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Theorem chpub 25482

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 25387	. . . . 5 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) ∈ ℝ)
2	1	adantr 481	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (ψ‘𝐴) ∈ ℝ)
3		chtcl 25372	. . . . 5 ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) ∈ ℝ)
4	3	adantr 481	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (θ‘𝐴) ∈ ℝ)
5	2, 4	resubcld 10922	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((ψ‘𝐴) − (θ‘𝐴)) ∈ ℝ)
6		simpl 483	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ)
7		0red 10497	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ∈ ℝ)
8		1red 10495	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 1 ∈ ℝ)
9		0lt1 11016	. . . . . . . . . 10 ⊢ 0 < 1
10	9	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 < 1)
11		simpr 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 1 ≤ 𝐴)
12	7, 8, 6, 10, 11	ltletrd 10653	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 < 𝐴)
13	6, 12	elrpd 12282	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℝ⁺)
14	13	rpge0d 12289	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ≤ 𝐴)
15	6, 14	resqrtcld 14615	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (√‘𝐴) ∈ ℝ)
16		ppifi 25369	. . . . 5 ⊢ ((√‘𝐴) ∈ ℝ → ((0[,](√‘𝐴)) ∩ ℙ) ∈ Fin)
17	15, 16	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((0[,](√‘𝐴)) ∩ ℙ) ∈ Fin)
18	13	adantr 481	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → 𝐴 ∈ ℝ⁺)
19	18	relogcld 24891	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (log‘𝐴) ∈ ℝ)
20	17, 19	fsumrecl 14928	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(log‘𝐴) ∈ ℝ)
21	13	relogcld 24891	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℝ)
22	15, 21	remulcld 10524	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((√‘𝐴) · (log‘𝐴)) ∈ ℝ)
23		ppifi 25369	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
24	23	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
25		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
26	25	elin2d 4103	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
27		prmnn 15851	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
28	26, 27	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
29	28	nnrpd 12283	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
30	29	relogcld 24891	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
31	21	adantr 481	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ)
32	28	nnred 11507	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
33		prmuz2 15873	. . . . . . . . . . . . 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 12173	. . . . . . . . . . . 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 24897	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ⁺)
38	31, 37	rerpdivcld 12316	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
39		reflcl 13020	. . . . . . . . 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 10524	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ ℝ)
42	41	recnd 10522	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ ℂ)
43	30	recnd 10522	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
44	24, 42, 43	fsumsub 14980	. . . . 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 11592	. . . . . . . . 9 ⊢ 0 ≤ 0
46	45	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ≤ 0)
47	8, 6, 6, 14, 11	lemul2ad 11434	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (𝐴 · 1) ≤ (𝐴 · 𝐴))
48	6	recnd 10522	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 𝐴 ∈ ℂ)
49	48	sqsqrtd 14637	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((√‘𝐴)↑2) = 𝐴)
50	48	mulid1d 10511	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (𝐴 · 1) = 𝐴)
51	49, 50	eqtr4d 2836	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((√‘𝐴)↑2) = (𝐴 · 1))
52	48	sqvald 13361	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (𝐴↑2) = (𝐴 · 𝐴))
53	47, 51, 52	3brtr4d 5000	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((√‘𝐴)↑2) ≤ (𝐴↑2))
54	6, 14	sqrtge0d 14618	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ≤ (√‘𝐴))
55	15, 6, 54, 14	le2sqd 13474	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((√‘𝐴) ≤ 𝐴 ↔ ((√‘𝐴)↑2) ≤ (𝐴↑2)))
56	53, 55	mpbird 258	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (√‘𝐴) ≤ 𝐴)
57		iccss 12658	. . . . . . . 8 ⊢ (((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (0 ≤ 0 ∧ (√‘𝐴) ≤ 𝐴)) → (0[,](√‘𝐴)) ⊆ (0[,]𝐴))
58	7, 6, 46, 56, 57	syl22anc 835	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (0[,](√‘𝐴)) ⊆ (0[,]𝐴))
59	58	ssrind 4138	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((0[,](√‘𝐴)) ∩ ℙ) ⊆ ((0[,]𝐴) ∩ ℙ))
60	59	sselda 3895	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
61	41, 30	resubcld 10922	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) ∈ ℝ)
62	61	recnd 10522	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) ∈ ℂ)
63	60, 62	syldan 591	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) ∈ ℂ)
64		eldifi 4030	. . . . . . . . . . . . . . 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	mulid2d 10512	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (1 · (log‘𝑝)) = (log‘𝑝))
67	25	elin1d 4102	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (0[,]𝐴))
68		0re 10496	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
69	6	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ)
70		elicc2 12655	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
71	68, 69, 70	sylancr 587	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
72	67, 71	mpbid 233	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴))
73	72	simp3d 1137	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝐴 ∈ ℝ⁺)
77	75, 76	logled 24895	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (𝑝 ≤ 𝐴 ↔ (log‘𝑝) ≤ (log‘𝐴)))
78	74, 77	mpbid 233	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (log‘𝑝) ≤ (log‘𝐴))
79	66, 78	eqbrtrd 4990	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (1 · (log‘𝑝)) ≤ (log‘𝐴))
80		1red 10495	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 1 ∈ ℝ)
81	21	adantr 481	. . . . . . . . . . . . 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 12334	. . . . . . . . . . . 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 233	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 1 ≤ ((log‘𝐴) / (log‘𝑝)))
85	6	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝐴 ∈ ℝ)
86	85	recnd 10522	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝐴 ∈ ℂ)
87	86	sqsqrtd 14637	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((√‘𝐴)↑2) = 𝐴)
88		eldifn 4031	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ)) → ¬ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ))
89	88	adantl 482	. . . . . . . . . . . . . . . . . . 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 4096	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ) ↔ (𝑝 ∈ (0[,](√‘𝐴)) ∧ 𝑝 ∈ ℙ))
92	91	rbaib 539	. . . . . . . . . . . . . . . . . . . . 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 10497	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 0 ∈ ℝ)
95	15	adantr 481	. . . . . . . . . . . . . . . . . . . . 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 11507	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝑝 ∈ ℝ)
98	75	rpge0d 12289	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 0 ≤ 𝑝)
99		elicc2 12655	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0 ∈ ℝ ∧ (√‘𝐴) ∈ ℝ) → (𝑝 ∈ (0[,](√‘𝐴)) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ (√‘𝐴))))
100		df-3an 1082	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ (√‘𝐴)) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (√‘𝐴)))
101	99, 100	syl6bb 288	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((0 ∈ ℝ ∧ (√‘𝐴) ∈ ℝ) → (𝑝 ∈ (0[,](√‘𝐴)) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (√‘𝐴))))
102	101	baibd 540	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((0 ∈ ℝ ∧ (√‘𝐴) ∈ ℝ) ∧ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝)) → (𝑝 ∈ (0[,](√‘𝐴)) ↔ 𝑝 ≤ (√‘𝐴)))
103	94, 95, 97, 98, 102	syl22anc 835	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (𝑝 ∈ (0[,](√‘𝐴)) ↔ 𝑝 ≤ (√‘𝐴)))
104	93, 103	bitrd 280	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ) ↔ 𝑝 ≤ (√‘𝐴)))
105	89, 104	mtbid 325	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ¬ 𝑝 ≤ (√‘𝐴))
106	95, 97	ltnled 10640	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((√‘𝐴) < 𝑝 ↔ ¬ 𝑝 ≤ (√‘𝐴)))
107	105, 106	mpbird 258	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (√‘𝐴) < 𝑝)
108	54	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 0 ≤ (√‘𝐴))
109	95, 97, 108, 98	lt2sqd 13473	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((√‘𝐴) < 𝑝 ↔ ((√‘𝐴)↑2) < (𝑝↑2)))
110	107, 109	mpbid 233	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((√‘𝐴)↑2) < (𝑝↑2))
111	87, 110	eqbrtrrd 4992	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → 𝐴 < (𝑝↑2))
112	96	nnsqcld 13459	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (𝑝↑2) ∈ ℕ)
113	112	nnrpd 12283	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (𝑝↑2) ∈ ℝ⁺)
114		logltb 24868	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (𝑝↑2) ∈ ℝ⁺) → (𝐴 < (𝑝↑2) ↔ (log‘𝐴) < (log‘(𝑝↑2))))
115	76, 113, 114	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (𝐴 < (𝑝↑2) ↔ (log‘𝐴) < (log‘(𝑝↑2))))
116	111, 115	mpbid 233	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (log‘𝐴) < (log‘(𝑝↑2)))
117		2z 11868	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℤ
118		relogexp 24864	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (log‘(𝑝↑2)) = (2 · (log‘𝑝)))
119	75, 117, 118	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (log‘(𝑝↑2)) = (2 · (log‘𝑝)))
120	116, 119	breqtrd 4994	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (log‘𝐴) < (2 · (log‘𝑝)))
121		2re 11565	. . . . . . . . . . . . . . 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 12337	. . . . . . . . . . . . 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 258	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝐴) / (log‘𝑝)) < 2)
125		df-2 11554	. . . . . . . . . . . 12 ⊢ 2 = (1 + 1)
126	124, 125	syl6breq 5009	. . . . . . . . . . 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 11866	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
129		flbi 13040	. . . . . . . . . . . 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 586	. . . . . . . . . . 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 709	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → (⌊‘((log‘𝐴) / (log‘𝑝))) = 1)
132	131	oveq2d 7039	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) = ((log‘𝑝) · 1))
133	65	mulid1d 10511	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝑝) · 1) = (log‘𝑝))
134	132, 133	eqtrd 2833	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) = (log‘𝑝))
135	134	oveq1d 7038	. . . . . . 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 10839	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ (((0[,]𝐴) ∩ ℙ) ∖ ((0[,](√‘𝐴)) ∩ ℙ))) → ((log‘𝑝) − (log‘𝑝)) = 0)
137	135, 136	eqtrd 2833	. . . . . 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 14919	. . . . 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 25480	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
140	139	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (ψ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))))
141		chtval 25373	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝))
142	141	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝))
143	140, 142	oveq12d 7041	. . . . 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 2844	. . . 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 591	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) − (log‘𝑝)) ∈ ℝ)
146	60, 41	syldan 591	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ ℝ)
147	60, 37	syldan 591	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ⁺)
148	147	rpge0d 12289	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → 0 ≤ (log‘𝑝))
149		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ))
150	149	elin2d 4103	. . . . . . . . . . 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 12283	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
153	152	relogcld 24891	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
154	146, 153	subge02d 11086	. . . . . . 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 233	. . . . . 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 591	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
157		flle 13023	. . . . . . . 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 591	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℝ)
160	159, 19, 147	lemuldiv2d 12335	. . . . . . 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 258	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)) → ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝)))) ≤ (log‘𝐴))
162	145, 146, 19, 155, 161	letrd 10650	. . . . 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 14991	. . . 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 4990	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((ψ‘𝐴) − (θ‘𝐴)) ≤ Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(log‘𝐴))
165	21	recnd 10522	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (log‘𝐴) ∈ ℂ)
166		fsumconst 14982	. . . . 5 ⊢ ((((0[,](√‘𝐴)) ∩ ℙ) ∈ Fin ∧ (log‘𝐴) ∈ ℂ) → Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(log‘𝐴) = ((♯‘((0[,](√‘𝐴)) ∩ ℙ)) · (log‘𝐴)))
167	17, 165, 166	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(log‘𝐴) = ((♯‘((0[,](√‘𝐴)) ∩ ℙ)) · (log‘𝐴)))
168		hashcl 13571	. . . . . . 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 11810	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (♯‘((0[,](√‘𝐴)) ∩ ℙ)) ∈ ℝ)
171		logge0 24873	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ≤ (log‘𝐴))
172		reflcl 13020	. . . . . . 7 ⊢ ((√‘𝐴) ∈ ℝ → (⌊‘(√‘𝐴)) ∈ ℝ)
173	15, 172	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (⌊‘(√‘𝐴)) ∈ ℝ)
174		fzfid 13195	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (1...(⌊‘(√‘𝐴))) ∈ Fin)
175		ppisval 25367	. . . . . . . . . . 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 4131	. . . . . . . . . . 11 ⊢ ((2...(⌊‘(√‘𝐴))) ∩ ℙ) ⊆ (2...(⌊‘(√‘𝐴)))
178		2eluzge1 12147	. . . . . . . . . . . 12 ⊢ 2 ∈ (ℤ_≥‘1)
179		fzss1 12800	. . . . . . . . . . . 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 3906	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((2...(⌊‘(√‘𝐴))) ∩ ℙ) ⊆ (1...(⌊‘(√‘𝐴))))
182	176, 181	eqsstrd 3932	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((0[,](√‘𝐴)) ∩ ℙ) ⊆ (1...(⌊‘(√‘𝐴))))
183		ssdomg 8410	. . . . . . . . 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 13592	. . . . . . . . 9 ⊢ ((((0[,](√‘𝐴)) ∩ ℙ) ∈ Fin ∧ (1...(⌊‘(√‘𝐴))) ∈ Fin) → ((♯‘((0[,](√‘𝐴)) ∩ ℙ)) ≤ (♯‘(1...(⌊‘(√‘𝐴)))) ↔ ((0[,](√‘𝐴)) ∩ ℙ) ≼ (1...(⌊‘(√‘𝐴)))))
186	17, 174, 185	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((♯‘((0[,](√‘𝐴)) ∩ ℙ)) ≤ (♯‘(1...(⌊‘(√‘𝐴)))) ↔ ((0[,](√‘𝐴)) ∩ ℙ) ≼ (1...(⌊‘(√‘𝐴)))))
187	184, 186	mpbird 258	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (♯‘((0[,](√‘𝐴)) ∩ ℙ)) ≤ (♯‘(1...(⌊‘(√‘𝐴)))))
188		flge0nn0 13044	. . . . . . . . 9 ⊢ (((√‘𝐴) ∈ ℝ ∧ 0 ≤ (√‘𝐴)) → (⌊‘(√‘𝐴)) ∈ ℕ₀)
189	15, 54, 188	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (⌊‘(√‘𝐴)) ∈ ℕ₀)
190		hashfz1 13560	. . . . . . . 8 ⊢ ((⌊‘(√‘𝐴)) ∈ ℕ₀ → (♯‘(1...(⌊‘(√‘𝐴)))) = (⌊‘(√‘𝐴)))
191	189, 190	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (♯‘(1...(⌊‘(√‘𝐴)))) = (⌊‘(√‘𝐴)))
192	187, 191	breqtrd 4994	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (♯‘((0[,](√‘𝐴)) ∩ ℙ)) ≤ (⌊‘(√‘𝐴)))
193		flle 13023	. . . . . . 7 ⊢ ((√‘𝐴) ∈ ℝ → (⌊‘(√‘𝐴)) ≤ (√‘𝐴))
194	15, 193	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (⌊‘(√‘𝐴)) ≤ (√‘𝐴))
195	170, 173, 15, 192, 194	letrd 10650	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (♯‘((0[,](√‘𝐴)) ∩ ℙ)) ≤ (√‘𝐴))
196	170, 15, 21, 171, 195	lemul1ad 11433	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((♯‘((0[,](√‘𝐴)) ∩ ℙ)) · (log‘𝐴)) ≤ ((√‘𝐴) · (log‘𝐴)))
197	167, 196	eqbrtrd 4990	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑝 ∈ ((0[,](√‘𝐴)) ∩ ℙ)(log‘𝐴) ≤ ((√‘𝐴) · (log‘𝐴)))
198	5, 20, 22, 164, 197	letrd 10650	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → ((ψ‘𝐴) − (θ‘𝐴)) ≤ ((√‘𝐴) · (log‘𝐴)))
199	2, 4, 22	lesubadd2d 11093	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (((ψ‘𝐴) − (θ‘𝐴)) ≤ ((√‘𝐴) · (log‘𝐴)) ↔ (ψ‘𝐴) ≤ ((θ‘𝐴) + ((√‘𝐴) · (log‘𝐴)))))
200	198, 199	mpbid 233	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 207 ∧ wa 396 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 ∖ cdif 3862 ∩ cin 3864 ⊆ wss 3865 class class class wbr 4968 ‘cfv 6232 (class class class)co 7023 ≼ cdom 8362 Fincfn 8364 ℂcc 10388 ℝcr 10389 0cc0 10390 1c1 10391 + caddc 10393 · cmul 10395 < clt 10528 ≤ cle 10529 − cmin 10723 / cdiv 11151 ℕcn 11492 2c2 11546 ℕ₀cn0 11751 ℤcz 11835 ℤ_≥cuz 12097 ℝ⁺crp 12243 [,]cicc 12595 ...cfz 12746 ⌊cfl 13014 ↑cexp 13283 ♯chash 13544 √csqrt 14430 Σcsu 14880 ℙcprime 15848 logclog 24823 θccht 25354 ψcchp 25356

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-inf2 8957 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 ax-addf 10469 ax-mulf 10470

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-fal 1538 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-of 7274 df-om 7444 df-1st 7552 df-2nd 7553 df-supp 7689 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-2o 7961 df-oadd 7964 df-er 8146 df-map 8265 df-pm 8266 df-ixp 8318 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-fsupp 8687 df-fi 8728 df-sup 8759 df-inf 8760 df-oi 8827 df-dju 9183 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-xnn0 11822 df-z 11836 df-dec 11953 df-uz 12098 df-q 12202 df-rp 12244 df-xneg 12361 df-xadd 12362 df-xmul 12363 df-ioo 12596 df-ioc 12597 df-ico 12598 df-icc 12599 df-fz 12747 df-fzo 12888 df-fl 13016 df-mod 13092 df-seq 13224 df-exp 13284 df-fac 13488 df-bc 13517 df-hash 13545 df-shft 14264 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-limsup 14666 df-clim 14683 df-rlim 14684 df-sum 14881 df-ef 15258 df-sin 15260 df-cos 15261 df-pi 15263 df-dvds 15445 df-gcd 15681 df-prm 15849 df-pc 16007 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-starv 16413 df-sca 16414 df-vsca 16415 df-ip 16416 df-tset 16417 df-ple 16418 df-ds 16420 df-unif 16421 df-hom 16422 df-cco 16423 df-rest 16529 df-topn 16530 df-0g 16548 df-gsum 16549 df-topgen 16550 df-pt 16551 df-prds 16554 df-xrs 16608 df-qtop 16613 df-imas 16614 df-xps 16616 df-mre 16690 df-mrc 16691 df-acs 16693 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-submnd 17779 df-mulg 17986 df-cntz 18192 df-cmn 18639 df-psmet 20223 df-xmet 20224 df-met 20225 df-bl 20226 df-mopn 20227 df-fbas 20228 df-fg 20229 df-cnfld 20232 df-top 21190 df-topon 21207 df-topsp 21229 df-bases 21242 df-cld 21315 df-ntr 21316 df-cls 21317 df-nei 21394 df-lp 21432 df-perf 21433 df-cn 21523 df-cnp 21524 df-haus 21611 df-tx 21858 df-hmeo 22051 df-fil 22142 df-fm 22234 df-flim 22235 df-flf 22236 df-xms 22617 df-ms 22618 df-tms 22619 df-cncf 23173 df-limc 24151 df-dv 24152 df-log 24825 df-cht 25360 df-vma 25361 df-chp 25362

This theorem is referenced by: chpchtlim 25741

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