aks6d1c7lem1 - Mathbox for metakunt

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Theorem aks6d1c7lem1 42141

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 12779	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
3	1, 2	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
4		0red 11153	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
5		3re 12242	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
6	5	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 3 ∈ ℝ)
7	3	zred 12614	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
8		3pos 12267	. . . . . . . . . . 11 ⊢ 0 < 3
9	8	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 3)
10		eluzle 12782	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
11	1, 10	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁)
12	4, 6, 7, 9, 11	ltletrd 11310	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝑁)
13	3, 12	jca 511	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
14		elnnz 12515	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
15	13, 14	sylibr 234	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
16	15	nnred 12177	. . . . . 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 42081	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
27	18, 26	eqeltrd 2828	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ ℕ₀)
28	27	nn0red 12480	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ ℝ)
29	27	nn0ge0d 12482	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝐷)
30	28, 29	resqrtcld 15360	. . . . . . . . 9 ⊢ (𝜑 → (√‘𝐷) ∈ ℝ)
31	30	flcld 13736	. . . . . . . 8 ⊢ (𝜑 → (⌊‘(√‘𝐷)) ∈ ℤ)
32	28, 29	sqrtge0d 15363	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (√‘𝐷))
33		0zd 12517	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
34		flge 13743	. . . . . . . . . 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 12518	. . . . . . 7 ⊢ ((⌊‘(√‘𝐷)) ∈ ℕ₀ ↔ ((⌊‘(√‘𝐷)) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘𝐷))))
39	37, 38	sylibr 234	. . . . . 6 ⊢ (𝜑 → (⌊‘(√‘𝐷)) ∈ ℕ₀)
40	16, 39	reexpcld 14104	. . . . 5 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ∈ ℝ)
41		2re 12236	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
42	41	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℝ)
43		2pos 12265	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
44	43	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 2)
45	15	nngt0d 12211	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑁)
46		1ne2 12365	. . . . . . . . . . . . . . . 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 41934	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
50	18, 28	eqeltrrd 2829	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ)
51	29, 18	breqtrd 5128	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
52	50, 51	resqrtcld 15360	. . . . . . . . . . . . 13 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ)
53	49, 52	remulcld 11180	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℝ)
54	53	flcld 13736	. . . . . . . . . . 11 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℤ)
55		1red 11151	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ)
56		0le1 11677	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 1
57	56	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ 1)
58	42	recnd 11178	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ∈ ℂ)
59	4, 44	gtned 11285	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ≠ 0)
60		logbid1 26654	. . . . . . . . . . . . . . . . 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 12541	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℤ
64	63	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ∈ ℤ)
65	42	leidd 11720	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≤ 2)
66		1nn0 12434	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℕ₀
67	41, 66	nn0addge1i 12466	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ≤ (2 + 1)
68		2p1e3 12299	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 + 1) = 3
69	67, 68	breqtri 5127	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ≤ 3
70	69	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ≤ 3)
71	42, 6, 7, 70, 11	letrd 11307	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≤ 𝑁)
72	64, 65, 42, 44, 7, 12, 71	logblebd 41937	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 2) ≤ (2 log_b 𝑁))
73	62, 72	eqbrtrd 5124	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ≤ (2 log_b 𝑁))
74	4, 55, 49, 57, 73	letrd 11307	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (2 log_b 𝑁))
75	50, 51	sqrtge0d 15363	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
76	49, 52, 74, 75	mulge0d 11731	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
77		flge 13743	. . . . . . . . . . . . 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 12518	. . . . . . . . . 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 12483	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℕ₀)
85	21	phicld 16718	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ)
86	85	nnred 12177	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℝ)
87	85	nnnn0d 12479	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ₀)
88	87	nn0ge0d 12482	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (ϕ‘𝑅))
89	86, 88	resqrtcld 15360	. . . . . . . . . . . 12 ⊢ (𝜑 → (√‘(ϕ‘𝑅)) ∈ ℝ)
90	89, 49	remulcld 11180	. . . . . . . . . . 11 ⊢ (𝜑 → ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ∈ ℝ)
91	90	flcld 13736	. . . . . . . . . 10 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℤ)
92	86, 88	sqrtge0d 15363	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (√‘(ϕ‘𝑅)))
93	89, 49, 92, 74	mulge0d 11731	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
94		flge 13743	. . . . . . . . . . . 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 12518	. . . . . . . . 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 12483	. . . . . . 7 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))) ∈ ℕ₀)
101	54	peano2zd 12617	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℤ)
102		1zzd 12540	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
103	102	znegcld 12616	. . . . . . . 8 ⊢ (𝜑 → -1 ∈ ℤ)
104	101, 103	zaddcld 12618	. . . . . . 7 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + -1) ∈ ℤ)
105		bccl 14263	. . . . . . 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 12480	. . . . 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 12483	. . . . . . 7 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))) ∈ ℕ₀)
109	26	nn0zd 12531	. . . . . . . 8 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℤ)
110	109, 103	zaddcld 12618	. . . . . . 7 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1) ∈ ℤ)
111		bccl 14263	. . . . . . 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 12480	. . . . 5 ⊢ (𝜑 → (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1)) ∈ ℝ)
114	52, 49	remulcld 11180	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ∈ ℝ)
115	114	flcld 13736	. . . . . . . . . . . 12 ⊢ (𝜑 → (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ∈ ℤ)
116	52, 49, 75, 74	mulge0d 11731	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))
117		flge 13743	. . . . . . . . . . . . . 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 12518	. . . . . . . . . . 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 12483	. . . . . . . . 9 ⊢ (𝜑 → (((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) + (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))) ∈ ℕ₀)
124		bccl 14263	. . . . . . . . 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 12480	. . . . . . 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 14263	. . . . . . . . 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 12480	. . . . . . 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 14104	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)) ∈ ℝ)
131		2nn0 12435	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ₀
132	131	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℕ₀)
133	132, 82	nn0mulcld 12484	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2 · (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) ∈ ℕ₀)
134	133, 83	nn0addcld 12483	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2 · (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))) + 1) ∈ ℕ₀)
135		bccl 14263	. . . . . . . . . . . . 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 12480	. . . . . . . . . . 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 11298	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ 2)
139	42, 138, 53	recxpcld 26608	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2↑_𝑐((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℝ)
140		reflcl 13734	. . . . . . . . . . . . . . . 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 11179	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℝ)
143	42, 138, 142	recxpcld 26608	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2↑_𝑐((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)) ∈ ℝ)
144		1le2 12366	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ≤ 2
145	144	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≤ 2)
146	55, 42, 7, 145, 71	letrd 11307	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ≤ 𝑁)
147		reflcl 13734	. . . . . . . . . . . . . . . . 17 ⊢ ((√‘𝐷) ∈ ℝ → (⌊‘(√‘𝐷)) ∈ ℝ)
148	30, 147	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘(√‘𝐷)) ∈ ℝ)
149	18	fveq2d 6844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (√‘𝐷) = (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
150	149	fveq2d 6844	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (⌊‘(√‘𝐷)) = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
151		flle 13737	. . . . . . . . . . . . . . . . . 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 5124	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘(√‘𝐷)) ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
154	7, 146, 148, 52, 153	cxplead 26606	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁↑_𝑐(⌊‘(√‘𝐷))) ≤ (𝑁↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
155	7	recnd 11178	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℂ)
156	4, 12	gtned 11285	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ≠ 0)
157	155, 156, 31	cxpexpzd 26596	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁↑_𝑐(⌊‘(√‘𝐷))) = (𝑁↑(⌊‘(√‘𝐷))))
158	59, 48	nelprd 4617	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 2 ∈ {0, 1})
159	58, 158	eldifd 3922	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2 ∈ (ℂ ∖ {0, 1}))
160	156	neneqd 2930	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ¬ 𝑁 = 0)
161		elsng 4599	. . . . . . . . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℂ ∖ {0}))
165		cxplogb 26672	. . . . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = ((2↑_𝑐(2 log_b 𝑁))↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
169	154, 157, 168	3brtr3d 5133	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ≤ ((2↑_𝑐(2 log_b 𝑁))↑_𝑐(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
170	42, 44	elrpd 12968	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℝ⁺)
171	52	recnd 11178	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℂ)
172		cxpmul 26573	. . . . . . . . . . . . . . 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 5130	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ≤ (2↑_𝑐((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
175		fllep1 13739	. . . . . . . . . . . . . . 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 11304	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 < 2)
178	84	nn0red 12480	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℝ)
179	42, 177, 53, 178	cxpled 26605	. . . . . . . . . . . . . 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 11307	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) ≤ (2↑_𝑐((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1)))
182		cxpexpz 26552	. . . . . . . . . . . . 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 5128	. . . . . . . . . . 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 11129	. . . . . . . . . . . . . . 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 13734	. . . . . . . . . . . . . 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 12480	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℝ)
191	42, 44, 6, 9, 48	relogbcld 41934	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 log_b 3) ∈ ℝ)
192	191	resqcld 14066	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2 log_b 3)↑2) ∈ ℝ)
193	49	recnd 11178	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℂ)
194	193	sqvald 14084	. . . . . . . . . . . . . . . . 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 42020	. . . . . . . . . . . . . . . . . . 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 11298	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≤ ((2 log_b 3)↑2))
200	6, 42, 59	redivcld 11986	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (3 / 2) ∈ ℝ)
201		2rp 12932	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ⁺
202	201	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 2 ∈ ℝ⁺)
203	4, 6, 9	ltled 11298	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 ≤ 3)
204	6, 202, 203	divge0d 13011	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 0 ≤ (3 / 2))
205		3lexlogpow2ineq1 42019	. . . . . . . . . . . . . . . . . . . . 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 11298	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (3 / 2) ≤ (2 log_b 3))
209	4, 200, 191, 204, 208	letrd 11307	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ≤ (2 log_b 3))
210	64, 65, 6, 9, 7, 12, 11	logblebd 41937	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 log_b 3) ≤ (2 log_b 𝑁))
211	191, 49, 132, 209, 210	leexp1ad 14117	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2 log_b 3)↑2) ≤ ((2 log_b 𝑁)↑2))
212	42, 192, 195, 199, 211	letrd 11307	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ≤ ((2 log_b 𝑁)↑2))
213	212, 194	breqtrd 5128	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≤ ((2 log_b 𝑁) · (2 log_b 𝑁)))
214		flge 13743	. . . . . . . . . . . . . . 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 11180	. . . . . . . . . . . . . 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 42084	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
220	171	sqvald 14084	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))↑2) = ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
221	26	nn0cnd 12481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℂ)
222	221	msqsqrtd 15385	. . . . . . . . . . . . . . . . . . 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 5128	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))↑2))
225	49, 52, 74, 75	lt2sqd 14197	. . . . . . . . . . . . . . . . 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 11298	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 𝑁) ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
228	49, 52, 49, 74, 227	lemul2ad 12099	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2 log_b 𝑁) · (2 log_b 𝑁)) ≤ ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
229		flwordi 13750	. . . . . . . . . . . . . 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 11307	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ≤ (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
232	54, 231	2ap1caineq 42106	. . . . . . . . . . 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 11308	. . . . . . . . . 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 12481	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) ∈ ℂ)
235	234	2timesd 12401	. . . . . . . . . . . 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 7384	. . . . . . . . . . 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 7384	. . . . . . . . . 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 5128	. . . . . . . . 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 11145	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
240	234, 234, 239	addassd 11172	. . . . . . . . . . 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 12481	. . . . . . . . . . . 12 ⊢ (𝜑 → ((⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) + 1) ∈ ℂ)
242	234, 241	addcomd 11352	. . . . . . . . . . 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 7384	. . . . . . . . 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 5128	. . . . . . . 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 11171	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)))
247	246	fveq2d 6844	. . . . . . . . . 10 ⊢ (𝜑 → (⌊‘((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))) = (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))))
248	247	oveq2d 7385	. . . . . . . . 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 7384	. . . . . . . 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 5128	. . . . . . 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 12480	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁))) ∈ ℝ)
252	99	nn0red 12480	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℝ)
253	17, 27	eqeltrrid 2833	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
254	253	nn0red 12480	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ)
255	253	nn0ge0d 12482	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
256	254, 255	resqrtcld 15360	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ)
257	256, 49	remulcld 11180	. . . . . . . . . 10 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ∈ ℝ)
258	15, 19, 20, 21, 22, 23, 24	aks6d1c4 42085	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ≤ (ϕ‘𝑅))
259	50, 51, 86, 88	sqrtled 15369	. . . . . . . . . . . 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 12098	. . . . . . . . . 10 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (2 log_b 𝑁)) ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
262		flwordi 13750	. . . . . . . . . 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 11769	. . . . . . . 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 42139	. . . . . . 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 11310	. . . . . 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 11510	. . . . . . . . 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 11515	. . . . . . . . 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 7385	. . . . . 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 5128	. . . . 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 12479	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑅 ∈ ℕ₀)
275	25	zncrng 21430	. . . . . . . . . . . . . . . . . . . . 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 21397	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ/nℤ‘𝑅) ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom (ℤ/nℤ‘𝑅)))
279		zringbas 21339	. . . . . . . . . . . . . . . . . . . . 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 6671	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐿 Fn ℤ)
284	15, 19, 20, 23	aks6d1c2p1 42079	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐸:(ℕ₀ × ℕ₀)⟶ℕ)
285		nnssz 12527	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℕ ⊆ ℤ
286	285	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ℕ ⊆ ℤ)
287	284, 286	fssd 6687	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐸:(ℕ₀ × ℕ₀)⟶ℤ)
288		frn 6677	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸:(ℕ₀ × ℕ₀)⟶ℤ → ran 𝐸 ⊆ ℤ)
289	287, 288	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ran 𝐸 ⊆ ℤ)
290	284	ffnd 6671	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐸 Fn (ℕ₀ × ℕ₀))
291		fnima 6630	. . . . . . . . . . . . . . . . . . . . 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 3975	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐸 “ (ℕ₀ × ℕ₀)) ⊆ ℤ ↔ ran 𝐸 ⊆ ℤ))
294	289, 293	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 “ (ℕ₀ × ℕ₀)) ⊆ ℤ)
295		vex 3448	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑘 ∈ V
296		vex 3448	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑙 ∈ V
297	295, 296	op1std 7957	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (1^st ‘𝑣) = 𝑘)
298	297	oveq2d 7385	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (𝑃↑(1^st ‘𝑣)) = (𝑃↑𝑘))
299	295, 296	op2ndd 7958	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → (2^nd ‘𝑣) = 𝑙)
300	299	oveq2d 7385	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑁 / 𝑃)↑(2^nd ‘𝑣)) = ((𝑁 / 𝑃)↑𝑙))
301	298, 300	oveq12d 7387	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = ⟨𝑘, 𝑙⟩ → ((𝑃↑(1^st ‘𝑣)) · ((𝑁 / 𝑃)↑(2^nd ‘𝑣))) = ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
302	301	mpompt 7483	. . . . . . . . . . . . . . . . . . . . . . 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 11144	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ V
307	306, 306	op1std 7957	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = ⟨0, 0⟩ → (1^st ‘𝑣) = 0)
308	307	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → (1^st ‘𝑣) = 0)
309	308	oveq2d 7385	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → (𝑃↑(1^st ‘𝑣)) = (𝑃↑0))
310	306, 306	op2ndd 7958	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = ⟨0, 0⟩ → (2^nd ‘𝑣) = 0)
311	310	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → (2^nd ‘𝑣) = 0)
312	311	oveq2d 7385	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → ((𝑁 / 𝑃)↑(2^nd ‘𝑣)) = ((𝑁 / 𝑃)↑0))
313	309, 312	oveq12d 7387	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑣 = ⟨0, 0⟩) → ((𝑃↑(1^st ‘𝑣)) · ((𝑁 / 𝑃)↑(2^nd ‘𝑣))) = ((𝑃↑0) · ((𝑁 / 𝑃)↑0)))
314		prmnn 16620	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
315	19, 314	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑃 ∈ ℕ)
316	315	nncnd 12178	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑃 ∈ ℂ)
317	316	exp0d 14081	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑃↑0) = 1)
318	315	nnne0d 12212	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑃 ≠ 0)
319	155, 316, 318	divcld 11934	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℂ)
320	319	exp0d 14081	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((𝑁 / 𝑃)↑0) = 1)
321	317, 320	oveq12d 7387	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑃↑0) · ((𝑁 / 𝑃)↑0)) = (1 · 1))
322	239	mulridd 11167	. . . . . . . . . . . . . . . . . . . . . . 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 12433	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ ℕ₀
327	326	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 ∈ ℕ₀)
328	327, 327	opelxpd 5670	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ⟨0, 0⟩ ∈ (ℕ₀ × ℕ₀))
329		1nn 12173	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℕ
330	329	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 1 ∈ ℕ)
331	305, 325, 328, 330	fvmptd 6957	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐸‘⟨0, 0⟩) = 1)
332		ssidd 3967	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (ℕ₀ × ℕ₀) ⊆ (ℕ₀ × ℕ₀))
333		fnfvima 7189	. . . . . . . . . . . . . . . . . . . 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 7189	. . . . . . . . . . . . . . . . . 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 6855	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℤRHom‘(ℤ/nℤ‘𝑅)) ∈ V)
340	338, 339	eqeltrd 2828	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐿 ∈ V)
341	340	imaexd 7872	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))) ∈ V)
342	337, 341	hashelne0d 14309	. . . . . . . . . . . . . . . 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 12432	. . . . . . . . . . . . . 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 12969	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ⁺)
348	347	rpsqrtcld 15354	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ⁺)
349	49, 52, 348, 226	ltmul1dd 13026	. . . . . . . . . 10 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) < ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
350	50, 51, 50, 51	sqrtmuld 15367	. . . . . . . . . . 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 5128	. . . . . . . . 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 5128	. . . . . . . 8 ⊢ (𝜑 → ((2 log_b 𝑁) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) < (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
355		fllt 13744	. . . . . . . . 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 41940	. . . . . . 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 11581	. . . . . . 7 ⊢ (𝜑 → -1 ∈ ℝ)
361		df-neg 11384	. . . . . . . . 9 ⊢ -1 = (0 − 1)
362	361	a1i 11	. . . . . . . 8 ⊢ (𝜑 → -1 = (0 − 1))
363	4	lem1d 12092	. . . . . . . 8 ⊢ (𝜑 → (0 − 1) ≤ 0)
364	362, 363	eqbrtrd 5124	. . . . . . 7 ⊢ (𝜑 → -1 ≤ 0)
365	360, 4, 252, 364, 96	letrd 11307	. . . . . 6 ⊢ (𝜑 → -1 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))
366	84, 26, 99, 103, 359, 365	bcle2d 42140	. . . . 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 11310	. . . 4 ⊢ (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1)))
368	221, 239	negsubd 11515	. . . . 5 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + -1) = ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1))
369	368	oveq2d 7385	. . . 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 5128	. . 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 7385	. . . 4 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))) = ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + 𝐴))
375	374	oveq1d 7384	. . 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 5128	. 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 7384	. . 3 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + 𝐴) = (𝐷 + 𝐴))
379	377	oveq1d 7384	. . 3 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1) = (𝐷 − 1))
380	378, 379	oveq12d 7387	. 2 ⊢ (𝜑 → (((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) + 𝐴)C((♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) − 1)) = ((𝐷 + 𝐴)C(𝐷 − 1)))
381	376, 380	breqtrd 5128	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 3444 ∖ cdif 3908 ⊆ wss 3911 {csn 4585 {cpr 4587 ⟨cop 4591 class class class wbr 5102 ↦ cmpt 5183 × cxp 5629 ran crn 5632 “ cima 5634 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ∈ cmpo 7371 1^st c1st 7945 2^nd c2nd 7946 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 < clt 11184 ≤ cle 11185 − cmin 11381 -cneg 11382 / cdiv 11811 ℕcn 12162 2c2 12217 3c3 12218 5c5 12220 ℕ₀cn0 12418 ℤcz 12505 ℤ_≥cuz 12769 ℝ⁺crp 12927 ⌊cfl 13728 ↑cexp 14002 Ccbc 14243 ♯chash 14271 √csqrt 15175 ∥ cdvds 16198 gcd cgcd 16440 ℙcprime 16617 od_ℤcodz 16709 ϕcphi 16710 Basecbs 17155 Ringcrg 20118 CRingccrg 20119 RingHom crh 20354 ℤ_ringczring 21332 ℤRHomczrh 21385 ℤ/nℤczn 21388 ↑_𝑐ccxp 26440 log_b clogb 26650

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8648 df-ec 8650 df-qs 8654 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15629 df-prod 15846 df-fallfac 15949 df-ef 16009 df-sin 16011 df-cos 16012 df-pi 16014 df-dvds 16199 df-gcd 16441 df-prm 16618 df-odz 16711 df-phi 16712 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-qus 17448 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-mhm 18686 df-submnd 18687 df-grp 18844 df-minusg 18845 df-sbg 18846 df-mulg 18976 df-subg 19031 df-nsg 19032 df-eqg 19033 df-ghm 19121 df-cntz 19225 df-cmn 19688 df-abl 19689 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 20744 df-lss 20814 df-lsp 20854 df-sra 21056 df-rgmod 21057 df-lidl 21094 df-rsp 21095 df-2idl 21136 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-zring 21333 df-zrh 21389 df-zn 21392 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-lp 22999 df-perf 23000 df-cn 23090 df-cnp 23091 df-haus 23178 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cncf 24747 df-limc 25743 df-dv 25744 df-log 26441 df-cxp 26442 df-logb 26651

This theorem is referenced by: aks6d1c7lem2 42142

W3C validator