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Theorem chpub 27171

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 27075	. . . . 5 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) ∈ ℝ)
2	1	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (ψ‘𝐴) ∈ ℝ)
3		chtcl 27060	. . . . 5 ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) ∈ ℝ)
4	3	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (θ‘𝐴) ∈ ℝ)
5	2, 4	resubcld 11567	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((ψ‘𝐴) − (θ‘𝐴)) ∈ ℝ)
6		simpl 482	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ)
7		0red 11136	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ∈ ℝ)
8		1red 11134	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ)
9		0lt1 11661	. . . . . . . . . 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 11295	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 < 𝐴)
13	6, 12	elrpd 12972	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ⁺)
14	13	rpge0d 12979	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ≤ 𝐴)
15	6, 14	resqrtcld 15369	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (√‘𝐴) ∈ ℝ)
16		ppifi 27057	. . . . 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 26575	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (log‘𝐴) ∈ ℝ)
20	17, 19	fsumrecl 15685	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(log‘𝐴) ∈ ℝ)
21	13	relogcld 26575	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ)
22	15, 21	remulcld 11164	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((√‘𝐴) · (log‘𝐴)) ∈ ℝ)
23		ppifi 27057	. . . . . . 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 4136	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
27		prmnn 16632	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
28	26, 27	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
29	28	nnrpd 12973	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
30	29	relogcld 26575	. . . . . . . 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 12178	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
33		prmuz2 16654	. . . . . . . . . . . . 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 12859	. . . . . . . . . . . 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 26581	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ⁺)
38	31, 37	rerpdivcld 13006	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
39		reflcl 13744	. . . . . . . . 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 11164	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ ℝ)
42	41	recnd 11162	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ ℂ)
43	30	recnd 11162	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
44	24, 42, 43	fsumsub 15739	. . . . 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 12271	. . . . . . . . 9 ⊢ 0 ≤ 0
46	45	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ≤ 0)
47	8, 6, 6, 14, 11	lemul2ad 12085	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (𝐴 · 1) ≤ (𝐴 · 𝐴))
48	6	recnd 11162	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℂ)
49	48	sqsqrtd 15393	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((√‘𝐴)↑2) = 𝐴)
50	48	mulridd 11151	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (𝐴 · 1) = 𝐴)
51	49, 50	eqtr4d 2773	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((√‘𝐴)↑2) = (𝐴 · 1))
52	48	sqvald 14094	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (𝐴↑2) = (𝐴 · 𝐴))
53	47, 51, 52	3brtr4d 5106	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((√‘𝐴)↑2) ≤ (𝐴↑2))
54	6, 14	sqrtge0d 15372	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ≤ (√‘𝐴))
55	15, 6, 54, 14	le2sqd 14208	. . . . . . . . 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 13356	. . . . . . . 8 ⊢ (((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (0 ≤ 0 ∧ (√‘𝐴) ≤ 𝐴)) → (0[,](√‘𝐴)) ⊆ (0[,]𝐴))
58	7, 6, 46, 56, 57	syl22anc 839	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (0[,](√‘𝐴)) ⊆ (0[,]𝐴))
59	58	ssrind 4174	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((0[,](√‘𝐴)) ∩ ℙ) ⊆ ((0[,]𝐴) ∩ ℙ))
60	59	sselda 3917	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
61	41, 30	resubcld 11567	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) ∈ ℝ)
62	61	recnd 11162	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) ∈ ℂ)
63	60, 62	syldan 592	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) ∈ ℂ)
64		eldifi 4063	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
65	64, 43	sylan2 594	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (log‘𝑝) ∈ ℂ)
66	65	mullidd 11152	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (1 · (log‘𝑝)) = (log‘𝑝))
67	25	elin1d 4135	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (0[,]𝐴))
68		0re 11135	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
69	6	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ)
70		elicc2 13353	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
71	68, 69, 70	sylancr 588	. . . . . . . . . . . . . . . . 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 1145	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ 𝐴)
74	64, 73	sylan2 594	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝑝 ≤ 𝐴)
75	64, 29	sylan2 594	. . . . . . . . . . . . . . 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 26579	. . . . . . . . . . . . . 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 5096	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (1 · (log‘𝑝)) ≤ (log‘𝐴))
80		1red 11134	. . . . . . . . . . . . 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 594	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (log‘𝑝) ∈ ℝ⁺)
83	80, 81, 82	lemuldivd 13024	. . . . . . . . . . . 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 11162	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝐴 ∈ ℂ)
87	86	sqsqrtd 15393	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((√‘𝐴)↑2) = 𝐴)
88		eldifn 4064	. . . . . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝑝 ∈ ℙ)
91		elin 3901	. . . . . . . . . . . . . . . . . . . . . 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 11136	. . . . . . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝑝 ∈ ℕ)
97	96	nnred 12178	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝑝 ∈ ℝ)
98	75	rpge0d 12979	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 0 ≤ 𝑝)
99		elicc2 13353	. . . . . . . . . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . . . . . . . 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 11282	. . . . . . . . . . . . . . . . . 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 14207	. . . . . . . . . . . . . . . . 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 5098	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝐴 < (𝑝↑2))
112	96	nnsqcld 14195	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (𝑝↑2) ∈ ℕ)
113	112	nnrpd 12973	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (𝑝↑2) ∈ ℝ⁺)
114		logltb 26552	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑝↑2) ∈ ℝ⁺) → (𝐴 < (𝑝↑2) ↔ (log‘𝐴) < (log‘(𝑝↑2))))
115	76, 113, 114	syl2anc 585	. . . . . . . . . . . . . . 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 12548	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℤ
118		relogexp 26548	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (log‘(𝑝↑2)) = (2 · (log‘𝑝)))
119	75, 117, 118	sylancl 587	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (log‘(𝑝↑2)) = (2 · (log‘𝑝)))
120	116, 119	breqtrd 5100	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (log‘𝐴) < (2 · (log‘𝑝)))
121		2re 12244	. . . . . . . . . . . . . . 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 13027	. . . . . . . . . . . . 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 12233	. . . . . . . . . . . 12 ⊢ 2 = (1 + 1)
126	124, 125	breqtrdi 5115	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝐴) / (log‘𝑝)) < (1 + 1))
127	64, 38	sylan2 594	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
128		1z 12546	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
129		flbi 13764	. . . . . . . . . . . 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 587	. . . . . . . . . . 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 714	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (⌊‘((log‘𝐴) / (log‘𝑝))) = 1)
132	131	oveq2d 7372	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) = ((log‘𝑝) · 1))
133	65	mulridd 11151	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝑝) · 1) = (log‘𝑝))
134	132, 133	eqtrd 2770	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) = (log‘𝑝))
135	134	oveq1d 7371	. . . . . . 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 11482	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝑝) − (log‘𝑝)) = 0)
137	135, 136	eqtrd 2770	. . . . . 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 15676	. . . . 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 27169	. . . . . . 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 27061	. . . . . . 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 7374	. . . . 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 2781	. . . 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 592	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) ∈ ℝ)
146	60, 41	syldan 592	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ ℝ)
147	60, 37	syldan 592	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ⁺)
148	147	rpge0d 12979	. . . . . . 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 4136	. . . . . . . . . . 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 12973	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
153	152	relogcld 26575	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
154	146, 153	subge02d 11731	. . . . . . 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 592	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
157		flle 13747	. . . . . . . 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 592	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℝ)
160	159, 19, 147	lemuldiv2d 13025	. . . . . . 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 11292	. . . . 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 15751	. . . 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 5096	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((ψ‘𝐴) − (θ‘𝐴)) ≤ Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(log‘𝐴))
165	21	recnd 11162	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℂ)
166		fsumconst 15741	. . . . 5 ⊢ ((((0[,](√‘𝐴)) ∩ ℙ) ∈ Fin ∧ (log‘𝐴) ∈ ℂ) → Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(log‘𝐴) = ((♯‘((0[,](√‘𝐴)) ∩ ℙ)) · (log‘𝐴)))
167	17, 165, 166	syl2anc 585	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(log‘𝐴) = ((♯‘((0[,](√‘𝐴)) ∩ ℙ)) · (log‘𝐴)))
168		hashcl 14307	. . . . . . 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 12488	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (♯‘((0[,](√‘𝐴)) ∩ ℙ)) ∈ ℝ)
171		logge0 26557	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ≤ (log‘𝐴))
172		reflcl 13744	. . . . . . 7 ⊢ ((√‘𝐴) ∈ ℝ → (⌊‘(√‘𝐴)) ∈ ℝ)
173	15, 172	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (⌊‘(√‘𝐴)) ∈ ℝ)
174		fzfid 13924	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1...(⌊‘(√‘𝐴))) ∈ Fin)
175		ppisval 27055	. . . . . . . . . . 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 4167	. . . . . . . . . . 11 ⊢ ((2...(⌊‘(√‘𝐴))) ∩ ℙ) ⊆ (2...(⌊‘(√‘𝐴)))
178		2eluzge1 12821	. . . . . . . . . . . 12 ⊢ 2 ∈ (ℤ_≥‘1)
179		fzss1 13506	. . . . . . . . . . . 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 3928	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((2...(⌊‘(√‘𝐴))) ∩ ℙ) ⊆ (1...(⌊‘(√‘𝐴))))
182	176, 181	eqsstrd 3951	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((0[,](√‘𝐴)) ∩ ℙ) ⊆ (1...(⌊‘(√‘𝐴))))
183		ssdomg 8936	. . . . . . . . 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 14330	. . . . . . . . 9 ⊢ ((((0[,](√‘𝐴)) ∩ ℙ) ∈ Fin ∧ (1...(⌊‘(√‘𝐴))) ∈ Fin) → ((♯‘((0[,](√‘𝐴)) ∩ ℙ)) ≤ (♯‘(1...(⌊‘(√‘𝐴)))) ↔ ((0[,](√‘𝐴)) ∩ ℙ) ≼ (1...(⌊‘(√‘𝐴)))))
186	17, 174, 185	syl2anc 585	. . . . . . . 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 13768	. . . . . . . . 9 ⊢ (((√‘𝐴) ∈ ℝ ∧ 0 ≤ (√‘𝐴)) → (⌊‘(√‘𝐴)) ∈ ℕ₀)
189	15, 54, 188	syl2anc 585	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (⌊‘(√‘𝐴)) ∈ ℕ₀)
190		hashfz1 14297	. . . . . . . 8 ⊢ ((⌊‘(√‘𝐴)) ∈ ℕ₀ → (♯‘(1...(⌊‘(√‘𝐴)))) = (⌊‘(√‘𝐴)))
191	189, 190	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (♯‘(1...(⌊‘(√‘𝐴)))) = (⌊‘(√‘𝐴)))
192	187, 191	breqtrd 5100	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (♯‘((0[,](√‘𝐴)) ∩ ℙ)) ≤ (⌊‘(√‘𝐴)))
193		flle 13747	. . . . . . 7 ⊢ ((√‘𝐴) ∈ ℝ → (⌊‘(√‘𝐴)) ≤ (√‘𝐴))
194	15, 193	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (⌊‘(√‘𝐴)) ≤ (√‘𝐴))
195	170, 173, 15, 192, 194	letrd 11292	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (♯‘((0[,](√‘𝐴)) ∩ ℙ)) ≤ (√‘𝐴))
196	170, 15, 21, 171, 195	lemul1ad 12084	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((♯‘((0[,](√‘𝐴)) ∩ ℙ)) · (log‘𝐴)) ≤ ((√‘𝐴) · (log‘𝐴)))
197	167, 196	eqbrtrd 5096	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(log‘𝐴) ≤ ((√‘𝐴) · (log‘𝐴)))
198	5, 20, 22, 164, 197	letrd 11292	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((ψ‘𝐴) − (θ‘𝐴)) ≤ ((√‘𝐴) · (log‘𝐴)))
199	2, 4, 22	lesubadd2d 11738	. 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 1542 ∈ wcel 2114 ∖ cdif 3882 ∩ cin 3884 ⊆ wss 3885 class class class wbr 5074 ‘cfv 6487 (class class class)co 7356 ≼ cdom 8880 Fincfn 8882 ℂcc 11025 ℝcr 11026 0cc0 11027 1c1 11028 + caddc 11030 · cmul 11032 < clt 11168 ≤ cle 11169 − cmin 11366 / cdiv 11796 ℕcn 12163 2c2 12225 ℕ₀cn0 12426 ℤcz 12513 ℤ_≥cuz 12777 ℝ⁺crp 12931 [,]cicc 13290 ...cfz 13450 ⌊cfl 13738 ↑cexp 14012 ♯chash 14281 √csqrt 15184 Σcsu 15637 ℙcprime 16629 logclog 26506 θccht 27042 ψcchp 27044

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-oadd 8398 df-er 8632 df-map 8764 df-pm 8765 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9264 df-fi 9313 df-sup 9344 df-inf 9345 df-oi 9414 df-dju 9814 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-xnn0 12500 df-z 12514 df-dec 12634 df-uz 12778 df-q 12888 df-rp 12932 df-xneg 13052 df-xadd 13053 df-xmul 13054 df-ioo 13291 df-ioc 13292 df-ico 13293 df-icc 13294 df-fz 13451 df-fzo 13598 df-fl 13740 df-mod 13818 df-seq 13953 df-exp 14013 df-fac 14225 df-bc 14254 df-hash 14282 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15638 df-ef 16021 df-sin 16023 df-cos 16024 df-pi 16026 df-dvds 16211 df-gcd 16453 df-prm 16630 df-pc 16797 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18741 df-mulg 19033 df-cntz 19281 df-cmn 19746 df-psmet 21333 df-xmet 21334 df-met 21335 df-bl 21336 df-mopn 21337 df-fbas 21338 df-fg 21339 df-cnfld 21342 df-top 22847 df-topon 22864 df-topsp 22886 df-bases 22899 df-cld 22972 df-ntr 22973 df-cls 22974 df-nei 23051 df-lp 23089 df-perf 23090 df-cn 23180 df-cnp 23181 df-haus 23268 df-tx 23515 df-hmeo 23708 df-fil 23799 df-fm 23891 df-flim 23892 df-flf 23893 df-xms 24273 df-ms 24274 df-tms 24275 df-cncf 24833 df-limc 25821 df-dv 25822 df-log 26508 df-cht 27048 df-vma 27049 df-chp 27050

This theorem is referenced by: chpchtlim 27430

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