Theorem chpchtsum 27453

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 14039	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin)
2		simpr 490	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
3	2	elin2d 4154	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
4		prmnn 16768	. . . . . . . . 9 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
5	3, 4	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
6	5	nnrpd 13086	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ+)
7	6	relogcld 26858	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
8	7	recnd 11264	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
9		fsumconst 15878	. . . . 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 596	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝) = ((♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝)))
11		simpl 488	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ)
12		1red 11236	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℝ)
13	5	nnred 12275	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
14		prmuz2 16790	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ≥‘2))
15	3, 14	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (ℤ≥‘2))
16		eluz2gt1 12972	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ℤ≥‘2) → 1 < 𝑝)
17	15, 16	syl 18	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝑝)
18	2	elin1d 4153	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (0[,]𝐴))
19		0re 11237	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
20		elicc2 13466	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
21	19, 11, 20	sylancr 599	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
22	18, 21	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴))
23	22	simp3d 1162	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ 𝐴)
24	12, 13, 11, 17, 23	ltletrd 11397	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝐴)
25	11, 24	rplogcld 26864	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ+)
26	13, 17	rplogcld 26864	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ+)
27	25, 26	rpdivcld 13105	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ+)
28	27	rpred 13088	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
29	27	rpge0d 13092	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ≤ ((log‘𝐴) / (log‘𝑝)))
30		flge0nn0 13883	. . . . . . 7 ⊢ ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 0 ≤ ((log‘𝐴) / (log‘𝑝))) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0)
31	28, 29, 30	syl2anc 596	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0)
32		hashfz1 14412	. . . . . 6 ⊢ ((⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0 → (♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) = (⌊‘((log‘𝐴) / (log‘𝑝))))
33	31, 32	syl 18	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) = (⌊‘((log‘𝐴) / (log‘𝑝))))
34	33	oveq1d 7431	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝)) = ((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝)))
35	28	flcld 13861	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
36	35	zcnd 12729	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℂ)
37	36, 8	mulcomd 11257	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝)) = ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
38	10, 34, 37	3eqtrrd 2802	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) = Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝))
39	38	sumeq2dv 15791	. 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 27452	. 2 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
41		simpl 488	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
42		0red 11238	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ∈ ℝ)
43		1red 11236	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 1 ∈ ℝ)
44		0lt1 11763	. . . . . . . . 9 ⊢ 0 < 1
45	44	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 < 1)
46		elfzuz2 13585	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...(⌊‘𝐴)) → (⌊‘𝐴) ∈ (ℤ≥‘1))
47		eluzle 12903	. . . . . . . . . . 11 ⊢ ((⌊‘𝐴) ∈ (ℤ≥‘1) → 1 ≤ (⌊‘𝐴))
48	47	adantl 487	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ≥‘1)) → 1 ≤ (⌊‘𝐴))
49		simpl 488	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ≥‘1)) → 𝐴 ∈ ℝ)
50		1z 12651	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
51		flge 13868	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℤ) → (1 ≤ 𝐴 ↔ 1 ≤ (⌊‘𝐴)))
52	49, 50, 51	sylancl 598	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ≥‘1)) → (1 ≤ 𝐴 ↔ 1 ≤ (⌊‘𝐴)))
53	48, 52	mpbird 260	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (⌊‘𝐴) ∈ (ℤ≥‘1)) → 1 ≤ 𝐴)
54	46, 53	sylan2 605	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 1 ≤ 𝐴)
55	42, 43, 41, 45, 54	ltletrd 11397	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 < 𝐴)
56	42, 41, 55	ltled 11385	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ≤ 𝐴)
57		elfznn 13610	. . . . . . . 8 ⊢ (𝑘 ∈ (1...(⌊‘𝐴)) → 𝑘 ∈ ℕ)
58	57	adantl 487	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝑘 ∈ ℕ)
59	58	nnrecred 12314	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (1 / 𝑘) ∈ ℝ)
60	41, 56, 59	recxpcld 26958	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (𝐴↑𝑐(1 / 𝑘)) ∈ ℝ)
61		chtval 27344	. . . . 5 ⊢ ((𝐴↑𝑐(1 / 𝑘)) ∈ ℝ → (θ‘(𝐴↑𝑐(1 / 𝑘))) = Σ𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)(log‘𝑝))
62	60, 61	syl 18	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (θ‘(𝐴↑𝑐(1 / 𝑘))) = Σ𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)(log‘𝑝))
63	62	sumeq2dv 15791	. . 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 27340	. . . 4 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
65		fzfid 14039	. . . 4 ⊢ (𝐴 ∈ ℝ → (1...(⌊‘𝐴)) ∈ Fin)
66		elinel2 4151	. . . . . . . 8 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ ℙ)
67		elfznn 13610	. . . . . . . 8 ⊢ (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
68	66, 67	anim12i 625	. . . . . . 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 11238	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ∈ ℝ)
71		inss2 4186	. . . . . . . . . . . . 13 ⊢ ((0[,]𝐴) ∩ ℙ) ⊆ ℙ
72	71	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ⊆ ℙ)
73	72	sselda 3934	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
74	73, 4	syl 18	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
75	74	nnred 12275	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
76	74	nngt0d 12312	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝑝)
77	70, 75, 11, 76, 23	ltletrd 11397	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝐴)
78	77	ex 418	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 0 < 𝐴))
79	78	adantrd 497	. . . . . 6 ⊢ (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 0 < 𝐴))
80	69, 79	jcad 522	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)))
81		elinel2 4151	. . . . . . . 8 ⊢ (𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ) → 𝑝 ∈ ℙ)
82	57, 81	anim12ci 626	. . . . . . 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 418	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝑘 ∈ (1...(⌊‘𝐴)) → 0 < 𝐴))
85	84	adantrd 497	. . . . . 6 ⊢ (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)) → 0 < 𝐴))
86	83, 85	jcad 522	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ)) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)))
87		elin 3918	. . . . . . . . 9 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ))
88		simprll 791	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℙ)
89	88	biantrud 541	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ)))
90		0red 11238	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ∈ ℝ)
91		simpl 488	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℝ)
92	88, 4	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℕ)
93	92	nnred 12275	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℝ)
94	92	nnnn0d 12592	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℕ0)
95	94	nn0ge0d 12595	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ 𝑝)
96		df-3an 1105	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ 𝐴))
97	20, 96	bitrdi 290	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ 𝐴)))
98	97	baibd 549	. . . . . . . . . . 11 ⊢ (((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝 ≤ 𝐴))
99	90, 91, 93, 95, 98	syl22anc 852	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝 ≤ 𝐴))
100	89, 99	bitr3d 284	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ) ↔ 𝑝 ≤ 𝐴))
101	87, 100	bitrid 286	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ 𝑝 ≤ 𝐴))
102		simprr 785	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 < 𝐴)
103	91, 102	elrpd 13085	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℝ+)
104	103	relogcld 26858	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘𝐴) ∈ ℝ)
105	88, 14	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ (ℤ≥‘2))
106	105, 16	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 1 < 𝑝)
107	93, 106	rplogcld 26864	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘𝑝) ∈ ℝ+)
108	104, 107	rerpdivcld 13119	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
109		simprlr 792	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℕ)
110	109	nnzd 12644	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℤ)
111		flge 13868	. . . . . . . . . 10 ⊢ ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
112	108, 110, 111	syl2anc 596	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
113	109	nnnn0d 12592	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℕ0)
114	92, 113	nnexpcld 14311	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑𝑘) ∈ ℕ)
115	114	nnrpd 13086	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑𝑘) ∈ ℝ+)
116	115, 103	logled 26862	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (log‘(𝑝↑𝑘)) ≤ (log‘𝐴)))
117	92	nnrpd 13086	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℝ+)
118		relogexp 26831	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℝ+ ∧ 𝑘 ∈ ℤ) → (log‘(𝑝↑𝑘)) = (𝑘 · (log‘𝑝)))
119	117, 110, 118	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (log‘(𝑝↑𝑘)) = (𝑘 · (log‘𝑝)))
120	119	breq1d 5117	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((log‘(𝑝↑𝑘)) ≤ (log‘𝐴) ↔ (𝑘 · (log‘𝑝)) ≤ (log‘𝐴)))
121	109	nnred 12275	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℝ)
122	121, 104, 107	lemuldivd 13137	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
123	116, 120, 122	3bitrd 308	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
124		nnuz 12929	. . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1)
125	109, 124	eleqtrdi 2872	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ (ℤ≥‘1))
126	108	flcld 13861	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
127		elfz5 13572	. . . . . . . . . 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 596	. . . . . . . . 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 315	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ (𝑝↑𝑘) ≤ 𝐴))
130	101, 129	anbi12d 644	. . . . . . 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 13861	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (⌊‘𝐴) ∈ ℤ)
132		elfz5 13572	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘1) ∧ (⌊‘𝐴) ∈ ℤ) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ (⌊‘𝐴)))
133	125, 131, 132	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ (⌊‘𝐴)))
134		flge 13868	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘 ≤ 𝐴 ↔ 𝑘 ≤ (⌊‘𝐴)))
135	91, 110, 134	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ≤ 𝐴 ↔ 𝑘 ≤ (⌊‘𝐴)))
136	133, 135	bitr4d 285	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ 𝐴))
137		elin 3918	. . . . . . . . . 10 ⊢ (𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ))
138	88	biantrud 541	. . . . . . . . . . . 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 13092	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ 𝐴)
140	109	nnrecred 12314	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (1 / 𝑘) ∈ ℝ)
141	91, 139, 140	recxpcld 26958	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴↑𝑐(1 / 𝑘)) ∈ ℝ)
142		elicc2 13466	. . . . . . . . . . . . . . 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 1105	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘))) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘))))
144	142, 143	bitrdi 290	. . . . . . . . . . . . . 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 549	. . . . . . . . . . . . 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 852	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ↔ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘))))
147	138, 146	bitr3d 284	. . . . . . . . . . 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 26959	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 0 ≤ (𝐴↑𝑐(1 / 𝑘)))
149	109	nnrpd 13086	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℝ+)
150	93, 95, 141, 148, 149	cxple2d 26962	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ≤ (𝐴↑𝑐(1 / 𝑘)) ↔ (𝑝↑𝑐𝑘) ≤ ((𝐴↑𝑐(1 / 𝑘))↑𝑐𝑘)))
151	92	nncnd 12276	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ∈ ℂ)
152		cxpexp 26903	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑝↑𝑐𝑘) = (𝑝↑𝑘))
153	151, 113, 152	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑𝑐𝑘) = (𝑝↑𝑘))
154	109	nncnd 12276	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ∈ ℂ)
155	109	nnne0d 12313	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ≠ 0)
156	154, 155	recid2d 12014	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((1 / 𝑘) · 𝑘) = 1)
157	156	oveq2d 7432	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴↑𝑐((1 / 𝑘) · 𝑘)) = (𝐴↑𝑐1))
158	103, 140, 154	cxpmuld 26972	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴↑𝑐((1 / 𝑘) · 𝑘)) = ((𝐴↑𝑐(1 / 𝑘))↑𝑐𝑘))
159	91	recnd 11264	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝐴 ∈ ℂ)
160	159	cxp1d 26941	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝐴↑𝑐1) = 𝐴)
161	157, 158, 160	3eqtr3d 2805	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝐴↑𝑐(1 / 𝑘))↑𝑐𝑘) = 𝐴)
162	153, 161	breq12d 5120	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑐𝑘) ≤ ((𝐴↑𝑐(1 / 𝑘))↑𝑐𝑘) ↔ (𝑝↑𝑘) ≤ 𝐴))
163	147, 150, 162	3bitrd 308	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ) ↔ (𝑝↑𝑘) ≤ 𝐴))
164	137, 163	bitrid 286	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝↑𝑘) ≤ 𝐴))
165	136, 164	anbi12d 644	. . . . . . . 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 12275	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑𝑘) ∈ ℝ)
167		bernneq3 14297	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ (ℤ≥‘2) ∧ 𝑘 ∈ ℕ0) → 𝑘 < (𝑝↑𝑘))
168	105, 113, 167	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 < (𝑝↑𝑘))
169	121, 166, 168	ltled 11385	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑘 ≤ (𝑝↑𝑘))
170		letr 11331	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℝ ∧ (𝑝↑𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑘 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑘 ≤ 𝐴))
171	121, 166, 91, 170	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑘 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑘 ≤ 𝐴))
172	169, 171	mpand 708	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 → 𝑘 ≤ 𝐴))
173	172	pm4.71rd 572	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑘 ≤ 𝐴 ∧ (𝑝↑𝑘) ≤ 𝐴)))
174	151	exp1d 14207	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑1) = 𝑝)
175	92	nnge1d 12311	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 1 ≤ 𝑝)
176	93, 175, 125	leexp2ad 14320	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → (𝑝↑1) ≤ (𝑝↑𝑘))
177	174, 176	eqbrtrrd 5133	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → 𝑝 ≤ (𝑝↑𝑘))
178		letr 11331	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℝ ∧ (𝑝↑𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑝 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑝 ≤ 𝐴))
179	93, 166, 91, 178	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑝 ≤ 𝐴))
180	177, 179	mpand 708	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 → 𝑝 ≤ 𝐴))
181	180	pm4.71rd 572	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 < 𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑝 ≤ 𝐴 ∧ (𝑝↑𝑘) ≤ 𝐴)))
182	165, 173, 181	3bitr2rd 311	. . . . . . 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 282	. . . . . 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 418	. . . . 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 382	. . . 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 730	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘𝑝) ∈ ℂ)
187	64, 65, 1, 185, 186	fsumcom2 15862	. . 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 2800	. 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 2807	1 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴↑𝑐(1 / 𝑘))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ∩ cin 3901   ⊆ wss 3902   class class class wbr 5107  ‘cfv 6537  (class class class)co 7416  Fincfn 8955  ℂcc 11125  ℝcr 11126  0cc0 11127  1c1 11128   · cmul 11132   < clt 11270   ≤ cle 11271   / cdiv 11898  ℕcn 12260  2c2 12322  ℕ₀cn0 12531  ℤcz 12618  ℤ_≥cuz 12890  ℝ⁺crp 13044  [,]cicc 13403  ...cfz 13563  ⌊cfl 13853  ↑cexp 14127  ♯chash 14396  Σcsu 15775  ℙcprime 16765  logclog 26789  ↑_𝑐ccxp 26790  θccht 27325  ψcchp 27327

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-inf2 9623  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205  ax-addf 11206

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7681  df-om 7866  df-1st 7989  df-2nd 7990  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-oadd 8462  df-er 8699  df-map 8831  df-pm 8832  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9335  df-fi 9384  df-sup 9415  df-inf 9416  df-oi 9485  df-dju 9909  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12532  df-z 12619  df-dec 12740  df-uz 12891  df-q 13001  df-rp 13045  df-xneg 13165  df-xadd 13166  df-xmul 13167  df-ioo 13404  df-ioc 13405  df-ico 13406  df-icc 13407  df-fz 13564  df-fzo 13712  df-fl 13855  df-mod 13933  df-seq 14068  df-exp 14128  df-fac 14340  df-bc 14369  df-hash 14397  df-shft 15142  df-cj 15188  df-re 15189  df-im 15190  df-sqrt 15324  df-abs 15325  df-limsup 15560  df-clim 15577  df-rlim 15578  df-sum 15776  df-ef 16157  df-sin 16159  df-cos 16160  df-pi 16162  df-dvds 16347  df-gcd 16589  df-prm 16766  df-pc 16933  df-struct 17243  df-sets 17260  df-slot 17278  df-ndx 17290  df-base 17306  df-ress 17327  df-plusg 17359  df-mulr 17360  df-starv 17361  df-sca 17362  df-vsca 17363  df-ip 17364  df-tset 17365  df-ple 17366  df-ds 17368  df-unif 17369  df-hom 17370  df-cco 17371  df-rest 17511  df-topn 17512  df-0g 17530  df-gsum 17531  df-topgen 17532  df-pt 17533  df-prds 17536  df-xrs 17592  df-qtop 17597  df-imas 17598  df-xps 17600  df-mre 17674  df-mrc 17675  df-acs 17677  df-mgm 18734  df-sgrp 18823  df-mnd 18839  df-submnd 18893  df-mulg 19192  df-cntz 19445  df-cmn 19910  df-psmet 21578  df-xmet 21579  df-met 21580  df-bl 21581  df-mopn 21582  df-fbas 21583  df-fg 21584  df-cnfld 21587  df-top 23120  df-topon 23137  df-topsp 23159  df-bases 23172  df-cld 23245  df-ntr 23246  df-cls 23247  df-nei 23324  df-lp 23362  df-perf 23363  df-cn 23453  df-cnp 23454  df-haus 23541  df-tx 23789  df-hmeo 23982  df-fil 24073  df-fm 24165  df-flim 24166  df-flf 24167  df-xms 24547  df-ms 24548  df-tms 24549  df-cncf 25107  df-limc 26095  df-dv 26096  df-log 26791  df-cxp 26792  df-cht 27331  df-vma 27332  df-chp 27333

This theorem is used by: (None)