aks6d1c7lem1 - Mathbox for metakunt

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Theorem aks6d1c7lem1 43005

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 12888	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
3	1, 2	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
4		0red 11226	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
5		3re 12336	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
6	5	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 3 ∈ ℝ)
7	3	zred 12716	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
8		3pos 12364	. . . . . . . . . . 11 ⊢ 0 < 3
9	8	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 3)
10		eluzle 12891	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
11	1, 10	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁)
12	4, 6, 7, 9, 11	ltletrd 11385	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝑁)
13	3, 12	jca 521	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
14		elnnz 12616	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
15	13, 14	sylibr 237	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
16	15	nnred 12263	. . . . . 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 2765	. . . . . . . . . . . . 13 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
26	15, 19, 20, 21, 22, 23, 24, 25	hashscontpowcl 42945	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
27	18, 26	eqeltrd 2865	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ ℕ₀)
28	27	nn0red 12581	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ ℝ)
29	27	nn0ge0d 12583	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝐷)
30	28, 29	resqrtcld 15493	. . . . . . . . 9 ⊢ (𝜑 → (√‘𝐷) ∈ ℝ)
31	30	flcld 13849	. . . . . . . 8 ⊢ (𝜑 → (⌊‘(√‘𝐷)) ∈ ℤ)
32	28, 29	sqrtge0d 15496	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (√‘𝐷))
33		0zd 12618	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
34		flge 13856	. . . . . . . . . 10 ⊢ (((√‘𝐷) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ (√‘𝐷) ↔ 0 ≤ (⌊‘(√‘𝐷))))
35	30, 33, 34	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (0 ≤ (√‘𝐷) ↔ 0 ≤ (⌊‘(√‘𝐷))))
36	32, 35	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (⌊‘(√‘𝐷)))
37	31, 36	jca 521	. . . . . . 7 ⊢ (𝜑 → ((⌊‘(√‘𝐷)) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘𝐷))))
38		elnn0z 12619	. . . . . . 7 ⊢ ((⌊‘(√‘𝐷)) ∈ ℕ₀ ↔ ((⌊‘(√‘𝐷)) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘𝐷))))
39	37, 38	sylibr 237	. . . . . 6 ⊢ (𝜑 → (⌊‘(√‘𝐷)) ∈ ℕ₀)
40	16, 39	reexpcld 14217	. . . . 5 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ∈ ℝ)
41		2re 12330	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
42	41	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℝ)
43		2pos 12360	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
44	43	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 2)
45	15	nngt0d 12300	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑁)
46		1ne2 12466	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 2
47	46	necomi 3014	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 1
48	47	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≠ 1)
49	42, 44, 16, 45, 48	relogbcld 42799	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
50	18, 28	eqeltrrd 2866	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ)
51	29, 18	breqtrd 5139	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
52	50, 51	resqrtcld 15493	. . . . . . . . . . . . 13 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ)
53	49, 52	remulcld 11254	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℝ)
54	53	flcld 13849	. . . . . . . . . . 11 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℤ)
55		1red 11224	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ)
56		0le1 11752	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 1
57	56	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ 1)
58	42	recnd 11252	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ∈ ℂ)
59	4, 44	gtned 11360	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ≠ 0)
60		logbid1 26984	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 log_b 2) = 1)
61	58, 59, 48, 60	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2 log_b 2) = 1)
62	61	eqcomd 2771	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 = (2 log_b 2))
63		2z 12641	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℤ
64	63	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ∈ ℤ)
65	42	leidd 11795	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≤ 2)
66		1nn0 12535	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℕ₀
67	41, 66	nn0addge1i 12567	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ≤ (2 + 1)
68		2p1e3 12397	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 + 1) = 3
69	67, 68	breqtri 5138	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≤ 3
70	69	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ≤ 3)
71	42, 6, 7, 70, 11	letrd 11382	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≤ 𝑁)
72	64, 65, 42, 44, 7, 12, 71	logblebd 42802	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 2) ≤ (2 log_b 𝑁))
73	62, 72	eqbrtrd 5135	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ≤ (2 log_b 𝑁))
74	4, 55, 49, 57, 73	letrd 11382	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (2 log_b 𝑁))
75	50, 51	sqrtge0d 15496	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
76	49, 52, 74, 75	mulge0d 11806	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
77		flge 13856	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ≤ ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ↔ 0 ≤ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
79	76, 78	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
80	54, 79	jca 521	. . . . . . . . . 10 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℤ ∧ 0 ≤ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))))
81		elnn0z 12619	. . . . . . . . . 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 237	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℕ₀)
83	66	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℕ₀)
84	82, 83	nn0addcld 12584	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℕ₀)
85	21	phicld 16853	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ)
86	85	nnred 12263	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℝ)
87	85	nnnn0d 12580	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ₀)
88	87	nn0ge0d 12583	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (ϕ‘𝑅))
89	86, 88	resqrtcld 15493	. . . . . . . . . . . 12 ⊢ (𝜑 → (√‘(ϕ‘𝑅)) ∈ ℝ)
90	89, 49	remulcld 11254	. . . . . . . . . . 11 ⊢ (𝜑 → ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ∈ ℝ)
91	90	flcld 13849	. . . . . . . . . 10 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℤ)
92	86, 88	sqrtge0d 15496	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (√‘(ϕ‘𝑅)))
93	89, 49, 92, 74	mulge0d 11806	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
94		flge 13856	. . . . . . . . . . . 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 596	. . . . . . . . . . 11 ⊢ (𝜑 → (0 ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ↔ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))))
96	93, 95	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))
97	91, 96	jca 521	. . . . . . . . 9 ⊢ (𝜑 → ((⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))))
98		elnn0z 12619	. . . . . . . . 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 237	. . . . . . . 8 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℕ₀)
100	84, 99	nn0addcld 12584	. . . . . . 7 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))) ∈ ℕ₀)
101	54	peano2zd 12719	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℤ)
102		1zzd 12640	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
103	102	znegcld 12718	. . . . . . . 8 ⊢ (𝜑 → -1 ∈ ℤ)
104	101, 103	zaddcld 12720	. . . . . . 7 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + -1) ∈ ℤ)
105		bccl 14376	. . . . . . 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 596	. . . . . 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 12581	. . . . 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 12584	. . . . . . 7 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))) ∈ ℕ₀)
109	26	nn0zd 12631	. . . . . . . 8 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℤ)
110	109, 103	zaddcld 12720	. . . . . . 7 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1) ∈ ℤ)
111		bccl 14376	. . . . . . 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 596	. . . . . 6 ⊢ (𝜑 → (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1)) ∈ ℕ₀)
113	112	nn0red 12581	. . . . 5 ⊢ (𝜑 → (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1)) ∈ ℝ)
114	52, 49	remulcld 11254	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ∈ ℝ)
115	114	flcld 13849	. . . . . . . . . . . 12 ⊢ (𝜑 → (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ∈ ℤ)
116	52, 49, 75, 74	mulge0d 11806	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))
117		flge 13856	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 ≤ ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ↔ 0 ≤ (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))))
119	116, 118	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))))
120	115, 119	jca 521	. . . . . . . . . . 11 ⊢ (𝜑 → ((⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))))
121		elnn0z 12619	. . . . . . . . . . 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 237	. . . . . . . . . 10 ⊢ (𝜑 → (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ∈ ℕ₀)
123	84, 122	nn0addcld 12584	. . . . . . . . 9 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))) ∈ ℕ₀)
124		bccl 14376	. . . . . . . . 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 596	. . . . . . . 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 12581	. . . . . . 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 14376	. . . . . . . . 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 596	. . . . . . . 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 12581	. . . . . . 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 14217	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)) ∈ ℝ)
131		2nn0 12536	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ₀
132	131	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℕ₀)
133	132, 82	nn0mulcld 12585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2 · (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) ∈ ℕ₀)
134	133, 83	nn0addcld 12584	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2 · (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1) ∈ ℕ₀)
135		bccl 14376	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 12581	. . . . . . . . . . 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 11373	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ 2)
139	42, 138, 53	recxpcld 26939	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2↑_𝑐((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℝ)
140		reflcl 13847	. . . . . . . . . . . . . . . 16 ⊢ (((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℝ → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℝ)
141	53, 140	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℝ)
142	141, 55	readdcld 11253	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℝ)
143	42, 138, 142	recxpcld 26939	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2↑_𝑐((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)) ∈ ℝ)
144		1le2 12467	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ≤ 2
145	144	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≤ 2)
146	55, 42, 7, 145, 71	letrd 11382	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ≤ 𝑁)
147		reflcl 13847	. . . . . . . . . . . . . . . . 17 ⊢ ((√‘𝐷) ∈ ℝ → (⌊‘(√‘𝐷)) ∈ ℝ)
148	30, 147	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘(√‘𝐷)) ∈ ℝ)
149	18	fveq2d 6889	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (√‘𝐷) = (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
150	149	fveq2d 6889	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (⌊‘(√‘𝐷)) = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
151		flle 13850	. . . . . . . . . . . . . . . . . 18 ⊢ ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
152	52, 151	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
153	150, 152	eqbrtrd 5135	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘(√‘𝐷)) ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
154	7, 146, 148, 52, 153	cxplead 26937	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁↑_𝑐(⌊‘(√‘𝐷))) ≤ (𝑁↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
155	7	recnd 11252	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℂ)
156	4, 12	gtned 11360	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ≠ 0)
157	155, 156, 31	cxpexpzd 26927	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁↑_𝑐(⌊‘(√‘𝐷))) = (𝑁↑(⌊‘(√‘𝐷))))
158	59, 48	nelprd 4625	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 2 ∈ {0, 1})
159	58, 158	eldifd 3917	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2 ∈ (ℂ ∖ {0, 1}))
160	156	neneqd 2965	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ¬ 𝑁 = 0)
161		elsng 4605	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ {0} ↔ 𝑁 = 0))
162	15, 161	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑁 ∈ {0} ↔ 𝑁 = 0))
163	160, 162	mtbird 328	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 𝑁 ∈ {0})
164	155, 163	eldifd 3917	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℂ ∖ {0}))
165		cxplogb 27002	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 𝑁 ∈ (ℂ ∖ {0})) → (2↑_𝑐(2 log_b 𝑁)) = 𝑁)
166	159, 164, 165	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2↑_𝑐(2 log_b 𝑁)) = 𝑁)
167	166	eqcomd 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 = (2↑_𝑐(2 log_b 𝑁)))
168	167	oveq1d 7434	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = ((2↑_𝑐(2 log_b 𝑁))↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
169	154, 157, 168	3brtr3d 5144	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ≤ ((2↑_𝑐(2 log_b 𝑁))↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
170	42, 44	elrpd 13073	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℝ⁺)
171	52	recnd 11252	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℂ)
172		cxpmul 26904	. . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . 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 5141	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ≤ (2↑_𝑐((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
175		fllep1 13852	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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 11379	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 < 2)
178	84	nn0red 12581	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℝ)
179	42, 177, 53, 178	cxpled 26936	. . . . . . . . . . . . . 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 235	. . . . . . . . . . . . 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 11382	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ≤ (2↑_𝑐((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)))
182		cxpexpz 26883	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 5139	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ≤ (2↑((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)))
185	49, 49	jca 521	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2 log_b 𝑁) ∈ ℝ ∧ (2 log_b 𝑁) ∈ ℝ))
186		remulcl 11200	. . . . . . . . . . . . . . 15 ⊢ (((2 log_b 𝑁) ∈ ℝ ∧ (2 log_b 𝑁) ∈ ℝ) → ((2 log_b 𝑁) · (2 log_b 𝑁)) ∈ ℝ)
187	185, 186	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2 log_b 𝑁) · (2 log_b 𝑁)) ∈ ℝ)
188		reflcl 13847	. . . . . . . . . . . . . 14 ⊢ (((2 log_b 𝑁) · (2 log_b 𝑁)) ∈ ℝ → (⌊‘((2 log_b 𝑁) · (2 log_b 𝑁))) ∈ ℝ)
189	187, 188	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (2 log_b 𝑁))) ∈ ℝ)
190	82	nn0red 12581	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℝ)
191	42, 44, 6, 9, 48	relogbcld 42799	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 log_b 3) ∈ ℝ)
192	191	resqcld 14179	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2 log_b 3)↑2) ∈ ℝ)
193	49	recnd 11252	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℂ)
194	193	sqvald 14197	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) = ((2 log_b 𝑁) · (2 log_b 𝑁)))
195	194, 187	eqeltrd 2865	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) ∈ ℝ)
196		3lexlogpow2ineq2 42884	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 < ((2 log_b 3)↑2) ∧ ((2 log_b 3)↑2) < 3)
197	196	simpli 489	. . . . . . . . . . . . . . . . . 18 ⊢ 2 < ((2 log_b 3)↑2)
198	197	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 < ((2 log_b 3)↑2))
199	42, 192, 198	ltled 11373	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≤ ((2 log_b 3)↑2))
200	6, 42, 59	redivcld 12058	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (3 / 2) ∈ ℝ)
201		2rp 13037	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ⁺
202	201	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 2 ∈ ℝ⁺)
203	4, 6, 9	ltled 11373	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 ≤ 3)
204	6, 202, 203	divge0d 13116	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 0 ≤ (3 / 2))
205		3lexlogpow2ineq1 42883	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((3 / 2) < (2 log_b 3) ∧ (2 log_b 3) < (5 / 3))
206	205	simpli 489	. . . . . . . . . . . . . . . . . . . 20 ⊢ (3 / 2) < (2 log_b 3)
207	206	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (3 / 2) < (2 log_b 3))
208	200, 191, 207	ltled 11373	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (3 / 2) ≤ (2 log_b 3))
209	4, 200, 191, 204, 208	letrd 11382	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ≤ (2 log_b 3))
210	64, 65, 6, 9, 7, 12, 11	logblebd 42802	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 log_b 3) ≤ (2 log_b 𝑁))
211	191, 49, 132, 209, 210	leexp1ad 14230	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2 log_b 3)↑2) ≤ ((2 log_b 𝑁)↑2))
212	42, 192, 195, 199, 211	letrd 11382	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ≤ ((2 log_b 𝑁)↑2))
213	212, 194	breqtrd 5139	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≤ ((2 log_b 𝑁) · (2 log_b 𝑁)))
214		flge 13856	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2 ≤ ((2 log_b 𝑁) · (2 log_b 𝑁)) ↔ 2 ≤ (⌊‘((2 log_b 𝑁) · (2 log_b 𝑁)))))
216	213, 215	mpbid 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ≤ (⌊‘((2 log_b 𝑁) · (2 log_b 𝑁))))
217	49, 49	remulcld 11254	. . . . . . . . . . . . . 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 42948	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
220	171	sqvald 14197	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))↑2) = ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
221	26	nn0cnd 12582	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℂ)
222	221	msqsqrtd 15518	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
223	220, 222	eqtr2d 2801	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))↑2))
224	219, 223	breqtrd 5139	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))↑2))
225	49, 52, 74, 75	lt2sqd 14310	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2 log_b 𝑁) < (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ↔ ((2 log_b 𝑁)↑2) < ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))↑2)))
226	224, 225	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2 log_b 𝑁) < (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
227	49, 52, 226	ltled 11373	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 𝑁) ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
228	49, 52, 49, 74, 227	lemul2ad 12170	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2 log_b 𝑁) · (2 log_b 𝑁)) ≤ ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
229		flwordi 13863	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 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 11382	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ≤ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
232	54, 231	2ap1caineq 42970	. . . . . . . . . . 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 11383	. . . . . . . . . 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 12582	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℂ)
235	234	2timesd 12502	. . . . . . . . . . . 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 7434	. . . . . . . . . . 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 7434	. . . . . . . . . 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 5139	. . . . . . . . 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 11217	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
240	234, 234, 239	addassd 11246	. . . . . . . . . . 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 12582	. . . . . . . . . . . 12 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℂ)
242	234, 241	addcomd 11427	. . . . . . . . . . 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 2800	. . . . . . . . . 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 7434	. . . . . . . . 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 5139	. . . . . . . 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 11245	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))
247	246	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) = (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))))
248	247	oveq2d 7435	. . . . . . . . 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 7434	. . . . . . . 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 5139	. . . . . . 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 12581	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ∈ ℝ)
252	99	nn0red 12581	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℝ)
253	17, 27	eqeltrrid 2870	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
254	253	nn0red 12581	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ)
255	253	nn0ge0d 12583	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
256	254, 255	resqrtcld 15493	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ)
257	256, 49	remulcld 11254	. . . . . . . . . 10 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ∈ ℝ)
258	15, 19, 20, 21, 22, 23, 24	aks6d1c4 42949	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ≤ (ϕ‘𝑅))
259	50, 51, 86, 88	sqrtled 15502	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ≤ (ϕ‘𝑅) ↔ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ≤ (√‘(ϕ‘𝑅))))
260	258, 259	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ≤ (√‘(ϕ‘𝑅)))
261	256, 89, 49, 74, 260	lemul1ad 12169	. . . . . . . . . 10 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
262		flwordi 13863	. . . . . . . . . 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 1398	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))
264	251, 252, 142, 263	leadd2dd 11844	. . . . . . . 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 43003	. . . . . . 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 11385	. . . . . 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 11585	. . . . . . . . 9 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) − 1) = (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
268	267	eqcomd 2771	. . . . . . . 8 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) = (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) − 1))
269	241, 239	negsubd 11590	. . . . . . . . 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 2771	. . . . . . . 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 2800	. . . . . . 7 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) = (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + -1))
272	271	oveq2d 7435	. . . . . 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 5139	. . . . 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 12580	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
275	25	zncrng 21744	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ ℕ₀ → (ℤ/nℤ‘𝑅) ∈ CRing)
276	274, 275	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (ℤ/nℤ‘𝑅) ∈ CRing)
277		crngring 20371	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ/nℤ‘𝑅) ∈ CRing → (ℤ/nℤ‘𝑅) ∈ Ring)
278	24	zrhrhm 21711	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ/nℤ‘𝑅) ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom (ℤ/nℤ‘𝑅)))
279		zringbas 21653	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℤ = (Base‘ℤ_ring)
280		eqid 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘(ℤ/nℤ‘𝑅)) = (Base‘(ℤ/nℤ‘𝑅))
281	279, 280	rhmf 20613	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿 ∈ (ℤ_ring RingHom (ℤ/nℤ‘𝑅)) → 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅)))
282	276, 277, 278, 281	4syl 20	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅)))
283	282	ffnd 6710	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐿 Fn ℤ)
284	15, 19, 20, 23	aks6d1c2p1 42943	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐸:(ℕ₀ × ℕ₀)⟶ℕ)
285		nnssz 12628	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℕ ⊆ ℤ
286	285	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ℕ ⊆ ℤ)
287	284, 286	fssd 6727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐸:(ℕ₀ × ℕ₀)⟶ℤ)
288		frn 6717	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸:(ℕ₀ × ℕ₀)⟶ℤ → ran 𝐸 ⊆ ℤ)
289	287, 288	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ran 𝐸 ⊆ ℤ)
290	284	ffnd 6710	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐸 Fn (ℕ₀ × ℕ₀))
291		fnima 6669	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐸 Fn (ℕ₀ × ℕ₀) → (𝐸 “ (ℕ₀ × ℕ₀)) = ran 𝐸)
292	290, 291	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐸 “ (ℕ₀ × ℕ₀)) = ran 𝐸)
293	292	sseq1d 3969	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐸 “ (ℕ₀ × ℕ₀)) ⊆ ℤ ↔ ran 𝐸 ⊆ ℤ))
294	289, 293	mpbird 260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 “ (ℕ₀ × ℕ₀)) ⊆ ℤ)
295		vex 3461	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑘 ∈ V
296		vex 3461	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑙 ∈ V
297	295, 296	op1std 8002	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (1^st ‘𝑣) = 𝑘)
298	297	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (𝑃↑(1^st ‘𝑣)) = (𝑃↑𝑘))
299	295, 296	op2ndd 8003	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (2^nd ‘𝑣) = 𝑙)
300	299	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑁 / 𝑃)↑(2^nd ‘𝑣)) = ((𝑁 / 𝑃)↑𝑙))
301	298, 300	oveq12d 7437	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑃↑(1^st ‘𝑣)) · ((𝑁 / 𝑃)↑(2^nd ‘𝑣))) = ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
302	301	mpompt 7533	. . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . 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 2788	. . . . . . . . . . . . . . . . . . . . 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 11215	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ V
307	306, 306	op1std 8002	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = ⟨0, 0⟩ → (1^st ‘𝑣) = 0)
308	307	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → (1^st ‘𝑣) = 0)
309	308	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → (𝑃↑(1^st ‘𝑣)) = (𝑃↑0))
310	306, 306	op2ndd 8003	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = ⟨0, 0⟩ → (2^nd ‘𝑣) = 0)
311	310	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → (2^nd ‘𝑣) = 0)
312	311	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → ((𝑁 / 𝑃)↑(2^nd ‘𝑣)) = ((𝑁 / 𝑃)↑0))
313	309, 312	oveq12d 7437	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → ((𝑃↑(1^st ‘𝑣)) · ((𝑁 / 𝑃)↑(2^nd ‘𝑣))) = ((𝑃↑0) · ((𝑁 / 𝑃)↑0)))
314		prmnn 16754	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
315	19, 314	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑃 ∈ ℕ)
316	315	nncnd 12264	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑃 ∈ ℂ)
317	316	exp0d 14194	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑃↑0) = 1)
318	315	nnne0d 12301	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑃 ≠ 0)
319	155, 316, 318	divcld 12006	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℂ)
320	319	exp0d 14194	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((𝑁 / 𝑃)↑0) = 1)
321	317, 320	oveq12d 7437	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑃↑0) · ((𝑁 / 𝑃)↑0)) = (1 · 1))
322	239	mulridd 11241	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1 · 1) = 1)
323	321, 322	eqtrd 2800	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((𝑃↑0) · ((𝑁 / 𝑃)↑0)) = 1)
324	323	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → ((𝑃↑0) · ((𝑁 / 𝑃)↑0)) = 1)
325	313, 324	eqtrd 2800	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → ((𝑃↑(1^st ‘𝑣)) · ((𝑁 / 𝑃)↑(2^nd ‘𝑣))) = 1)
326		0nn0 12534	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ ℕ₀
327	326	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 ∈ ℕ₀)
328	327, 327	opelxpd 5702	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ⟨0, 0⟩ ∈ (ℕ₀ × ℕ₀))
329		1nn 12259	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℕ
330	329	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 1 ∈ ℕ)
331	305, 325, 328, 330	fvmptd 7001	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐸‘⟨0, 0⟩) = 1)
332		ssidd 3961	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (ℕ₀ × ℕ₀) ⊆ (ℕ₀ × ℕ₀))
333		fnfvima 7238	. . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐸‘⟨0, 0⟩) ∈ (𝐸 “ (ℕ₀ × ℕ₀)))
335	331, 334	eqeltrrd 2866	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ (𝐸 “ (ℕ₀ × ℕ₀)))
336		fnfvima 7238	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐿 Fn ℤ ∧ (𝐸 “ (ℕ₀ × ℕ₀)) ⊆ ℤ ∧ 1 ∈ (𝐸 “ (ℕ₀ × ℕ₀))) → (𝐿‘1) ∈ (𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))
337	283, 294, 335, 336	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐿‘1) ∈ (𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))
338	24	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅)))
339		fvexd 6900	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℤRHom‘(ℤ/nℤ‘𝑅)) ∈ V)
340	338, 339	eqeltrd 2865	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐿 ∈ V)
341	340	imaexd 7919	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))) ∈ V)
342	337, 341	hashelne0d 14422	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = 0)
343	342	neqned 2967	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ≠ 0)
344	26, 343	jca 521	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀ ∧ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ≠ 0))
345		elnnne0 12533	. . . . . . . . . . . . . 14 ⊢ ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ ↔ ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀ ∧ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ≠ 0))
346	344, 345	sylibr 237	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ)
347	346	nnrpd 13074	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ⁺)
348	347	rpsqrtcld 15487	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ⁺)
349	49, 52, 348, 226	ltmul1dd 13131	. . . . . . . . . 10 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) < ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
350	50, 51, 50, 51	sqrtmuld 15500	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
351	350	eqcomd 2771	. . . . . . . . . 10 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
352	349, 351	breqtrd 5139	. . . . . . . . 9 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) < (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
353	350, 222	eqtrd 2800	. . . . . . . . 9 ⊢ (𝜑 → (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
354	352, 353	breqtrd 5139	. . . . . . . 8 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) < (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
355		fllt 13857	. . . . . . . . 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 596	. . . . . . . 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 235	. . . . . . 7 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) < (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
358	54, 109	zltp1led 12664	. . . . . . 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 235	. . . . . 6 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
360	55	renegcld 11656	. . . . . . 7 ⊢ (𝜑 → -1 ∈ ℝ)
361		df-neg 11459	. . . . . . . . 9 ⊢ -1 = (0 − 1)
362	361	a1i 11	. . . . . . . 8 ⊢ (𝜑 → -1 = (0 − 1))
363	4	lem1d 12163	. . . . . . . 8 ⊢ (𝜑 → (0 − 1) ≤ 0)
364	362, 363	eqbrtrd 5135	. . . . . . 7 ⊢ (𝜑 → -1 ≤ 0)
365	360, 4, 252, 364, 96	letrd 11382	. . . . . 6 ⊢ (𝜑 → -1 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))
366	84, 26, 99, 103, 359, 365	bcle2d 43004	. . . . 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 11385	. . . 4 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1)))
368	221, 239	negsubd 11590	. . . . 5 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1) = ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1))
369	368	oveq2d 7435	. . . 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 5139	. . 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 2774	. . . . . 6 ⊢ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) = 𝐴
373	372	a1i 11	. . . . 5 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) = 𝐴)
374	373	oveq2d 7435	. . . 4 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))) = ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + 𝐴))
375	374	oveq1d 7434	. . 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 5139	. 2 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + 𝐴)C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1)))
377	18	eqcomd 2771	. . . 4 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = 𝐷)
378	377	oveq1d 7434	. . 3 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + 𝐴) = (𝐷 + 𝐴))
379	377	oveq1d 7434	. . 3 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1) = (𝐷 − 1))
380	378, 379	oveq12d 7437	. 2 ⊢ (𝜑 → (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + 𝐴)C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1)) = ((𝐷 + 𝐴)C(𝐷 − 1)))
381	376, 380	breqtrd 5139	1 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < ((𝐷 + 𝐴)C(𝐷 − 1)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 Vcvv 3457 ∖ cdif 3903 ⊆ wss 3906 {csn 4591 {cpr 4593 ⟨cop 4597 class class class wbr 5111 ↦ cmpt 5194 × cxp 5661 ran crn 5664 “ cima 5666 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ∈ cmpo 7421 1^st c1st 7990 2^nd c2nd 7991 ℂcc 11113 ℝcr 11114 0cc0 11115 1c1 11116 + caddc 11118 · cmul 11120 < clt 11258 ≤ cle 11259 − cmin 11456 -cneg 11457 / cdiv 11886 ℕcn 12248 2c2 12310 3c3 12311 5c5 12313 ℕ₀cn0 12519 ℤcz 12606 ℤ_≥cuz 12878 ℝ⁺crp 13032 ⌊cfl 13841 ↑cexp 14115 Ccbc 14356 ♯chash 14384 √csqrt 15308 ∥ cdvds 16332 gcd cgcd 16574 ℙcprime 16751 od_ℤcodz 16844 ϕcphi 16845 Basecbs 17291 Ringcrg 20359 CRingccrg 20360 RingHom crh 20597 ℤ_ringczring 21646 ℤRHomczrh 21699 ℤ/nℤczn 21702 ↑_𝑐ccxp 26771 log_b clogb 26980

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193 ax-addf 11194 ax-mulf 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-tpos 8228 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-er 8700 df-ec 8702 df-qs 8706 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-dju 9903 df-card 9941 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-xnn0 12593 df-z 12607 df-dec 12728 df-uz 12879 df-q 12989 df-rp 13033 df-xneg 13153 df-xadd 13154 df-xmul 13155 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13552 df-fzo 13700 df-fl 13843 df-mod 13921 df-seq 14056 df-exp 14116 df-fac 14328 df-bc 14357 df-hash 14385 df-shft 15128 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-limsup 15546 df-clim 15563 df-rlim 15564 df-sum 15762 df-prod 15981 df-fallfac 16084 df-ef 16143 df-sin 16145 df-cos 16146 df-pi 16148 df-dvds 16333 df-gcd 16575 df-prm 16752 df-odz 16846 df-phi 16847 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-starv 17347 df-sca 17348 df-vsca 17349 df-ip 17350 df-tset 17351 df-ple 17352 df-ds 17354 df-unif 17355 df-hom 17356 df-cco 17357 df-rest 17497 df-topn 17498 df-0g 17516 df-gsum 17517 df-topgen 17518 df-pt 17519 df-prds 17522 df-xrs 17578 df-qtop 17583 df-imas 17584 df-qus 17585 df-xps 17586 df-mre 17660 df-mrc 17661 df-acs 17663 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-mhm 18878 df-submnd 18879 df-grp 19047 df-minusg 19048 df-sbg 19049 df-mulg 19178 df-subg 19233 df-nsg 19234 df-eqg 19235 df-ghm 19328 df-cntz 19431 df-cmn 19896 df-abl 19897 df-mgp 20261 df-rng 20275 df-ur 20308 df-ring 20361 df-cring 20362 df-oppr 20465 df-dvdsr 20485 df-unit 20486 df-rhm 20600 df-subrng 20695 df-subrg 20719 df-lmod 21033 df-lss 21103 df-lsp 21143 df-sra 21344 df-rgmod 21345 df-lidl 21382 df-rsp 21383 df-2idl 21439 df-psmet 21564 df-xmet 21565 df-met 21566 df-bl 21567 df-mopn 21568 df-fbas 21569 df-fg 21570 df-cnfld 21573 df-zring 21647 df-zrh 21703 df-zn 21706 df-top 23101 df-topon 23118 df-topsp 23140 df-bases 23153 df-cld 23226 df-ntr 23227 df-cls 23228 df-nei 23305 df-lp 23343 df-perf 23344 df-cn 23434 df-cnp 23435 df-haus 23522 df-tx 23770 df-hmeo 23963 df-fil 24054 df-fm 24146 df-flim 24147 df-flf 24148 df-xms 24528 df-ms 24529 df-tms 24530 df-cncf 25088 df-limc 26076 df-dv 26077 df-log 26772 df-cxp 26773 df-logb 26981

This theorem is used by: aks6d1c7lem2 43006

W3C validator