ wss 3911   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369  Fincfn 8895  ℂcc 11042  ℝcr 11043  0cc0 11044  1c1 11045   · cmul 11049   < clt 11184   ≤ cle 11185   / cdiv 11811  ℕcn 12162  2c2 12217  ℕ₀cn0 12418  ℤcz 12505  ℤ_≥cuz 12769  ℝ⁺crp 12927  [,]cicc 13285  ...cfz 13444  ⌊cfl 13728  ↑cexp 14002  ♯chash 14271  Σcsu 15628  ℙcprime 16617  logclog 26496  ↑_𝑐ccxp 26497  θccht 27034  ψcchp 27036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122  ax-addf 11123

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-om 7823  df-1st 7947  df-2nd 7948  df-supp 8117  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-oadd 8415  df-er 8648  df-map 8778  df-pm 8779  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9289  df-fi 9338  df-sup 9369  df-inf 9370  df-oi 9439  df-dju 9830  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-q 12884  df-rp 12928  df-xneg 13048  df-xadd 13049  df-xmul 13050  df-ioo 13286  df-ioc 13287  df-ico 13288  df-icc 13289  df-fz 13445  df-fzo 13592  df-fl 13730  df-mod 13808  df-seq 13943  df-exp 14003  df-fac 14215  df-bc 14244  df-hash 14272  df-shft 15009  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-limsup 15413  df-clim 15430  df-rlim 15431  df-sum 15629  df-ef 16009  df-sin 16011  df-cos 16012  df-pi 16014  df-dvds 16199  df-gcd 16441  df-prm 16618  df-pc 16784  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-starv 17211  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-hom 17220  df-cco 17221  df-rest 17361  df-topn 17362  df-0g 17380  df-gsum 17381  df-topgen 17382  df-pt 17383  df-prds 17386  df-xrs 17441  df-qtop 17446  df-imas 17447  df-xps 17449  df-mre 17523  df-mrc 17524  df-acs 17526  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-submnd 18693  df-mulg 18982  df-cntz 19231  df-cmn 19696  df-psmet 21288  df-xmet 21289  df-met 21290  df-bl 21291  df-mopn 21292  df-fbas 21293  df-fg 21294  df-cnfld 21297  df-top 22814  df-topon 22831  df-topsp 22853  df-bases 22866  df-cld 22939  df-ntr 22940  df-cls 22941  df-nei 23018  df-lp 23056  df-perf 23057  df-cn 23147  df-cnp 23148  df-haus 23235  df-tx 23482  df-hmeo 23675  df-fil 23766  df-fm 23858  df-flim 23859  df-flf 23860  df-xms 24241  df-ms 24242  df-tms 24243  df-cncf 24804  df-limc 25800  df-dv 25801  df-log 26498  df-cxp 26499  df-cht 27040  df-vma 27041  df-chp 27042

This theorem is referenced by: (None)