aks6d1c7lem1 - Mathbox for metakunt

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Theorem aks6d1c7lem1 42153

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 12745	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
3	1, 2	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
4		0red 11118	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
5		3re 12208	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
6	5	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 3 ∈ ℝ)
7	3	zred 12580	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
8		3pos 12233	. . . . . . . . . . 11 ⊢ 0 < 3
9	8	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 3)
10		eluzle 12748	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
11	1, 10	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁)
12	4, 6, 7, 9, 11	ltletrd 11276	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝑁)
13	3, 12	jca 511	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
14		elnnz 12481	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
15	13, 14	sylibr 234	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
16	15	nnred 12143	. . . . . 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 2729	. . . . . . . . . . . . 13 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
26	15, 19, 20, 21, 22, 23, 24, 25	hashscontpowcl 42093	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
27	18, 26	eqeltrd 2828	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ ℕ₀)
28	27	nn0red 12446	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ ℝ)
29	27	nn0ge0d 12448	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝐷)
30	28, 29	resqrtcld 15325	. . . . . . . . 9 ⊢ (𝜑 → (√‘𝐷) ∈ ℝ)
31	30	flcld 13702	. . . . . . . 8 ⊢ (𝜑 → (⌊‘(√‘𝐷)) ∈ ℤ)
32	28, 29	sqrtge0d 15328	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (√‘𝐷))
33		0zd 12483	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
34		flge 13709	. . . . . . . . . 10 ⊢ (((√‘𝐷) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ (√‘𝐷) ↔ 0 ≤ (⌊‘(√‘𝐷))))
35	30, 33, 34	syl2anc 584	. . . . . . . . 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 12484	. . . . . . 7 ⊢ ((⌊‘(√‘𝐷)) ∈ ℕ₀ ↔ ((⌊‘(√‘𝐷)) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘𝐷))))
39	37, 38	sylibr 234	. . . . . 6 ⊢ (𝜑 → (⌊‘(√‘𝐷)) ∈ ℕ₀)
40	16, 39	reexpcld 14070	. . . . 5 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ∈ ℝ)
41		2re 12202	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
42	41	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℝ)
43		2pos 12231	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
44	43	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 2)
45	15	nngt0d 12177	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑁)
46		1ne2 12331	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ 2
47	46	necomi 2979	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 1
48	47	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≠ 1)
49	42, 44, 16, 45, 48	relogbcld 41946	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
50	18, 28	eqeltrrd 2829	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ)
51	29, 18	breqtrd 5118	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
52	50, 51	resqrtcld 15325	. . . . . . . . . . . . 13 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ)
53	49, 52	remulcld 11145	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℝ)
54	53	flcld 13702	. . . . . . . . . . 11 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℤ)
55		1red 11116	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ)
56		0le1 11643	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 1
57	56	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ 1)
58	42	recnd 11143	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ∈ ℂ)
59	4, 44	gtned 11251	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ≠ 0)
60		logbid1 26676	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 log_b 2) = 1)
61	58, 59, 48, 60	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2 log_b 2) = 1)
62	61	eqcomd 2735	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 = (2 log_b 2))
63		2z 12507	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℤ
64	63	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ∈ ℤ)
65	42	leidd 11686	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≤ 2)
66		1nn0 12400	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℕ₀
67	41, 66	nn0addge1i 12432	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ≤ (2 + 1)
68		2p1e3 12265	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 + 1) = 3
69	67, 68	breqtri 5117	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≤ 3
70	69	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ≤ 3)
71	42, 6, 7, 70, 11	letrd 11273	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≤ 𝑁)
72	64, 65, 42, 44, 7, 12, 71	logblebd 41949	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 2) ≤ (2 log_b 𝑁))
73	62, 72	eqbrtrd 5114	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ≤ (2 log_b 𝑁))
74	4, 55, 49, 57, 73	letrd 11273	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (2 log_b 𝑁))
75	50, 51	sqrtge0d 15328	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
76	49, 52, 74, 75	mulge0d 11697	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
77		flge 13709	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 12484	. . . . . . . . . 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 12449	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℕ₀)
85	21	phicld 16683	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ)
86	85	nnred 12143	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℝ)
87	85	nnnn0d 12445	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ₀)
88	87	nn0ge0d 12448	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (ϕ‘𝑅))
89	86, 88	resqrtcld 15325	. . . . . . . . . . . 12 ⊢ (𝜑 → (√‘(ϕ‘𝑅)) ∈ ℝ)
90	89, 49	remulcld 11145	. . . . . . . . . . 11 ⊢ (𝜑 → ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ∈ ℝ)
91	90	flcld 13702	. . . . . . . . . 10 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℤ)
92	86, 88	sqrtge0d 15328	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (√‘(ϕ‘𝑅)))
93	89, 49, 92, 74	mulge0d 11697	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
94		flge 13709	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 12484	. . . . . . . . 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 12449	. . . . . . 7 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))) ∈ ℕ₀)
101	54	peano2zd 12583	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℤ)
102		1zzd 12506	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
103	102	znegcld 12582	. . . . . . . 8 ⊢ (𝜑 → -1 ∈ ℤ)
104	101, 103	zaddcld 12584	. . . . . . 7 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + -1) ∈ ℤ)
105		bccl 14229	. . . . . . 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 584	. . . . . 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 12446	. . . . 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 12449	. . . . . . 7 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))) ∈ ℕ₀)
109	26	nn0zd 12497	. . . . . . . 8 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℤ)
110	109, 103	zaddcld 12584	. . . . . . 7 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1) ∈ ℤ)
111		bccl 14229	. . . . . . 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 584	. . . . . 6 ⊢ (𝜑 → (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1)) ∈ ℕ₀)
113	112	nn0red 12446	. . . . 5 ⊢ (𝜑 → (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1)) ∈ ℝ)
114	52, 49	remulcld 11145	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ∈ ℝ)
115	114	flcld 13702	. . . . . . . . . . . 12 ⊢ (𝜑 → (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ∈ ℤ)
116	52, 49, 75, 74	mulge0d 11697	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))
117		flge 13709	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 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 12484	. . . . . . . . . . 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 12449	. . . . . . . . 9 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))) ∈ ℕ₀)
124		bccl 14229	. . . . . . . . 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 584	. . . . . . . 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 12446	. . . . . . 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 14229	. . . . . . . . 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 584	. . . . . . . 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 12446	. . . . . . 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 14070	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)) ∈ ℝ)
131		2nn0 12401	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ₀
132	131	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℕ₀)
133	132, 82	nn0mulcld 12450	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2 · (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) ∈ ℕ₀)
134	133, 83	nn0addcld 12449	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2 · (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1) ∈ ℕ₀)
135		bccl 14229	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 12446	. . . . . . . . . . 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 11264	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ 2)
139	42, 138, 53	recxpcld 26630	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2↑_𝑐((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℝ)
140		reflcl 13700	. . . . . . . . . . . . . . . 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 11144	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℝ)
143	42, 138, 142	recxpcld 26630	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2↑_𝑐((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)) ∈ ℝ)
144		1le2 12332	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ≤ 2
145	144	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≤ 2)
146	55, 42, 7, 145, 71	letrd 11273	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ≤ 𝑁)
147		reflcl 13700	. . . . . . . . . . . . . . . . 17 ⊢ ((√‘𝐷) ∈ ℝ → (⌊‘(√‘𝐷)) ∈ ℝ)
148	30, 147	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘(√‘𝐷)) ∈ ℝ)
149	18	fveq2d 6826	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (√‘𝐷) = (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
150	149	fveq2d 6826	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (⌊‘(√‘𝐷)) = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
151		flle 13703	. . . . . . . . . . . . . . . . . 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 5114	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘(√‘𝐷)) ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
154	7, 146, 148, 52, 153	cxplead 26628	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁↑_𝑐(⌊‘(√‘𝐷))) ≤ (𝑁↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
155	7	recnd 11143	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℂ)
156	4, 12	gtned 11251	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ≠ 0)
157	155, 156, 31	cxpexpzd 26618	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁↑_𝑐(⌊‘(√‘𝐷))) = (𝑁↑(⌊‘(√‘𝐷))))
158	59, 48	nelprd 4609	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 2 ∈ {0, 1})
159	58, 158	eldifd 3914	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2 ∈ (ℂ ∖ {0, 1}))
160	156	neneqd 2930	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ¬ 𝑁 = 0)
161		elsng 4591	. . . . . . . . . . . . . . . . . . . . 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 3914	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℂ ∖ {0}))
165		cxplogb 26694	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 𝑁 ∈ (ℂ ∖ {0})) → (2↑_𝑐(2 log_b 𝑁)) = 𝑁)
166	159, 164, 165	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2↑_𝑐(2 log_b 𝑁)) = 𝑁)
167	166	eqcomd 2735	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 = (2↑_𝑐(2 log_b 𝑁)))
168	167	oveq1d 7364	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = ((2↑_𝑐(2 log_b 𝑁))↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
169	154, 157, 168	3brtr3d 5123	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ≤ ((2↑_𝑐(2 log_b 𝑁))↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
170	42, 44	elrpd 12934	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℝ⁺)
171	52	recnd 11143	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℂ)
172		cxpmul 26595	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 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 5120	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ≤ (2↑_𝑐((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
175		fllep1 13705	. . . . . . . . . . . . . . 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 11270	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 < 2)
178	84	nn0red 12446	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℝ)
179	42, 177, 53, 178	cxpled 26627	. . . . . . . . . . . . . 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 11273	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ≤ (2↑_𝑐((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)))
182		cxpexpz 26574	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 5118	. . . . . . . . . . 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 11094	. . . . . . . . . . . . . . 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 13700	. . . . . . . . . . . . . 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 12446	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℝ)
191	42, 44, 6, 9, 48	relogbcld 41946	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 log_b 3) ∈ ℝ)
192	191	resqcld 14032	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2 log_b 3)↑2) ∈ ℝ)
193	49	recnd 11143	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℂ)
194	193	sqvald 14050	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) = ((2 log_b 𝑁) · (2 log_b 𝑁)))
195	194, 187	eqeltrd 2828	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) ∈ ℝ)
196		3lexlogpow2ineq2 42032	. . . . . . . . . . . . . . . . . . 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 11264	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≤ ((2 log_b 3)↑2))
200	6, 42, 59	redivcld 11952	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (3 / 2) ∈ ℝ)
201		2rp 12898	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ⁺
202	201	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 2 ∈ ℝ⁺)
203	4, 6, 9	ltled 11264	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 ≤ 3)
204	6, 202, 203	divge0d 12977	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 0 ≤ (3 / 2))
205		3lexlogpow2ineq1 42031	. . . . . . . . . . . . . . . . . . . . 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 11264	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (3 / 2) ≤ (2 log_b 3))
209	4, 200, 191, 204, 208	letrd 11273	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ≤ (2 log_b 3))
210	64, 65, 6, 9, 7, 12, 11	logblebd 41949	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 log_b 3) ≤ (2 log_b 𝑁))
211	191, 49, 132, 209, 210	leexp1ad 14083	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2 log_b 3)↑2) ≤ ((2 log_b 𝑁)↑2))
212	42, 192, 195, 199, 211	letrd 11273	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ≤ ((2 log_b 𝑁)↑2))
213	212, 194	breqtrd 5118	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≤ ((2 log_b 𝑁) · (2 log_b 𝑁)))
214		flge 13709	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 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 11145	. . . . . . . . . . . . . 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 42096	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
220	171	sqvald 14050	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))↑2) = ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
221	26	nn0cnd 12447	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℂ)
222	221	msqsqrtd 15350	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
223	220, 222	eqtr2d 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))↑2))
224	219, 223	breqtrd 5118	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))↑2))
225	49, 52, 74, 75	lt2sqd 14163	. . . . . . . . . . . . . . . . 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 11264	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 𝑁) ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
228	49, 52, 49, 74, 227	lemul2ad 12065	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2 log_b 𝑁) · (2 log_b 𝑁)) ≤ ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
229		flwordi 13716	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 11273	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ≤ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
232	54, 231	2ap1caineq 42118	. . . . . . . . . . 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 11274	. . . . . . . . . 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 12447	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℂ)
235	234	2timesd 12367	. . . . . . . . . . . 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 7364	. . . . . . . . . . 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 7364	. . . . . . . . . 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 5118	. . . . . . . . 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 11110	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
240	234, 234, 239	addassd 11137	. . . . . . . . . . 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 12447	. . . . . . . . . . . 12 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℂ)
242	234, 241	addcomd 11318	. . . . . . . . . . 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 2764	. . . . . . . . . 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 7364	. . . . . . . . 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 5118	. . . . . . . 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 11136	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))
247	246	fveq2d 6826	. . . . . . . . . 10 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) = (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))))
248	247	oveq2d 7365	. . . . . . . . 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 7364	. . . . . . . 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 5118	. . . . . . 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 12446	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ∈ ℝ)
252	99	nn0red 12446	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℝ)
253	17, 27	eqeltrrid 2833	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
254	253	nn0red 12446	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ)
255	253	nn0ge0d 12448	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
256	254, 255	resqrtcld 15325	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ)
257	256, 49	remulcld 11145	. . . . . . . . . 10 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ∈ ℝ)
258	15, 19, 20, 21, 22, 23, 24	aks6d1c4 42097	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ≤ (ϕ‘𝑅))
259	50, 51, 86, 88	sqrtled 15334	. . . . . . . . . . . 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 12064	. . . . . . . . . 10 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
262		flwordi 13716	. . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))
264	251, 252, 142, 263	leadd2dd 11735	. . . . . . . 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 42151	. . . . . . 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 11276	. . . . . 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 11476	. . . . . . . . 9 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) − 1) = (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
268	267	eqcomd 2735	. . . . . . . 8 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) = (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) − 1))
269	241, 239	negsubd 11481	. . . . . . . . 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 2735	. . . . . . . 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 2764	. . . . . . 7 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) = (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + -1))
272	271	oveq2d 7365	. . . . . 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 5118	. . . . 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 12445	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
275	25	zncrng 21451	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ ℕ₀ → (ℤ/nℤ‘𝑅) ∈ CRing)
276	274, 275	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (ℤ/nℤ‘𝑅) ∈ CRing)
277		crngring 20130	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ/nℤ‘𝑅) ∈ CRing → (ℤ/nℤ‘𝑅) ∈ Ring)
278	24	zrhrhm 21418	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ/nℤ‘𝑅) ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom (ℤ/nℤ‘𝑅)))
279		zringbas 21360	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℤ = (Base‘ℤ_ring)
280		eqid 2729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘(ℤ/nℤ‘𝑅)) = (Base‘(ℤ/nℤ‘𝑅))
281	279, 280	rhmf 20370	. . . . . . . . . . . . . . . . . . . 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 6653	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐿 Fn ℤ)
284	15, 19, 20, 23	aks6d1c2p1 42091	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐸:(ℕ₀ × ℕ₀)⟶ℕ)
285		nnssz 12493	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℕ ⊆ ℤ
286	285	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ℕ ⊆ ℤ)
287	284, 286	fssd 6669	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐸:(ℕ₀ × ℕ₀)⟶ℤ)
288		frn 6659	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸:(ℕ₀ × ℕ₀)⟶ℤ → ran 𝐸 ⊆ ℤ)
289	287, 288	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ran 𝐸 ⊆ ℤ)
290	284	ffnd 6653	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐸 Fn (ℕ₀ × ℕ₀))
291		fnima 6612	. . . . . . . . . . . . . . . . . . . . 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 3967	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐸 “ (ℕ₀ × ℕ₀)) ⊆ ℤ ↔ ran 𝐸 ⊆ ℤ))
294	289, 293	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 “ (ℕ₀ × ℕ₀)) ⊆ ℤ)
295		vex 3440	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑘 ∈ V
296		vex 3440	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑙 ∈ V
297	295, 296	op1std 7934	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (1^st ‘𝑣) = 𝑘)
298	297	oveq2d 7365	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (𝑃↑(1^st ‘𝑣)) = (𝑃↑𝑘))
299	295, 296	op2ndd 7935	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (2^nd ‘𝑣) = 𝑙)
300	299	oveq2d 7365	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑁 / 𝑃)↑(2^nd ‘𝑣)) = ((𝑁 / 𝑃)↑𝑙))
301	298, 300	oveq12d 7367	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑃↑(1^st ‘𝑣)) · ((𝑁 / 𝑃)↑(2^nd ‘𝑣))) = ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
302	301	mpompt 7463	. . . . . . . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . . . . . . . . . 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 11109	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ V
307	306, 306	op1std 7934	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = ⟨0, 0⟩ → (1^st ‘𝑣) = 0)
308	307	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → (1^st ‘𝑣) = 0)
309	308	oveq2d 7365	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → (𝑃↑(1^st ‘𝑣)) = (𝑃↑0))
310	306, 306	op2ndd 7935	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = ⟨0, 0⟩ → (2^nd ‘𝑣) = 0)
311	310	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → (2^nd ‘𝑣) = 0)
312	311	oveq2d 7365	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → ((𝑁 / 𝑃)↑(2^nd ‘𝑣)) = ((𝑁 / 𝑃)↑0))
313	309, 312	oveq12d 7367	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → ((𝑃↑(1^st ‘𝑣)) · ((𝑁 / 𝑃)↑(2^nd ‘𝑣))) = ((𝑃↑0) · ((𝑁 / 𝑃)↑0)))
314		prmnn 16585	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
315	19, 314	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑃 ∈ ℕ)
316	315	nncnd 12144	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑃 ∈ ℂ)
317	316	exp0d 14047	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑃↑0) = 1)
318	315	nnne0d 12178	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑃 ≠ 0)
319	155, 316, 318	divcld 11900	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℂ)
320	319	exp0d 14047	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((𝑁 / 𝑃)↑0) = 1)
321	317, 320	oveq12d 7367	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑃↑0) · ((𝑁 / 𝑃)↑0)) = (1 · 1))
322	239	mulridd 11132	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (1 · 1) = 1)
323	321, 322	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → ((𝑃↑(1^st ‘𝑣)) · ((𝑁 / 𝑃)↑(2^nd ‘𝑣))) = 1)
326		0nn0 12399	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ ℕ₀
327	326	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 ∈ ℕ₀)
328	327, 327	opelxpd 5658	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ⟨0, 0⟩ ∈ (ℕ₀ × ℕ₀))
329		1nn 12139	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℕ
330	329	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 1 ∈ ℕ)
331	305, 325, 328, 330	fvmptd 6937	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐸‘⟨0, 0⟩) = 1)
332		ssidd 3959	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (ℕ₀ × ℕ₀) ⊆ (ℕ₀ × ℕ₀))
333		fnfvima 7169	. . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐸‘⟨0, 0⟩) ∈ (𝐸 “ (ℕ₀ × ℕ₀)))
335	331, 334	eqeltrrd 2829	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ (𝐸 “ (ℕ₀ × ℕ₀)))
336		fnfvima 7169	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐿 Fn ℤ ∧ (𝐸 “ (ℕ₀ × ℕ₀)) ⊆ ℤ ∧ 1 ∈ (𝐸 “ (ℕ₀ × ℕ₀))) → (𝐿‘1) ∈ (𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))
337	283, 294, 335, 336	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐿‘1) ∈ (𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))
338	24	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅)))
339		fvexd 6837	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℤRHom‘(ℤ/nℤ‘𝑅)) ∈ V)
340	338, 339	eqeltrd 2828	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐿 ∈ V)
341	340	imaexd 7849	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))) ∈ V)
342	337, 341	hashelne0d 14275	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = 0)
343	342	neqned 2932	. . . . . . . . . . . . . . 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 12398	. . . . . . . . . . . . . 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 12935	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ⁺)
348	347	rpsqrtcld 15319	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ⁺)
349	49, 52, 348, 226	ltmul1dd 12992	. . . . . . . . . 10 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) < ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
350	50, 51, 50, 51	sqrtmuld 15332	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
351	350	eqcomd 2735	. . . . . . . . . 10 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
352	349, 351	breqtrd 5118	. . . . . . . . 9 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) < (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
353	350, 222	eqtrd 2764	. . . . . . . . 9 ⊢ (𝜑 → (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
354	352, 353	breqtrd 5118	. . . . . . . 8 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) < (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
355		fllt 13710	. . . . . . . . 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 584	. . . . . . . 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 41952	. . . . . . 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 11547	. . . . . . 7 ⊢ (𝜑 → -1 ∈ ℝ)
361		df-neg 11350	. . . . . . . . 9 ⊢ -1 = (0 − 1)
362	361	a1i 11	. . . . . . . 8 ⊢ (𝜑 → -1 = (0 − 1))
363	4	lem1d 12058	. . . . . . . 8 ⊢ (𝜑 → (0 − 1) ≤ 0)
364	362, 363	eqbrtrd 5114	. . . . . . 7 ⊢ (𝜑 → -1 ≤ 0)
365	360, 4, 252, 364, 96	letrd 11273	. . . . . 6 ⊢ (𝜑 → -1 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))
366	84, 26, 99, 103, 359, 365	bcle2d 42152	. . . . 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 11276	. . . 4 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1)))
368	221, 239	negsubd 11481	. . . . 5 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1) = ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1))
369	368	oveq2d 7365	. . . 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 5118	. . 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 2738	. . . . . 6 ⊢ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) = 𝐴
373	372	a1i 11	. . . . 5 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) = 𝐴)
374	373	oveq2d 7365	. . . 4 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))) = ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + 𝐴))
375	374	oveq1d 7364	. . 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 5118	. 2 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + 𝐴)C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1)))
377	18	eqcomd 2735	. . . 4 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = 𝐷)
378	377	oveq1d 7364	. . 3 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + 𝐴) = (𝐷 + 𝐴))
379	377	oveq1d 7364	. . 3 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1) = (𝐷 − 1))
380	378, 379	oveq12d 7367	. 2 ⊢ (𝜑 → (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + 𝐴)C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1)) = ((𝐷 + 𝐴)C(𝐷 − 1)))
381	376, 380	breqtrd 5118	1 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < ((𝐷 + 𝐴)C(𝐷 − 1)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 Vcvv 3436 ∖ cdif 3900 ⊆ wss 3903 {csn 4577 {cpr 4579 ⟨cop 4583 class class class wbr 5092 ↦ cmpt 5173 × cxp 5617 ran crn 5620 “ cima 5622 Fn wfn 6477 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ∈ cmpo 7351 1^st c1st 7922 2^nd c2nd 7923 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 < clt 11149 ≤ cle 11150 − cmin 11347 -cneg 11348 / cdiv 11777 ℕcn 12128 2c2 12183 3c3 12184 5c5 12186 ℕ₀cn0 12384 ℤcz 12471 ℤ_≥cuz 12735 ℝ⁺crp 12893 ⌊cfl 13694 ↑cexp 13968 Ccbc 14209 ♯chash 14237 √csqrt 15140 ∥ cdvds 16163 gcd cgcd 16405 ℙcprime 16582 od_ℤcodz 16674 ϕcphi 16675 Basecbs 17120 Ringcrg 20118 CRingccrg 20119 RingHom crh 20354 ℤ_ringczring 21353 ℤRHomczrh 21406 ℤ/nℤczn 21409 ↑_𝑐ccxp 26462 log_b clogb 26672

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-tpos 8159 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-er 8625 df-ec 8627 df-qs 8631 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9797 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-xnn0 12458 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ioc 13253 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-prod 15811 df-fallfac 15914 df-ef 15974 df-sin 15976 df-cos 15977 df-pi 15979 df-dvds 16164 df-gcd 16406 df-prm 16583 df-odz 16676 df-phi 16677 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-qus 17413 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-mhm 18657 df-submnd 18658 df-grp 18815 df-minusg 18816 df-sbg 18817 df-mulg 18947 df-subg 19002 df-nsg 19003 df-eqg 19004 df-ghm 19092 df-cntz 19196 df-cmn 19661 df-abl 19662 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-rhm 20357 df-subrng 20431 df-subrg 20455 df-lmod 20765 df-lss 20835 df-lsp 20875 df-sra 21077 df-rgmod 21078 df-lidl 21115 df-rsp 21116 df-2idl 21157 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-zring 21354 df-zrh 21410 df-zn 21413 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-limc 25765 df-dv 25766 df-log 26463 df-cxp 26464 df-logb 26673

This theorem is referenced by: aks6d1c7lem2 42154

W3C validator