aks6d1c7lem1 - Mathbox for metakunt

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Theorem aks6d1c7lem1 42633

Description: The last set of inequalities of Claim 7 of Theorem 6.1 https://www3.nd.edu/%7eandyp/notes/AKS.pdf. (Contributed by metakunt, 12-May-2025.)

**Hypotheses**
Ref	Expression
aks6d1c7lem1.1	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c7lem1.2	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks6d1c7lem1.3	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
aks6d1c7lem1.4	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks6d1c7lem1.5	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c7lem1.6	⊢ 𝐸 = (𝑘 ∈ ℕ₀, 𝑙 ∈ ℕ₀ ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
aks6d1c7lem1.7	⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
aks6d1c7lem1.8	⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))
aks6d1c7lem1.9	⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
aks6d1c7lem1.10	⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))

**Assertion**
Ref	Expression
aks6d1c7lem1	⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < ((𝐷 + 𝐴)C(𝐷 − 1)))

Distinct variable groups: 𝑘,𝑁,𝑙 𝑃,𝑘,𝑙 𝜑,𝑘,𝑙 Allowed substitution hints: 𝐴(𝑘,𝑙) 𝐷(𝑘,𝑙) 𝑅(𝑘,𝑙) 𝐸(𝑘,𝑙) 𝐿(𝑘,𝑙)

Proof of Theorem aks6d1c7lem1 Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aks6d1c7lem1.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
2		eluzelz 12789	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
3	1, 2	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
4		0red 11138	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
5		3re 12252	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
6	5	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 3 ∈ ℝ)
7	3	zred 12624	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
8		3pos 12277	. . . . . . . . . . 11 ⊢ 0 < 3
9	8	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 3)
10		eluzle 12792	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
11	1, 10	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁)
12	4, 6, 7, 9, 11	ltletrd 11297	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝑁)
13	3, 12	jca 511	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
14		elnnz 12525	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
15	13, 14	sylibr 234	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
16	15	nnred 12180	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17		aks6d1c7lem1.8	. . . . . . . . . . . . 13 ⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
19		aks6d1c7lem1.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℙ)
20		aks6d1c7lem1.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
21		aks6d1c7lem1.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℕ)
22		aks6d1c7lem1.5	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
23		aks6d1c7lem1.6	. . . . . . . . . . . . 13 ⊢ 𝐸 = (𝑘 ∈ ℕ₀, 𝑙 ∈ ℕ₀ ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
24		aks6d1c7lem1.7	. . . . . . . . . . . . 13 ⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
25		eqid 2737	. . . . . . . . . . . . 13 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
26	15, 19, 20, 21, 22, 23, 24, 25	hashscontpowcl 42573	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
27	18, 26	eqeltrd 2837	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ ℕ₀)
28	27	nn0red 12490	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ ℝ)
29	27	nn0ge0d 12492	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝐷)
30	28, 29	resqrtcld 15371	. . . . . . . . 9 ⊢ (𝜑 → (√‘𝐷) ∈ ℝ)
31	30	flcld 13748	. . . . . . . 8 ⊢ (𝜑 → (⌊‘(√‘𝐷)) ∈ ℤ)
32	28, 29	sqrtge0d 15374	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (√‘𝐷))
33		0zd 12527	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
34		flge 13755	. . . . . . . . . 10 ⊢ (((√‘𝐷) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ (√‘𝐷) ↔ 0 ≤ (⌊‘(√‘𝐷))))
35	30, 33, 34	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (0 ≤ (√‘𝐷) ↔ 0 ≤ (⌊‘(√‘𝐷))))
36	32, 35	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (⌊‘(√‘𝐷)))
37	31, 36	jca 511	. . . . . . 7 ⊢ (𝜑 → ((⌊‘(√‘𝐷)) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘𝐷))))
38		elnn0z 12528	. . . . . . 7 ⊢ ((⌊‘(√‘𝐷)) ∈ ℕ₀ ↔ ((⌊‘(√‘𝐷)) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘𝐷))))
39	37, 38	sylibr 234	. . . . . 6 ⊢ (𝜑 → (⌊‘(√‘𝐷)) ∈ ℕ₀)
40	16, 39	reexpcld 14116	. . . . 5 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ∈ ℝ)
41		2re 12246	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
42	41	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℝ)
43		2pos 12275	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
44	43	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 2)
45	15	nngt0d 12217	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑁)
46		1ne2 12375	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 2
47	46	necomi 2987	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 1
48	47	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≠ 1)
49	42, 44, 16, 45, 48	relogbcld 42427	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
50	18, 28	eqeltrrd 2838	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ)
51	29, 18	breqtrd 5112	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
52	50, 51	resqrtcld 15371	. . . . . . . . . . . . 13 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ)
53	49, 52	remulcld 11166	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℝ)
54	53	flcld 13748	. . . . . . . . . . 11 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℤ)
55		1red 11136	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ)
56		0le1 11664	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 1
57	56	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ 1)
58	42	recnd 11164	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ∈ ℂ)
59	4, 44	gtned 11272	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ≠ 0)
60		logbid1 26745	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 log_b 2) = 1)
61	58, 59, 48, 60	syl3anc 1374	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2 log_b 2) = 1)
62	61	eqcomd 2743	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 = (2 log_b 2))
63		2z 12550	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℤ
64	63	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ∈ ℤ)
65	42	leidd 11707	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≤ 2)
66		1nn0 12444	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℕ₀
67	41, 66	nn0addge1i 12476	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ≤ (2 + 1)
68		2p1e3 12309	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 + 1) = 3
69	67, 68	breqtri 5111	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≤ 3
70	69	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ≤ 3)
71	42, 6, 7, 70, 11	letrd 11294	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≤ 𝑁)
72	64, 65, 42, 44, 7, 12, 71	logblebd 42430	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 2) ≤ (2 log_b 𝑁))
73	62, 72	eqbrtrd 5108	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ≤ (2 log_b 𝑁))
74	4, 55, 49, 57, 73	letrd 11294	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (2 log_b 𝑁))
75	50, 51	sqrtge0d 15374	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
76	49, 52, 74, 75	mulge0d 11718	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
77		flge 13755	. . . . . . . . . . . . 13 ⊢ ((((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ↔ 0 ≤ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
78	53, 33, 77	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ≤ ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ↔ 0 ≤ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
79	76, 78	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
80	54, 79	jca 511	. . . . . . . . . 10 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℤ ∧ 0 ≤ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
81		elnn0z 12528	. . . . . . . . . 10 ⊢ ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℕ₀ ↔ ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℤ ∧ 0 ≤ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
82	80, 81	sylibr 234	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℕ₀)
83	66	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℕ₀)
84	82, 83	nn0addcld 12493	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℕ₀)
85	21	phicld 16733	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ)
86	85	nnred 12180	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℝ)
87	85	nnnn0d 12489	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ₀)
88	87	nn0ge0d 12492	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (ϕ‘𝑅))
89	86, 88	resqrtcld 15371	. . . . . . . . . . . 12 ⊢ (𝜑 → (√‘(ϕ‘𝑅)) ∈ ℝ)
90	89, 49	remulcld 11166	. . . . . . . . . . 11 ⊢ (𝜑 → ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ∈ ℝ)
91	90	flcld 13748	. . . . . . . . . 10 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℤ)
92	86, 88	sqrtge0d 15374	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (√‘(ϕ‘𝑅)))
93	89, 49, 92, 74	mulge0d 11718	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
94		flge 13755	. . . . . . . . . . . 12 ⊢ ((((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ↔ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))))
95	90, 33, 94	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → (0 ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ↔ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))))
96	93, 95	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))
97	91, 96	jca 511	. . . . . . . . 9 ⊢ (𝜑 → ((⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))))
98		elnn0z 12528	. . . . . . . . 9 ⊢ ((⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℕ₀ ↔ ((⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))))
99	97, 98	sylibr 234	. . . . . . . 8 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℕ₀)
100	84, 99	nn0addcld 12493	. . . . . . 7 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))) ∈ ℕ₀)
101	54	peano2zd 12627	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℤ)
102		1zzd 12549	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
103	102	znegcld 12626	. . . . . . . 8 ⊢ (𝜑 → -1 ∈ ℤ)
104	101, 103	zaddcld 12628	. . . . . . 7 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + -1) ∈ ℤ)
105		bccl 14275	. . . . . . 7 ⊢ (((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))) ∈ ℕ₀ ∧ (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + -1) ∈ ℤ) → ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C(((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + -1)) ∈ ℕ₀)
106	100, 104, 105	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C(((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + -1)) ∈ ℕ₀)
107	106	nn0red 12490	. . . . 5 ⊢ (𝜑 → ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C(((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + -1)) ∈ ℝ)
108	26, 99	nn0addcld 12493	. . . . . . 7 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))) ∈ ℕ₀)
109	26	nn0zd 12540	. . . . . . . 8 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℤ)
110	109, 103	zaddcld 12628	. . . . . . 7 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1) ∈ ℤ)
111		bccl 14275	. . . . . . 7 ⊢ ((((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))) ∈ ℕ₀ ∧ ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1) ∈ ℤ) → (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1)) ∈ ℕ₀)
112	108, 110, 111	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1)) ∈ ℕ₀)
113	112	nn0red 12490	. . . . 5 ⊢ (𝜑 → (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1)) ∈ ℝ)
114	52, 49	remulcld 11166	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ∈ ℝ)
115	114	flcld 13748	. . . . . . . . . . . 12 ⊢ (𝜑 → (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ∈ ℤ)
116	52, 49, 75, 74	mulge0d 11718	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))
117		flge 13755	. . . . . . . . . . . . . 14 ⊢ ((((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ↔ 0 ≤ (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))))
118	114, 33, 117	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 ≤ ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ↔ 0 ≤ (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))))
119	116, 118	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))))
120	115, 119	jca 511	. . . . . . . . . . 11 ⊢ (𝜑 → ((⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))))
121		elnn0z 12528	. . . . . . . . . . 11 ⊢ ((⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ∈ ℕ₀ ↔ ((⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))))
122	120, 121	sylibr 234	. . . . . . . . . 10 ⊢ (𝜑 → (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ∈ ℕ₀)
123	84, 122	nn0addcld 12493	. . . . . . . . 9 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))) ∈ ℕ₀)
124		bccl 14275	. . . . . . . . 9 ⊢ (((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))) ∈ ℕ₀ ∧ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℤ) → ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))))C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) ∈ ℕ₀)
125	123, 54, 124	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))))C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) ∈ ℕ₀)
126	125	nn0red 12490	. . . . . . 7 ⊢ (𝜑 → ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))))C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) ∈ ℝ)
127		bccl 14275	. . . . . . . . 9 ⊢ (((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))) ∈ ℕ₀ ∧ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℤ) → ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) ∈ ℕ₀)
128	100, 54, 127	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) ∈ ℕ₀)
129	128	nn0red 12490	. . . . . . 7 ⊢ (𝜑 → ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) ∈ ℝ)
130	42, 84	reexpcld 14116	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)) ∈ ℝ)
131		2nn0 12445	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ₀
132	131	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℕ₀)
133	132, 82	nn0mulcld 12494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2 · (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) ∈ ℕ₀)
134	133, 83	nn0addcld 12493	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2 · (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1) ∈ ℕ₀)
135		bccl 14275	. . . . . . . . . . . . 13 ⊢ ((((2 · (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1) ∈ ℕ₀ ∧ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℤ) → (((2 · (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1)C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) ∈ ℕ₀)
136	134, 54, 135	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (((2 · (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1)C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) ∈ ℕ₀)
137	136	nn0red 12490	. . . . . . . . . . 11 ⊢ (𝜑 → (((2 · (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1)C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) ∈ ℝ)
138	4, 42, 44	ltled 11285	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ 2)
139	42, 138, 53	recxpcld 26700	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2↑_𝑐((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℝ)
140		reflcl 13746	. . . . . . . . . . . . . . . 16 ⊢ (((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℝ → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℝ)
141	53, 140	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℝ)
142	141, 55	readdcld 11165	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℝ)
143	42, 138, 142	recxpcld 26700	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2↑_𝑐((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)) ∈ ℝ)
144		1le2 12376	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ≤ 2
145	144	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≤ 2)
146	55, 42, 7, 145, 71	letrd 11294	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ≤ 𝑁)
147		reflcl 13746	. . . . . . . . . . . . . . . . 17 ⊢ ((√‘𝐷) ∈ ℝ → (⌊‘(√‘𝐷)) ∈ ℝ)
148	30, 147	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘(√‘𝐷)) ∈ ℝ)
149	18	fveq2d 6838	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (√‘𝐷) = (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
150	149	fveq2d 6838	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (⌊‘(√‘𝐷)) = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
151		flle 13749	. . . . . . . . . . . . . . . . . 18 ⊢ ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
152	52, 151	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
153	150, 152	eqbrtrd 5108	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘(√‘𝐷)) ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
154	7, 146, 148, 52, 153	cxplead 26698	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁↑_𝑐(⌊‘(√‘𝐷))) ≤ (𝑁↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
155	7	recnd 11164	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℂ)
156	4, 12	gtned 11272	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ≠ 0)
157	155, 156, 31	cxpexpzd 26688	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁↑_𝑐(⌊‘(√‘𝐷))) = (𝑁↑(⌊‘(√‘𝐷))))
158	59, 48	nelprd 4602	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 2 ∈ {0, 1})
159	58, 158	eldifd 3901	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2 ∈ (ℂ ∖ {0, 1}))
160	156	neneqd 2938	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ¬ 𝑁 = 0)
161		elsng 4582	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ {0} ↔ 𝑁 = 0))
162	15, 161	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑁 ∈ {0} ↔ 𝑁 = 0))
163	160, 162	mtbird 325	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 𝑁 ∈ {0})
164	155, 163	eldifd 3901	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℂ ∖ {0}))
165		cxplogb 26763	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 𝑁 ∈ (ℂ ∖ {0})) → (2↑_𝑐(2 log_b 𝑁)) = 𝑁)
166	159, 164, 165	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2↑_𝑐(2 log_b 𝑁)) = 𝑁)
167	166	eqcomd 2743	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 = (2↑_𝑐(2 log_b 𝑁)))
168	167	oveq1d 7375	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = ((2↑_𝑐(2 log_b 𝑁))↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
169	154, 157, 168	3brtr3d 5117	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ≤ ((2↑_𝑐(2 log_b 𝑁))↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
170	42, 44	elrpd 12974	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℝ⁺)
171	52	recnd 11164	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℂ)
172		cxpmul 26665	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℝ⁺ ∧ (2 log_b 𝑁) ∈ ℝ ∧ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℂ) → (2↑_𝑐((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) = ((2↑_𝑐(2 log_b 𝑁))↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
173	170, 49, 171, 172	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2↑_𝑐((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) = ((2↑_𝑐(2 log_b 𝑁))↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
174	169, 173	breqtrrd 5114	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ≤ (2↑_𝑐((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
175		fllep1 13751	. . . . . . . . . . . . . . 15 ⊢ (((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℝ → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ≤ ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1))
176	53, 175	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ≤ ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1))
177	55, 42, 145, 48	leneltd 11291	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 < 2)
178	84	nn0red 12490	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℝ)
179	42, 177, 53, 178	cxpled 26697	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ≤ ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ↔ (2↑_𝑐((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ≤ (2↑_𝑐((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1))))
180	176, 179	mpbid 232	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2↑_𝑐((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ≤ (2↑_𝑐((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)))
181	40, 139, 143, 174, 180	letrd 11294	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ≤ (2↑_𝑐((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)))
182		cxpexpz 26644	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℤ) → (2↑_𝑐((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)) = (2↑((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)))
183	58, 59, 101, 182	syl3anc 1374	. . . . . . . . . . . 12 ⊢ (𝜑 → (2↑_𝑐((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)) = (2↑((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)))
184	181, 183	breqtrd 5112	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ≤ (2↑((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)))
185	49, 49	jca 511	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2 log_b 𝑁) ∈ ℝ ∧ (2 log_b 𝑁) ∈ ℝ))
186		remulcl 11114	. . . . . . . . . . . . . . 15 ⊢ (((2 log_b 𝑁) ∈ ℝ ∧ (2 log_b 𝑁) ∈ ℝ) → ((2 log_b 𝑁) · (2 log_b 𝑁)) ∈ ℝ)
187	185, 186	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2 log_b 𝑁) · (2 log_b 𝑁)) ∈ ℝ)
188		reflcl 13746	. . . . . . . . . . . . . 14 ⊢ (((2 log_b 𝑁) · (2 log_b 𝑁)) ∈ ℝ → (⌊‘((2 log_b 𝑁) · (2 log_b 𝑁))) ∈ ℝ)
189	187, 188	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (2 log_b 𝑁))) ∈ ℝ)
190	82	nn0red 12490	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℝ)
191	42, 44, 6, 9, 48	relogbcld 42427	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 log_b 3) ∈ ℝ)
192	191	resqcld 14078	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2 log_b 3)↑2) ∈ ℝ)
193	49	recnd 11164	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℂ)
194	193	sqvald 14096	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) = ((2 log_b 𝑁) · (2 log_b 𝑁)))
195	194, 187	eqeltrd 2837	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) ∈ ℝ)
196		3lexlogpow2ineq2 42512	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 < ((2 log_b 3)↑2) ∧ ((2 log_b 3)↑2) < 3)
197	196	simpli 483	. . . . . . . . . . . . . . . . . 18 ⊢ 2 < ((2 log_b 3)↑2)
198	197	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 < ((2 log_b 3)↑2))
199	42, 192, 198	ltled 11285	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≤ ((2 log_b 3)↑2))
200	6, 42, 59	redivcld 11974	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (3 / 2) ∈ ℝ)
201		2rp 12938	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ⁺
202	201	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 2 ∈ ℝ⁺)
203	4, 6, 9	ltled 11285	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 ≤ 3)
204	6, 202, 203	divge0d 13017	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 0 ≤ (3 / 2))
205		3lexlogpow2ineq1 42511	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((3 / 2) < (2 log_b 3) ∧ (2 log_b 3) < (5 / 3))
206	205	simpli 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (3 / 2) < (2 log_b 3)
207	206	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (3 / 2) < (2 log_b 3))
208	200, 191, 207	ltled 11285	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (3 / 2) ≤ (2 log_b 3))
209	4, 200, 191, 204, 208	letrd 11294	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ≤ (2 log_b 3))
210	64, 65, 6, 9, 7, 12, 11	logblebd 42430	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 log_b 3) ≤ (2 log_b 𝑁))
211	191, 49, 132, 209, 210	leexp1ad 14129	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2 log_b 3)↑2) ≤ ((2 log_b 𝑁)↑2))
212	42, 192, 195, 199, 211	letrd 11294	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ≤ ((2 log_b 𝑁)↑2))
213	212, 194	breqtrd 5112	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≤ ((2 log_b 𝑁) · (2 log_b 𝑁)))
214		flge 13755	. . . . . . . . . . . . . . 15 ⊢ ((((2 log_b 𝑁) · (2 log_b 𝑁)) ∈ ℝ ∧ 2 ∈ ℤ) → (2 ≤ ((2 log_b 𝑁) · (2 log_b 𝑁)) ↔ 2 ≤ (⌊‘((2 log_b 𝑁) · (2 log_b 𝑁)))))
215	187, 64, 214	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2 ≤ ((2 log_b 𝑁) · (2 log_b 𝑁)) ↔ 2 ≤ (⌊‘((2 log_b 𝑁) · (2 log_b 𝑁)))))
216	213, 215	mpbid 232	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ≤ (⌊‘((2 log_b 𝑁) · (2 log_b 𝑁))))
217	49, 49	remulcld 11166	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2 log_b 𝑁) · (2 log_b 𝑁)) ∈ ℝ)
218		aks6d1c7lem1.10	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))
219	15, 19, 20, 21, 22, 23, 24, 25, 218	aks6d1c3 42576	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
220	171	sqvald 14096	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))↑2) = ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
221	26	nn0cnd 12491	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℂ)
222	221	msqsqrtd 15396	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
223	220, 222	eqtr2d 2773	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))↑2))
224	219, 223	breqtrd 5112	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))↑2))
225	49, 52, 74, 75	lt2sqd 14209	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2 log_b 𝑁) < (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ↔ ((2 log_b 𝑁)↑2) < ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))↑2)))
226	224, 225	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2 log_b 𝑁) < (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
227	49, 52, 226	ltled 11285	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 𝑁) ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
228	49, 52, 49, 74, 227	lemul2ad 12087	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2 log_b 𝑁) · (2 log_b 𝑁)) ≤ ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
229		flwordi 13762	. . . . . . . . . . . . . 14 ⊢ ((((2 log_b 𝑁) · (2 log_b 𝑁)) ∈ ℝ ∧ ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℝ ∧ ((2 log_b 𝑁) · (2 log_b 𝑁)) ≤ ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) → (⌊‘((2 log_b 𝑁) · (2 log_b 𝑁))) ≤ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
230	217, 53, 228, 229	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (2 log_b 𝑁))) ≤ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
231	42, 189, 190, 216, 230	letrd 11294	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ≤ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
232	54, 231	2ap1caineq 42598	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)) < (((2 · (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1)C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
233	40, 130, 137, 184, 232	lelttrd 11295	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < (((2 · (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1)C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
234	82	nn0cnd 12491	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℂ)
235	234	2timesd 12411	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 · (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) = ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
236	235	oveq1d 7375	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 · (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1) = (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1))
237	236	oveq1d 7375	. . . . . . . . . 10 ⊢ (𝜑 → (((2 · (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1)C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) = ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1)C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
238	233, 237	breqtrd 5112	. . . . . . . . 9 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1)C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
239		1cnd 11130	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
240	234, 234, 239	addassd 11158	. . . . . . . . . . 11 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1) = ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)))
241	84	nn0cnd 12491	. . . . . . . . . . . 12 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℂ)
242	234, 241	addcomd 11339	. . . . . . . . . . 11 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)) = (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
243	240, 242	eqtrd 2772	. . . . . . . . . 10 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1) = (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
244	243	oveq1d 7375	. . . . . . . . 9 ⊢ (𝜑 → ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1)C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) = ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
245	238, 244	breqtrd 5112	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
246	193, 171	mulcomd 11157	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))
247	246	fveq2d 6838	. . . . . . . . . 10 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) = (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))))
248	247	oveq2d 7376	. . . . . . . . 9 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) = (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))))
249	248	oveq1d 7375	. . . . . . . 8 ⊢ (𝜑 → ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) = ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))))C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
250	245, 249	breqtrd 5112	. . . . . . 7 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))))C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
251	122	nn0red 12490	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ∈ ℝ)
252	99	nn0red 12490	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℝ)
253	17, 27	eqeltrrid 2842	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
254	253	nn0red 12490	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ)
255	253	nn0ge0d 12492	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
256	254, 255	resqrtcld 15371	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ)
257	256, 49	remulcld 11166	. . . . . . . . . 10 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ∈ ℝ)
258	15, 19, 20, 21, 22, 23, 24	aks6d1c4 42577	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ≤ (ϕ‘𝑅))
259	50, 51, 86, 88	sqrtled 15380	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ≤ (ϕ‘𝑅) ↔ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ≤ (√‘(ϕ‘𝑅))))
260	258, 259	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ≤ (√‘(ϕ‘𝑅)))
261	256, 89, 49, 74, 260	lemul1ad 12086	. . . . . . . . . 10 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
262		flwordi 13762	. . . . . . . . . 10 ⊢ ((((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ∈ ℝ ∧ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ∈ ℝ ∧ ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) → (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))
263	257, 90, 261, 262	syl3anc 1374	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))
264	251, 252, 142, 263	leadd2dd 11756	. . . . . . . 8 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))) ≤ (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))))
265	123, 100, 54, 264	bcled 42631	. . . . . . 7 ⊢ (𝜑 → ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))))C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) ≤ ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
266	40, 126, 129, 250, 265	ltletrd 11297	. . . . . 6 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
267	234, 239	pncand 11497	. . . . . . . . 9 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) − 1) = (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
268	267	eqcomd 2743	. . . . . . . 8 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) = (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) − 1))
269	241, 239	negsubd 11502	. . . . . . . . 9 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + -1) = (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) − 1))
270	269	eqcomd 2743	. . . . . . . 8 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) − 1) = (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + -1))
271	268, 270	eqtrd 2772	. . . . . . 7 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) = (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + -1))
272	271	oveq2d 7376	. . . . . 6 ⊢ (𝜑 → ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C(⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) = ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C(((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + -1)))
273	266, 272	breqtrd 5112	. . . . 5 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C(((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + -1)))
274	21	nnnn0d 12489	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
275	25	zncrng 21534	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ ℕ₀ → (ℤ/nℤ‘𝑅) ∈ CRing)
276	274, 275	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (ℤ/nℤ‘𝑅) ∈ CRing)
277		crngring 20217	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ/nℤ‘𝑅) ∈ CRing → (ℤ/nℤ‘𝑅) ∈ Ring)
278	24	zrhrhm 21501	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ/nℤ‘𝑅) ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom (ℤ/nℤ‘𝑅)))
279		zringbas 21443	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℤ = (Base‘ℤ_ring)
280		eqid 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘(ℤ/nℤ‘𝑅)) = (Base‘(ℤ/nℤ‘𝑅))
281	279, 280	rhmf 20455	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿 ∈ (ℤ_ring RingHom (ℤ/nℤ‘𝑅)) → 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅)))
282	276, 277, 278, 281	4syl 19	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅)))
283	282	ffnd 6663	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐿 Fn ℤ)
284	15, 19, 20, 23	aks6d1c2p1 42571	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐸:(ℕ₀ × ℕ₀)⟶ℕ)
285		nnssz 12537	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℕ ⊆ ℤ
286	285	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ℕ ⊆ ℤ)
287	284, 286	fssd 6679	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐸:(ℕ₀ × ℕ₀)⟶ℤ)
288		frn 6669	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸:(ℕ₀ × ℕ₀)⟶ℤ → ran 𝐸 ⊆ ℤ)
289	287, 288	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ran 𝐸 ⊆ ℤ)
290	284	ffnd 6663	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐸 Fn (ℕ₀ × ℕ₀))
291		fnima 6622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐸 Fn (ℕ₀ × ℕ₀) → (𝐸 “ (ℕ₀ × ℕ₀)) = ran 𝐸)
292	290, 291	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐸 “ (ℕ₀ × ℕ₀)) = ran 𝐸)
293	292	sseq1d 3954	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐸 “ (ℕ₀ × ℕ₀)) ⊆ ℤ ↔ ran 𝐸 ⊆ ℤ))
294	289, 293	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 “ (ℕ₀ × ℕ₀)) ⊆ ℤ)
295		vex 3434	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑘 ∈ V
296		vex 3434	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑙 ∈ V
297	295, 296	op1std 7945	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (1^st ‘𝑣) = 𝑘)
298	297	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (𝑃↑(1^st ‘𝑣)) = (𝑃↑𝑘))
299	295, 296	op2ndd 7946	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (2^nd ‘𝑣) = 𝑙)
300	299	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑁 / 𝑃)↑(2^nd ‘𝑣)) = ((𝑁 / 𝑃)↑𝑙))
301	298, 300	oveq12d 7378	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑃↑(1^st ‘𝑣)) · ((𝑁 / 𝑃)↑(2^nd ‘𝑣))) = ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
302	301	mpompt 7474	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 ∈ (ℕ₀ × ℕ₀) ↦ ((𝑃↑(1^st ‘𝑣)) · ((𝑁 / 𝑃)↑(2^nd ‘𝑣)))) = (𝑘 ∈ ℕ₀, 𝑙 ∈ ℕ₀ ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
303	302	eqcomi 2746	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ ℕ₀, 𝑙 ∈ ℕ₀ ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) = (𝑣 ∈ (ℕ₀ × ℕ₀) ↦ ((𝑃↑(1^st ‘𝑣)) · ((𝑁 / 𝑃)↑(2^nd ‘𝑣))))
304	23, 303	eqtri 2760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐸 = (𝑣 ∈ (ℕ₀ × ℕ₀) ↦ ((𝑃↑(1^st ‘𝑣)) · ((𝑁 / 𝑃)↑(2^nd ‘𝑣))))
305	304	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐸 = (𝑣 ∈ (ℕ₀ × ℕ₀) ↦ ((𝑃↑(1^st ‘𝑣)) · ((𝑁 / 𝑃)↑(2^nd ‘𝑣)))))
306		c0ex 11129	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ V
307	306, 306	op1std 7945	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = ⟨0, 0⟩ → (1^st ‘𝑣) = 0)
308	307	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → (1^st ‘𝑣) = 0)
309	308	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → (𝑃↑(1^st ‘𝑣)) = (𝑃↑0))
310	306, 306	op2ndd 7946	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = ⟨0, 0⟩ → (2^nd ‘𝑣) = 0)
311	310	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → (2^nd ‘𝑣) = 0)
312	311	oveq2d 7376	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → ((𝑁 / 𝑃)↑(2^nd ‘𝑣)) = ((𝑁 / 𝑃)↑0))
313	309, 312	oveq12d 7378	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → ((𝑃↑(1^st ‘𝑣)) · ((𝑁 / 𝑃)↑(2^nd ‘𝑣))) = ((𝑃↑0) · ((𝑁 / 𝑃)↑0)))
314		prmnn 16634	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
315	19, 314	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑃 ∈ ℕ)
316	315	nncnd 12181	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑃 ∈ ℂ)
317	316	exp0d 14093	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑃↑0) = 1)
318	315	nnne0d 12218	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑃 ≠ 0)
319	155, 316, 318	divcld 11922	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℂ)
320	319	exp0d 14093	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((𝑁 / 𝑃)↑0) = 1)
321	317, 320	oveq12d 7378	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑃↑0) · ((𝑁 / 𝑃)↑0)) = (1 · 1))
322	239	mulridd 11153	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1 · 1) = 1)
323	321, 322	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((𝑃↑0) · ((𝑁 / 𝑃)↑0)) = 1)
324	323	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → ((𝑃↑0) · ((𝑁 / 𝑃)↑0)) = 1)
325	313, 324	eqtrd 2772	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → ((𝑃↑(1^st ‘𝑣)) · ((𝑁 / 𝑃)↑(2^nd ‘𝑣))) = 1)
326		0nn0 12443	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ ℕ₀
327	326	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 ∈ ℕ₀)
328	327, 327	opelxpd 5663	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ⟨0, 0⟩ ∈ (ℕ₀ × ℕ₀))
329		1nn 12176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℕ
330	329	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 1 ∈ ℕ)
331	305, 325, 328, 330	fvmptd 6949	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐸‘⟨0, 0⟩) = 1)
332		ssidd 3946	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (ℕ₀ × ℕ₀) ⊆ (ℕ₀ × ℕ₀))
333		fnfvima 7181	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 Fn (ℕ₀ × ℕ₀) ∧ (ℕ₀ × ℕ₀) ⊆ (ℕ₀ × ℕ₀) ∧ ⟨0, 0⟩ ∈ (ℕ₀ × ℕ₀)) → (𝐸‘⟨0, 0⟩) ∈ (𝐸 “ (ℕ₀ × ℕ₀)))
334	290, 332, 328, 333	syl3anc 1374	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐸‘⟨0, 0⟩) ∈ (𝐸 “ (ℕ₀ × ℕ₀)))
335	331, 334	eqeltrrd 2838	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ (𝐸 “ (ℕ₀ × ℕ₀)))
336		fnfvima 7181	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐿 Fn ℤ ∧ (𝐸 “ (ℕ₀ × ℕ₀)) ⊆ ℤ ∧ 1 ∈ (𝐸 “ (ℕ₀ × ℕ₀))) → (𝐿‘1) ∈ (𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))
337	283, 294, 335, 336	syl3anc 1374	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐿‘1) ∈ (𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))
338	24	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅)))
339		fvexd 6849	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℤRHom‘(ℤ/nℤ‘𝑅)) ∈ V)
340	338, 339	eqeltrd 2837	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐿 ∈ V)
341	340	imaexd 7860	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))) ∈ V)
342	337, 341	hashelne0d 14321	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = 0)
343	342	neqned 2940	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ≠ 0)
344	26, 343	jca 511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀ ∧ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ≠ 0))
345		elnnne0 12442	. . . . . . . . . . . . . 14 ⊢ ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ ↔ ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀ ∧ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ≠ 0))
346	344, 345	sylibr 234	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ)
347	346	nnrpd 12975	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ⁺)
348	347	rpsqrtcld 15365	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ⁺)
349	49, 52, 348, 226	ltmul1dd 13032	. . . . . . . . . 10 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) < ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
350	50, 51, 50, 51	sqrtmuld 15378	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
351	350	eqcomd 2743	. . . . . . . . . 10 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
352	349, 351	breqtrd 5112	. . . . . . . . 9 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) < (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
353	350, 222	eqtrd 2772	. . . . . . . . 9 ⊢ (𝜑 → (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
354	352, 353	breqtrd 5112	. . . . . . . 8 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) < (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
355		fllt 13756	. . . . . . . . 9 ⊢ ((((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℝ ∧ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℤ) → (((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) < (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ↔ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) < (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
356	53, 109, 355	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) < (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ↔ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) < (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
357	354, 356	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) < (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
358	54, 109	zltp1led 12573	. . . . . . 7 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) < (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ↔ ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
359	357, 358	mpbid 232	. . . . . 6 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
360	55	renegcld 11568	. . . . . . 7 ⊢ (𝜑 → -1 ∈ ℝ)
361		df-neg 11371	. . . . . . . . 9 ⊢ -1 = (0 − 1)
362	361	a1i 11	. . . . . . . 8 ⊢ (𝜑 → -1 = (0 − 1))
363	4	lem1d 12080	. . . . . . . 8 ⊢ (𝜑 → (0 − 1) ≤ 0)
364	362, 363	eqbrtrd 5108	. . . . . . 7 ⊢ (𝜑 → -1 ≤ 0)
365	360, 4, 252, 364, 96	letrd 11294	. . . . . 6 ⊢ (𝜑 → -1 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))
366	84, 26, 99, 103, 359, 365	bcle2d 42632	. . . . 5 ⊢ (𝜑 → ((((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C(((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + -1)) ≤ (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1)))
367	40, 107, 113, 273, 366	ltletrd 11297	. . . 4 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1)))
368	221, 239	negsubd 11502	. . . . 5 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1) = ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1))
369	368	oveq2d 7376	. . . 4 ⊢ (𝜑 → (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1)) = (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1)))
370	367, 369	breqtrd 5112	. . 3 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1)))
371		aks6d1c7lem1.9	. . . . . . 7 ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
372	371	eqcomi 2746	. . . . . 6 ⊢ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) = 𝐴
373	372	a1i 11	. . . . 5 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) = 𝐴)
374	373	oveq2d 7376	. . . 4 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))) = ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + 𝐴))
375	374	oveq1d 7375	. . 3 ⊢ (𝜑 → (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1)) = (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + 𝐴)C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1)))
376	370, 375	breqtrd 5112	. 2 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + 𝐴)C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1)))
377	18	eqcomd 2743	. . . 4 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = 𝐷)
378	377	oveq1d 7375	. . 3 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + 𝐴) = (𝐷 + 𝐴))
379	377	oveq1d 7375	. . 3 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1) = (𝐷 − 1))
380	378, 379	oveq12d 7378	. 2 ⊢ (𝜑 → (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + 𝐴)C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1)) = ((𝐷 + 𝐴)C(𝐷 − 1)))
381	376, 380	breqtrd 5112	1 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < ((𝐷 + 𝐴)C(𝐷 − 1)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3430 ∖ cdif 3887 ⊆ wss 3890 {csn 4568 {cpr 4570 ⟨cop 4574 class class class wbr 5086 ↦ cmpt 5167 × cxp 5622 ran crn 5625 “ cima 5627 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 ∈ cmpo 7362 1^st c1st 7933 2^nd c2nd 7934 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 < clt 11170 ≤ cle 11171 − cmin 11368 -cneg 11369 / cdiv 11798 ℕcn 12165 2c2 12227 3c3 12228 5c5 12230 ℕ₀cn0 12428 ℤcz 12515 ℤ_≥cuz 12779 ℝ⁺crp 12933 ⌊cfl 13740 ↑cexp 14014 Ccbc 14255 ♯chash 14283 √csqrt 15186 ∥ cdvds 16212 gcd cgcd 16454 ℙcprime 16631 od_ℤcodz 16724 ϕcphi 16725 Basecbs 17170 Ringcrg 20205 CRingccrg 20206 RingHom crh 20440 ℤ_ringczring 21436 ℤRHomczrh 21489 ℤ/nℤczn 21492 ↑_𝑐ccxp 26532 log_b clogb 26741

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 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-tpos 8169 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-oadd 8402 df-er 8636 df-ec 8638 df-qs 8642 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-dju 9816 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-xnn0 12502 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-prod 15860 df-fallfac 15963 df-ef 16023 df-sin 16025 df-cos 16026 df-pi 16028 df-dvds 16213 df-gcd 16455 df-prm 16632 df-odz 16726 df-phi 16727 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-qus 17464 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-nsg 19091 df-eqg 19092 df-ghm 19179 df-cntz 19283 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-cring 20208 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-rhm 20443 df-subrng 20514 df-subrg 20538 df-lmod 20848 df-lss 20918 df-lsp 20958 df-sra 21160 df-rgmod 21161 df-lidl 21198 df-rsp 21199 df-2idl 21240 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-fbas 21341 df-fg 21342 df-cnfld 21345 df-zring 21437 df-zrh 21493 df-zn 21496 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cld 22994 df-ntr 22995 df-cls 22996 df-nei 23073 df-lp 23111 df-perf 23112 df-cn 23202 df-cnp 23203 df-haus 23290 df-tx 23537 df-hmeo 23730 df-fil 23821 df-fm 23913 df-flim 23914 df-flf 23915 df-xms 24295 df-ms 24296 df-tms 24297 df-cncf 24855 df-limc 25843 df-dv 25844 df-log 26533 df-cxp 26534 df-logb 26742

This theorem is referenced by: aks6d1c7lem2 42634

W3C validator