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Theorem chpub 27282

Description: An upper bound on the second Chebyshev function. (Contributed by Mario Carneiro, 8-Apr-2016.)

**Assertion**
Ref	Expression
chpub	⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (ψ‘𝐴) ≤ ((θ‘𝐴) + ((√‘𝐴) · (log‘𝐴))))

Proof of Theorem chpub Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		chpcl 27185	. . . . 5 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) ∈ ℝ)
2	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (ψ‘𝐴) ∈ ℝ)
3		chtcl 27170	. . . . 5 ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) ∈ ℝ)
4	3	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (θ‘𝐴) ∈ ℝ)
5	2, 4	resubcld 11718	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((ψ‘𝐴) − (θ‘𝐴)) ∈ ℝ)
6		simpl 482	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ)
7		0red 11293	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ∈ ℝ)
8		1red 11291	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ)
9		0lt1 11812	. . . . . . . . . 10 ⊢ 0 < 1
10	9	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 < 1)
11		simpr 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 1 ≤ 𝐴)
12	7, 8, 6, 10, 11	ltletrd 11450	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 < 𝐴)
13	6, 12	elrpd 13096	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ⁺)
14	13	rpge0d 13103	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ≤ 𝐴)
15	6, 14	resqrtcld 15466	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (√‘𝐴) ∈ ℝ)
16		ppifi 27167	. . . . 5 ⊢ ((√‘𝐴) ∈ ℝ → ((0[,](√‘𝐴)) ∩ ℙ) ∈ Fin)
17	15, 16	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((0[,](√‘𝐴)) ∩ ℙ) ∈ Fin)
18	13	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → 𝐴 ∈ ℝ⁺)
19	18	relogcld 26683	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (log‘𝐴) ∈ ℝ)
20	17, 19	fsumrecl 15782	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(log‘𝐴) ∈ ℝ)
21	13	relogcld 26683	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ)
22	15, 21	remulcld 11320	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((√‘𝐴) · (log‘𝐴)) ∈ ℝ)
23		ppifi 27167	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
24	23	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
25		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
26	25	elin2d 4228	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
27		prmnn 16721	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
28	26, 27	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
29	28	nnrpd 13097	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
30	29	relogcld 26683	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
31	21	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ)
32	28	nnred 12308	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
33		prmuz2 16743	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
34	26, 33	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (ℤ_≥‘2))
35		eluz2gt1 12985	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 1 < 𝑝)
36	34, 35	syl 17	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝑝)
37	32, 36	rplogcld 26689	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ⁺)
38	31, 37	rerpdivcld 13130	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
39		reflcl 13847	. . . . . . . . 9 ⊢ (((log‘𝐴) / (log‘𝑝)) ∈ ℝ → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℝ)
40	38, 39	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℝ)
41	30, 40	remulcld 11320	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ ℝ)
42	41	recnd 11318	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ ℂ)
43	30	recnd 11318	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
44	24, 42, 43	fsumsub 15836	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) = (Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)))
45		0le0 12394	. . . . . . . . 9 ⊢ 0 ≤ 0
46	45	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ≤ 0)
47	8, 6, 6, 14, 11	lemul2ad 12235	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (𝐴 · 1) ≤ (𝐴 · 𝐴))
48	6	recnd 11318	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℂ)
49	48	sqsqrtd 15488	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((√‘𝐴)↑2) = 𝐴)
50	48	mulridd 11307	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (𝐴 · 1) = 𝐴)
51	49, 50	eqtr4d 2783	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((√‘𝐴)↑2) = (𝐴 · 1))
52	48	sqvald 14193	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (𝐴↑2) = (𝐴 · 𝐴))
53	47, 51, 52	3brtr4d 5198	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((√‘𝐴)↑2) ≤ (𝐴↑2))
54	6, 14	sqrtge0d 15469	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ≤ (√‘𝐴))
55	15, 6, 54, 14	le2sqd 14306	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((√‘𝐴) ≤ 𝐴 ↔ ((√‘𝐴)↑2) ≤ (𝐴↑2)))
56	53, 55	mpbird 257	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (√‘𝐴) ≤ 𝐴)
57		iccss 13475	. . . . . . . 8 ⊢ (((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (0 ≤ 0 ∧ (√‘𝐴) ≤ 𝐴)) → (0[,](√‘𝐴)) ⊆ (0[,]𝐴))
58	7, 6, 46, 56, 57	syl22anc 838	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (0[,](√‘𝐴)) ⊆ (0[,]𝐴))
59	58	ssrind 4265	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((0[,](√‘𝐴)) ∩ ℙ) ⊆ ((0[,]𝐴) ∩ ℙ))
60	59	sselda 4008	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
61	41, 30	resubcld 11718	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) ∈ ℝ)
62	61	recnd 11318	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) ∈ ℂ)
63	60, 62	syldan 590	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) ∈ ℂ)
64		eldifi 4154	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
65	64, 43	sylan2 592	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (log‘𝑝) ∈ ℂ)
66	65	mullidd 11308	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (1 · (log‘𝑝)) = (log‘𝑝))
67	25	elin1d 4227	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (0[,]𝐴))
68		0re 11292	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
69	6	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ)
70		elicc2 13472	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
71	68, 69, 70	sylancr 586	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
72	67, 71	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴))
73	72	simp3d 1144	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ 𝐴)
74	64, 73	sylan2 592	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝑝 ≤ 𝐴)
75	64, 29	sylan2 592	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝑝 ∈ ℝ⁺)
76	13	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝐴 ∈ ℝ⁺)
77	75, 76	logled 26687	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (𝑝 ≤ 𝐴 ↔ (log‘𝑝) ≤ (log‘𝐴)))
78	74, 77	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (log‘𝑝) ≤ (log‘𝐴))
79	66, 78	eqbrtrd 5188	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (1 · (log‘𝑝)) ≤ (log‘𝐴))
80		1red 11291	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 1 ∈ ℝ)
81	21	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (log‘𝐴) ∈ ℝ)
82	64, 37	sylan2 592	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (log‘𝑝) ∈ ℝ⁺)
83	80, 81, 82	lemuldivd 13148	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((1 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 1 ≤ ((log‘𝐴) / (log‘𝑝))))
84	79, 83	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 1 ≤ ((log‘𝐴) / (log‘𝑝)))
85	6	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝐴 ∈ ℝ)
86	85	recnd 11318	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝐴 ∈ ℂ)
87	86	sqsqrtd 15488	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((√‘𝐴)↑2) = 𝐴)
88		eldifn 4155	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ)) → ¬ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ))
89	88	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ¬ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ))
90	64, 26	sylan2 592	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝑝 ∈ ℙ)
91		elin 3992	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ) ↔ (𝑝 ∈ (0[,](√‘𝐴)) ∧ 𝑝 ∈ ℙ))
92	91	rbaib 538	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ ℙ → (𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ) ↔ 𝑝 ∈ (0[,](√‘𝐴))))
93	90, 92	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ) ↔ 𝑝 ∈ (0[,](√‘𝐴))))
94		0red 11293	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 0 ∈ ℝ)
95	15	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (√‘𝐴) ∈ ℝ)
96	64, 28	sylan2 592	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝑝 ∈ ℕ)
97	96	nnred 12308	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝑝 ∈ ℝ)
98	75	rpge0d 13103	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 0 ≤ 𝑝)
99		elicc2 13472	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0 ∈ ℝ ∧ (√‘𝐴) ∈ ℝ) → (𝑝 ∈ (0[,](√‘𝐴)) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ (√‘𝐴))))
100		df-3an 1089	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ (√‘𝐴)) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (√‘𝐴)))
101	99, 100	bitrdi 287	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((0 ∈ ℝ ∧ (√‘𝐴) ∈ ℝ) → (𝑝 ∈ (0[,](√‘𝐴)) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (√‘𝐴))))
102	101	baibd 539	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((0 ∈ ℝ ∧ (√‘𝐴) ∈ ℝ) ∧ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝)) → (𝑝 ∈ (0[,](√‘𝐴)) ↔ 𝑝 ≤ (√‘𝐴)))
103	94, 95, 97, 98, 102	syl22anc 838	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (𝑝 ∈ (0[,](√‘𝐴)) ↔ 𝑝 ≤ (√‘𝐴)))
104	93, 103	bitrd 279	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ) ↔ 𝑝 ≤ (√‘𝐴)))
105	89, 104	mtbid 324	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ¬ 𝑝 ≤ (√‘𝐴))
106	95, 97	ltnled 11437	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((√‘𝐴) < 𝑝 ↔ ¬ 𝑝 ≤ (√‘𝐴)))
107	105, 106	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (√‘𝐴) < 𝑝)
108	54	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 0 ≤ (√‘𝐴))
109	95, 97, 108, 98	lt2sqd 14305	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((√‘𝐴) < 𝑝 ↔ ((√‘𝐴)↑2) < (𝑝↑2)))
110	107, 109	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((√‘𝐴)↑2) < (𝑝↑2))
111	87, 110	eqbrtrrd 5190	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝐴 < (𝑝↑2))
112	96	nnsqcld 14293	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (𝑝↑2) ∈ ℕ)
113	112	nnrpd 13097	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (𝑝↑2) ∈ ℝ⁺)
114		logltb 26660	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑝↑2) ∈ ℝ⁺) → (𝐴 < (𝑝↑2) ↔ (log‘𝐴) < (log‘(𝑝↑2))))
115	76, 113, 114	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (𝐴 < (𝑝↑2) ↔ (log‘𝐴) < (log‘(𝑝↑2))))
116	111, 115	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (log‘𝐴) < (log‘(𝑝↑2)))
117		2z 12675	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℤ
118		relogexp 26656	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (log‘(𝑝↑2)) = (2 · (log‘𝑝)))
119	75, 117, 118	sylancl 585	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (log‘(𝑝↑2)) = (2 · (log‘𝑝)))
120	116, 119	breqtrd 5192	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (log‘𝐴) < (2 · (log‘𝑝)))
121		2re 12367	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
122	121	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 2 ∈ ℝ)
123	81, 122, 82	ltdivmul2d 13151	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (((log‘𝐴) / (log‘𝑝)) < 2 ↔ (log‘𝐴) < (2 · (log‘𝑝))))
124	120, 123	mpbird 257	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝐴) / (log‘𝑝)) < 2)
125		df-2 12356	. . . . . . . . . . . 12 ⊢ 2 = (1 + 1)
126	124, 125	breqtrdi 5207	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝐴) / (log‘𝑝)) < (1 + 1))
127	64, 38	sylan2 592	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
128		1z 12673	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
129		flbi 13867	. . . . . . . . . . . 12 ⊢ ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 1 ∈ ℤ) → ((⌊‘((log‘𝐴) / (log‘𝑝))) = 1 ↔ (1 ≤ ((log‘𝐴) / (log‘𝑝)) ∧ ((log‘𝐴) / (log‘𝑝)) < (1 + 1))))
130	127, 128, 129	sylancl 585	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((⌊‘((log‘𝐴) / (log‘𝑝))) = 1 ↔ (1 ≤ ((log‘𝐴) / (log‘𝑝)) ∧ ((log‘𝐴) / (log‘𝑝)) < (1 + 1))))
131	84, 126, 130	mpbir2and 712	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (⌊‘((log‘𝐴) / (log‘𝑝))) = 1)
132	131	oveq2d 7464	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) = ((log‘𝑝) · 1))
133	65	mulridd 11307	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝑝) · 1) = (log‘𝑝))
134	132, 133	eqtrd 2780	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) = (log‘𝑝))
135	134	oveq1d 7463	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) = ((log‘𝑝) − (log‘𝑝)))
136	65	subidd 11635	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝑝) − (log‘𝑝)) = 0)
137	135, 136	eqtrd 2780	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) = 0)
138	59, 63, 137, 24	fsumss 15773	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)))
139		chpval2 27280	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
140	139	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (ψ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
141		chtval 27171	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝))
142	141	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝))
143	140, 142	oveq12d 7466	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((ψ‘𝐴) − (θ‘𝐴)) = (Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)))
144	44, 138, 143	3eqtr4rd 2791	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((ψ‘𝐴) − (θ‘𝐴)) = Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)))
145	60, 61	syldan 590	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) ∈ ℝ)
146	60, 41	syldan 590	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ ℝ)
147	60, 37	syldan 590	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ⁺)
148	147	rpge0d 13103	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → 0 ≤ (log‘𝑝))
149		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ))
150	149	elin2d 4228	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → 𝑝 ∈ ℙ)
151	150, 27	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → 𝑝 ∈ ℕ)
152	151	nnrpd 13097	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
153	152	relogcld 26683	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
154	146, 153	subge02d 11882	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (0 ≤ (log‘𝑝) ↔ (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) ≤ ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝))))))
155	148, 154	mpbid 232	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) ≤ ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
156	60, 38	syldan 590	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
157		flle 13850	. . . . . . . 8 ⊢ (((log‘𝐴) / (log‘𝑝)) ∈ ℝ → (⌊‘((log‘𝐴) / (log‘𝑝))) ≤ ((log‘𝐴) / (log‘𝑝)))
158	156, 157	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ≤ ((log‘𝐴) / (log‘𝑝)))
159	60, 40	syldan 590	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℝ)
160	159, 19, 147	lemuldiv2d 13149	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) ≤ (log‘𝐴) ↔ (⌊‘((log‘𝐴) / (log‘𝑝))) ≤ ((log‘𝐴) / (log‘𝑝))))
161	158, 160	mpbird 257	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) ≤ (log‘𝐴))
162	145, 146, 19, 155, 161	letrd 11447	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) ≤ (log‘𝐴))
163	17, 145, 19, 162	fsumle 15847	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) ≤ Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(log‘𝐴))
164	144, 163	eqbrtrd 5188	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((ψ‘𝐴) − (θ‘𝐴)) ≤ Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(log‘𝐴))
165	21	recnd 11318	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℂ)
166		fsumconst 15838	. . . . 5 ⊢ ((((0[,](√‘𝐴)) ∩ ℙ) ∈ Fin ∧ (log‘𝐴) ∈ ℂ) → Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(log‘𝐴) = ((♯‘((0[,](√‘𝐴)) ∩ ℙ)) · (log‘𝐴)))
167	17, 165, 166	syl2anc 583	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(log‘𝐴) = ((♯‘((0[,](√‘𝐴)) ∩ ℙ)) · (log‘𝐴)))
168		hashcl 14405	. . . . . . 7 ⊢ (((0[,](√‘𝐴)) ∩ ℙ) ∈ Fin → (♯‘((0[,](√‘𝐴)) ∩ ℙ)) ∈ ℕ₀)
169	17, 168	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (♯‘((0[,](√‘𝐴)) ∩ ℙ)) ∈ ℕ₀)
170	169	nn0red 12614	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (♯‘((0[,](√‘𝐴)) ∩ ℙ)) ∈ ℝ)
171		logge0 26665	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ≤ (log‘𝐴))
172		reflcl 13847	. . . . . . 7 ⊢ ((√‘𝐴) ∈ ℝ → (⌊‘(√‘𝐴)) ∈ ℝ)
173	15, 172	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (⌊‘(√‘𝐴)) ∈ ℝ)
174		fzfid 14024	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1...(⌊‘(√‘𝐴))) ∈ Fin)
175		ppisval 27165	. . . . . . . . . . 11 ⊢ ((√‘𝐴) ∈ ℝ → ((0[,](√‘𝐴)) ∩ ℙ) = ((2...(⌊‘(√‘𝐴))) ∩ ℙ))
176	15, 175	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((0[,](√‘𝐴)) ∩ ℙ) = ((2...(⌊‘(√‘𝐴))) ∩ ℙ))
177		inss1 4258	. . . . . . . . . . 11 ⊢ ((2...(⌊‘(√‘𝐴))) ∩ ℙ) ⊆ (2...(⌊‘(√‘𝐴)))
178		2eluzge1 12959	. . . . . . . . . . . 12 ⊢ 2 ∈ (ℤ_≥‘1)
179		fzss1 13623	. . . . . . . . . . . 12 ⊢ (2 ∈ (ℤ_≥‘1) → (2...(⌊‘(√‘𝐴))) ⊆ (1...(⌊‘(√‘𝐴))))
180	178, 179	mp1i 13	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (2...(⌊‘(√‘𝐴))) ⊆ (1...(⌊‘(√‘𝐴))))
181	177, 180	sstrid 4020	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((2...(⌊‘(√‘𝐴))) ∩ ℙ) ⊆ (1...(⌊‘(√‘𝐴))))
182	176, 181	eqsstrd 4047	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((0[,](√‘𝐴)) ∩ ℙ) ⊆ (1...(⌊‘(√‘𝐴))))
183		ssdomg 9060	. . . . . . . . 9 ⊢ ((1...(⌊‘(√‘𝐴))) ∈ Fin → (((0[,](√‘𝐴)) ∩ ℙ) ⊆ (1...(⌊‘(√‘𝐴))) → ((0[,](√‘𝐴)) ∩ ℙ) ≼ (1...(⌊‘(√‘𝐴)))))
184	174, 182, 183	sylc 65	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((0[,](√‘𝐴)) ∩ ℙ) ≼ (1...(⌊‘(√‘𝐴))))
185		hashdom 14428	. . . . . . . . 9 ⊢ ((((0[,](√‘𝐴)) ∩ ℙ) ∈ Fin ∧ (1...(⌊‘(√‘𝐴))) ∈ Fin) → ((♯‘((0[,](√‘𝐴)) ∩ ℙ)) ≤ (♯‘(1...(⌊‘(√‘𝐴)))) ↔ ((0[,](√‘𝐴)) ∩ ℙ) ≼ (1...(⌊‘(√‘𝐴)))))
186	17, 174, 185	syl2anc 583	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((♯‘((0[,](√‘𝐴)) ∩ ℙ)) ≤ (♯‘(1...(⌊‘(√‘𝐴)))) ↔ ((0[,](√‘𝐴)) ∩ ℙ) ≼ (1...(⌊‘(√‘𝐴)))))
187	184, 186	mpbird 257	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (♯‘((0[,](√‘𝐴)) ∩ ℙ)) ≤ (♯‘(1...(⌊‘(√‘𝐴)))))
188		flge0nn0 13871	. . . . . . . . 9 ⊢ (((√‘𝐴) ∈ ℝ ∧ 0 ≤ (√‘𝐴)) → (⌊‘(√‘𝐴)) ∈ ℕ₀)
189	15, 54, 188	syl2anc 583	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (⌊‘(√‘𝐴)) ∈ ℕ₀)
190		hashfz1 14395	. . . . . . . 8 ⊢ ((⌊‘(√‘𝐴)) ∈ ℕ₀ → (♯‘(1...(⌊‘(√‘𝐴)))) = (⌊‘(√‘𝐴)))
191	189, 190	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (♯‘(1...(⌊‘(√‘𝐴)))) = (⌊‘(√‘𝐴)))
192	187, 191	breqtrd 5192	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (♯‘((0[,](√‘𝐴)) ∩ ℙ)) ≤ (⌊‘(√‘𝐴)))
193		flle 13850	. . . . . . 7 ⊢ ((√‘𝐴) ∈ ℝ → (⌊‘(√‘𝐴)) ≤ (√‘𝐴))
194	15, 193	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (⌊‘(√‘𝐴)) ≤ (√‘𝐴))
195	170, 173, 15, 192, 194	letrd 11447	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (♯‘((0[,](√‘𝐴)) ∩ ℙ)) ≤ (√‘𝐴))
196	170, 15, 21, 171, 195	lemul1ad 12234	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((♯‘((0[,](√‘𝐴)) ∩ ℙ)) · (log‘𝐴)) ≤ ((√‘𝐴) · (log‘𝐴)))
197	167, 196	eqbrtrd 5188	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(log‘𝐴) ≤ ((√‘𝐴) · (log‘𝐴)))
198	5, 20, 22, 164, 197	letrd 11447	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((ψ‘𝐴) − (θ‘𝐴)) ≤ ((√‘𝐴) · (log‘𝐴)))
199	2, 4, 22	lesubadd2d 11889	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (((ψ‘𝐴) − (θ‘𝐴)) ≤ ((√‘𝐴) · (log‘𝐴)) ↔ (ψ‘𝐴) ≤ ((θ‘𝐴) + ((√‘𝐴) · (log‘𝐴)))))
200	198, 199	mpbid 232	1 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (ψ‘𝐴) ≤ ((θ‘𝐴) + ((√‘𝐴) · (log‘𝐴))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∖ cdif 3973 ∩ cin 3975 ⊆ wss 3976 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ≼ cdom 9001 Fincfn 9003 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 ℝ⁺crp 13057 [,]cicc 13410 ...cfz 13567 ⌊cfl 13841 ↑cexp 14112 ♯chash 14379 √csqrt 15282 Σcsu 15734 ℙcprime 16718 logclog 26614 θccht 27152 ψcchp 27154

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-dvds 16303 df-gcd 16541 df-prm 16719 df-pc 16884 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922 df-log 26616 df-cht 27158 df-vma 27159 df-chp 27160

This theorem is referenced by: chpchtlim 27541

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