Theorem chpchtsum 27240

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 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
3	2	elin2d 4199	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
4		prmnn 16670	. . . . . . . . 9 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
5	3, 4	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
6	5	nnrpd 13063	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ+)
7	6	relogcld 26642	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
8	7	recnd 11288	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
9		fsumconst 15789	. . . . 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 582	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝) = ((♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝)))
11		simpl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ)
12		1red 11261	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℝ)
13	5	nnred 12274	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
14		prmuz2 16692	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ≥‘2))
15	3, 14	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (ℤ≥‘2))
16		eluz2gt1 12951	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ℤ≥‘2) → 1 < 𝑝)
17	15, 16	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝑝)
18	2	elin1d 4198	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (0[,]𝐴))
19		0re 11262	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
20		elicc2 13438	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
21	19, 11, 20	sylancr 585	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
22	18, 21	mpbid 231	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴))
23	22	simp3d 1141	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ 𝐴)
24	12, 13, 11, 17, 23	ltletrd 11420	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝐴)
25	11, 24	rplogcld 26648	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ+)
26	13, 17	rplogcld 26648	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ+)
27	25, 26	rpdivcld 13082	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ+)
28	27	rpred 13065	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
29	27	rpge0d 13069	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ≤ ((log‘𝐴) / (log‘𝑝)))
30		flge0nn0 13835	. . . . . . 7 ⊢ ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 0 ≤ ((log‘𝐴) / (log‘𝑝))) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0)
31	28, 29, 30	syl2anc 582	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0)
32		hashfz1 14358	. . . . . 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 7438	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝)) = ((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝)))
35	28	flcld 13813	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
36	35	zcnd 12714	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℂ)
37	36, 8	mulcomd 11281	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝)) = ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
38	10, 34, 37	3eqtrrd 2770	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) = Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝))
39	38	sumeq2dv 15702	. 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 27239	. 2 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
41		simpl 481	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
42		0red 11263	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ∈ ℝ)
43		1red 11261	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℝ)
44		0lt1 11782	. . . . . . . . 9 ⊢ 0 < 1
45	44	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 < 1)
46		elfzuz2 13555	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...(⌊‘𝐴)) → (⌊‘𝐴) ∈ (ℤ≥‘1))
47		eluzle 12882	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ (ℤ≥‘1) → 1 ≤ (⌊‘𝐴))
48	47	adantl 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ≥‘1)) → 1 ≤ (⌊‘𝐴))
49		simpl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ≥‘1)) → 𝐴 ∈ ℝ)
50		1z 12639	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
51		flge 13820	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℤ) → (1 ≤ 𝐴 ↔ 1 ≤ (⌊‘𝐴)))
52	49, 50, 51	sylancl 584	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ≥‘1)) → (1 ≤ 𝐴 ↔ 1 ≤ (⌊‘𝐴)))
53	48, 52	mpbird 256	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ≥‘1)) → 1 ≤ 𝐴)
54	46, 53	sylan2 591	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 1 ≤ 𝐴)
55	42, 43, 41, 45, 54	ltletrd 11420	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 < 𝐴)
56	42, 41, 55	ltled 11408	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ≤ 𝐴)
57		elfznn 13579	. . . . . . . 8 ⊢ (𝑘 ∈ (1...(⌊‘𝐴)) → 𝑘 ∈ ℕ)
58	57	adantl 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝑘 ∈ ℕ)
59	58	nnrecred 12310	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (1 / 𝑘) ∈ ℝ)
60	41, 56, 59	recxpcld 26742	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (𝐴↑𝑐(1 / 𝑘)) ∈ ℝ)
61		chtval 27130	. . . . 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 15702	. . 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 27126	. . . 4 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
65		fzfid 13988	. . . 4 ⊢ (𝐴 ∈ ℝ → (1...(⌊‘𝐴)) ∈ Fin)
66		elinel2 4196	. . . . . . . 8 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ ℙ)
67		elfznn 13579	. . . . . . . 8 ⊢ (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
68	66, 67	anim12i 611	. . . . . . 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 11263	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ∈ ℝ)
71		inss2 4230	. . . . . . . . . . . . 13 ⊢ ((0[,]𝐴) ∩ ℙ) ⊆ ℙ
72	71	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ⊆ ℙ)
73	72	sselda 3978	. . . . . . . . . . 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 12274	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
76	74	nngt0d 12308	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝑝)
77	70, 75, 11, 76, 23	ltletrd 11420	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝐴)
78	77	ex 411	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 0 < 𝐴))
79	78	adantrd 490	. . . . . 6 ⊢ (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 0 < 𝐴))
80	69, 79	jcad 511	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)))
81		elinel2 4196	. . . . . . . 8 ⊢ (𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ) → 𝑝 ∈ ℙ)
82	57, 81	anim12ci 612	. . . . . . 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 411	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝑘 ∈ (1...(⌊‘𝐴)) → 0 < 𝐴))
85	84	adantrd 490	. . . . . 6 ⊢ (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)) → 0 < 𝐴))
86	83, 85	jcad 511	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)))
87		elin 3962	. . . . . . . . 9 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ))
88		simprll 777	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℙ)
89	88	biantrud 530	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ)))
90		0red 11263	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ∈ ℝ)
91		simpl 481	. . . . . . . . . . 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 12274	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℝ)
94	92	nnnn0d 12579	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℕ0)
95	94	nn0ge0d 12582	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ 𝑝)
96		df-3an 1086	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ 𝐴))
97	20, 96	bitrdi 286	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ 𝐴)))
98	97	baibd 538	. . . . . . . . . . 11 ⊢ (((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝 ≤ 𝐴))
99	90, 91, 93, 95, 98	syl22anc 837	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝 ≤ 𝐴))
100	89, 99	bitr3d 280	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ) ↔ 𝑝 ≤ 𝐴))
101	87, 100	bitrid 282	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ 𝑝 ≤ 𝐴))
102		simprr 771	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 < 𝐴)
103	91, 102	elrpd 13062	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℝ+)
104	103	relogcld 26642	. . . . . . . . . . 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 26648	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘𝑝) ∈ ℝ+)
108	104, 107	rerpdivcld 13096	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
109		simprlr 778	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℕ)
110	109	nnzd 12632	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℤ)
111		flge 13820	. . . . . . . . . 10 ⊢ ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
112	108, 110, 111	syl2anc 582	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
113	109	nnnn0d 12579	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℕ0)
114	92, 113	nnexpcld 14257	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑𝑘) ∈ ℕ)
115	114	nnrpd 13063	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑𝑘) ∈ ℝ+)
116	115, 103	logled 26646	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (log‘(𝑝↑𝑘)) ≤ (log‘𝐴)))
117	92	nnrpd 13063	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℝ+)
118		relogexp 26615	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℝ+ ∧ 𝑘 ∈ ℤ) → (log‘(𝑝↑𝑘)) = (𝑘 · (log‘𝑝)))
119	117, 110, 118	syl2anc 582	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘(𝑝↑𝑘)) = (𝑘 · (log‘𝑝)))
120	119	breq1d 5162	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((log‘(𝑝↑𝑘)) ≤ (log‘𝐴) ↔ (𝑘 · (log‘𝑝)) ≤ (log‘𝐴)))
121	109	nnred 12274	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℝ)
122	121, 104, 107	lemuldivd 13114	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
123	116, 120, 122	3bitrd 304	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
124		nnuz 12912	. . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1)
125	109, 124	eleqtrdi 2835	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ (ℤ≥‘1))
126	108	flcld 13813	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
127		elfz5 13542	. . . . . . . . . 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 582	. . . . . . . . 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 311	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ (𝑝↑𝑘) ≤ 𝐴))
130	101, 129	anbi12d 630	. . . . . . 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 13813	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (⌊‘𝐴) ∈ ℤ)
132		elfz5 13542	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘1) ∧ (⌊‘𝐴) ∈ ℤ) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ (⌊‘𝐴)))
133	125, 131, 132	syl2anc 582	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ (⌊‘𝐴)))
134		flge 13820	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘 ≤ 𝐴 ↔ 𝑘 ≤ (⌊‘𝐴)))
135	91, 110, 134	syl2anc 582	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ≤ 𝐴 ↔ 𝑘 ≤ (⌊‘𝐴)))
136	133, 135	bitr4d 281	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ 𝐴))
137		elin 3962	. . . . . . . . . 10 ⊢ (𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ))
138	88	biantrud 530	. . . . . . . . . . . 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 13069	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ 𝐴)
140	109	nnrecred 12310	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (1 / 𝑘) ∈ ℝ)
141	91, 139, 140	recxpcld 26742	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴↑𝑐(1 / 𝑘)) ∈ ℝ)
142		elicc2 13438	. . . . . . . . . . . . . . 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 1086	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘))) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘))))
144	142, 143	bitrdi 286	. . . . . . . . . . . . . 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 538	. . . . . . . . . . . . 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 837	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ↔ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘))))
147	138, 146	bitr3d 280	. . . . . . . . . . 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 26743	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ (𝐴↑𝑐(1 / 𝑘)))
149	109	nnrpd 13063	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℝ+)
150	93, 95, 141, 148, 149	cxple2d 26746	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ≤ (𝐴↑𝑐(1 / 𝑘)) ↔ (𝑝↑𝑐𝑘) ≤ ((𝐴↑𝑐(1 / 𝑘))↑𝑐𝑘)))
151	92	nncnd 12275	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℂ)
152		cxpexp 26687	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑝↑𝑐𝑘) = (𝑝↑𝑘))
153	151, 113, 152	syl2anc 582	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑𝑐𝑘) = (𝑝↑𝑘))
154	109	nncnd 12275	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℂ)
155	109	nnne0d 12309	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ≠ 0)
156	154, 155	recid2d 12033	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((1 / 𝑘) · 𝑘) = 1)
157	156	oveq2d 7439	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴↑𝑐((1 / 𝑘) · 𝑘)) = (𝐴↑𝑐1))
158	103, 140, 154	cxpmuld 26756	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴↑𝑐((1 / 𝑘) · 𝑘)) = ((𝐴↑𝑐(1 / 𝑘))↑𝑐𝑘))
159	91	recnd 11288	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℂ)
160	159	cxp1d 26725	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴↑𝑐1) = 𝐴)
161	157, 158, 160	3eqtr3d 2773	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝐴↑𝑐(1 / 𝑘))↑𝑐𝑘) = 𝐴)
162	153, 161	breq12d 5165	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑐𝑘) ≤ ((𝐴↑𝑐(1 / 𝑘))↑𝑐𝑘) ↔ (𝑝↑𝑘) ≤ 𝐴))
163	147, 150, 162	3bitrd 304	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ) ↔ (𝑝↑𝑘) ≤ 𝐴))
164	137, 163	bitrid 282	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝↑𝑘) ≤ 𝐴))
165	136, 164	anbi12d 630	. . . . . . . 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 12274	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑𝑘) ∈ ℝ)
167		bernneq3 14243	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ (ℤ≥‘2) ∧ 𝑘 ∈ ℕ0) → 𝑘 < (𝑝↑𝑘))
168	105, 113, 167	syl2anc 582	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 < (𝑝↑𝑘))
169	121, 166, 168	ltled 11408	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ≤ (𝑝↑𝑘))
170		letr 11354	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℝ ∧ (𝑝↑𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑘 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑘 ≤ 𝐴))
171	121, 166, 91, 170	syl3anc 1368	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑘 ≤ 𝐴))
172	169, 171	mpand 693	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 → 𝑘 ≤ 𝐴))
173	172	pm4.71rd 561	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑘 ≤ 𝐴 ∧ (𝑝↑𝑘) ≤ 𝐴)))
174	151	exp1d 14155	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑1) = 𝑝)
175	92	nnge1d 12307	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 1 ≤ 𝑝)
176	93, 175, 125	leexp2ad 14266	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑1) ≤ (𝑝↑𝑘))
177	174, 176	eqbrtrrd 5176	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ≤ (𝑝↑𝑘))
178		letr 11354	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℝ ∧ (𝑝↑𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑝 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑝 ≤ 𝐴))
179	93, 166, 91, 178	syl3anc 1368	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑝 ≤ 𝐴))
180	177, 179	mpand 693	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 → 𝑝 ≤ 𝐴))
181	180	pm4.71rd 561	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑝 ≤ 𝐴 ∧ (𝑝↑𝑘) ≤ 𝐴)))
182	165, 173, 181	3bitr2rd 307	. . . . . . 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 278	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ))))
184	183	ex 411	. . . . 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 378	. . . 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 715	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘𝑝) ∈ ℂ)
187	64, 65, 1, 185, 186	fsumcom2 15773	. . 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 2768	. 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 2775	1 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴↑𝑐(1 / 𝑘))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ∩ cin 3945   ⊆ wss 3946   class class class wbr 5152  ‘cfv 6553  (class class class)co 7423  Fincfn 8973  ℂcc 11152  ℝcr 11153  0cc0 11154  1c1 11155   · cmul 11159   < clt 11294   ≤ cle 11295   / cdiv 11917  ℕcn 12259  2c2 12314  ℕ₀cn0 12519  ℤcz 12605  ℤ_≥cuz 12869  ℝ⁺crp 13023  [,]cicc 13376  ...cfz 13533  ⌊cfl 13805  ↑cexp 14076  ♯chash 14342  Σcsu 15685  ℙcprime 16667  logclog 26573  ↑_𝑐ccxp 26574  θccht 27111  ψcchp 27113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745  ax-inf2 9680  ax-cnex 11210  ax-resscn 11211  ax-1cn 11212  ax-icn 11213  ax-addcl 11214  ax-addrcl 11215  ax-mulcl 11216  ax-mulrcl 11217  ax-mulcom 11218  ax-addass 11219  ax-mulass 11220  ax-distr 11221  ax-i2m1 11222  ax-1ne0 11223  ax-1rid 11224  ax-rnegex 11225  ax-rrecex 11226  ax-cnre 11227  ax-pre-lttri 11228  ax-pre-lttrn 11229  ax-pre-ltadd 11230  ax-pre-mulgt0 11231  ax-pre-sup 11232  ax-addf 11233

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5579  df-eprel 5585  df-po 5593  df-so 5594  df-fr 5636  df-se 5637  df-we 5638  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-pred 6311  df-ord 6378  df-on 6379  df-lim 6380  df-suc 6381  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7379  df-ov 7426  df-oprab 7427  df-mpo 7428  df-of 7689  df-om 7876  df-1st 8002  df-2nd 8003  df-supp 8174  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-2o 8496  df-oadd 8499  df-er 8733  df-map 8856  df-pm 8857  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fsupp 9402  df-fi 9450  df-sup 9481  df-inf 9482  df-oi 9549  df-dju 9940  df-card 9978  df-pnf 11296  df-mnf 11297  df-xr 11298  df-ltxr 11299  df-le 11300  df-sub 11492  df-neg 11493  df-div 11918  df-nn 12260  df-2 12322  df-3 12323  df-4 12324  df-5 12325  df-6 12326  df-7 12327  df-8 12328  df-9 12329  df-n0 12520  df-z 12606  df-dec 12725  df-uz 12870  df-q 12980  df-rp 13024  df-xneg 13141  df-xadd 13142  df-xmul 13143  df-ioo 13377  df-ioc 13378  df-ico 13379  df-icc 13380  df-fz 13534  df-fzo 13677  df-fl 13807  df-mod 13885  df-seq 14017  df-exp 14077  df-fac 14286  df-bc 14315  df-hash 14343  df-shft 15067  df-cj 15099  df-re 15100  df-im 15101  df-sqrt 15235  df-abs 15236  df-limsup 15468  df-clim 15485  df-rlim 15486  df-sum 15686  df-ef 16064  df-sin 16066  df-cos 16067  df-pi 16069  df-dvds 16252  df-gcd 16490  df-prm 16668  df-pc 16834  df-struct 17144  df-sets 17161  df-slot 17179  df-ndx 17191  df-base 17209  df-ress 17238  df-plusg 17274  df-mulr 17275  df-starv 17276  df-sca 17277  df-vsca 17278  df-ip 17279  df-tset 17280  df-ple 17281  df-ds 17283  df-unif 17284  df-hom 17285  df-cco 17286  df-rest 17432  df-topn 17433  df-0g 17451  df-gsum 17452  df-topgen 17453  df-pt 17454  df-prds 17457  df-xrs 17512  df-qtop 17517  df-imas 17518  df-xps 17520  df-mre 17594  df-mrc 17595  df-acs 17597  df-mgm 18628  df-sgrp 18707  df-mnd 18723  df-submnd 18769  df-mulg 19057  df-cntz 19306  df-cmn 19775  df-psmet 21327  df-xmet 21328  df-met 21329  df-bl 21330  df-mopn 21331  df-fbas 21332  df-fg 21333  df-cnfld 21336  df-top 22879  df-topon 22896  df-topsp 22918  df-bases 22932  df-cld 23006  df-ntr 23007  df-cls 23008  df-nei 23085  df-lp 23123  df-perf 23124  df-cn 23214  df-cnp 23215  df-haus 23302  df-tx 23549  df-hmeo 23742  df-fil 23833  df-fm 23925  df-flim 23926  df-flf 23927  df-xms 24309  df-ms 24310  df-tms 24311  df-cncf 24881  df-limc 25878  df-dv 25879  df-log 26575  df-cxp 26576  df-cht 27117  df-vma 27118  df-chp 27119

This theorem is referenced by: (None)