aks4d1p1p4 - Mathbox for metakunt

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Theorem aks4d1p1p4 41597

Description: Technical step for inequality. The hard work is in to prove the final hypothesis. (Contributed by metakunt, 19-Aug-2024.)

**Hypotheses**
Ref	Expression
aks4d1p1p4.1	⊢ (𝜑 → 𝑁 ∈ ℕ)
aks4d1p1p4.2	⊢ 𝐴 = ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1))
aks4d1p1p4.3	⊢ 𝐵 = (⌈‘((2 log_b 𝑁)↑5))
aks4d1p1p4.4	⊢ (𝜑 → 3 ≤ 𝑁)
aks4d1p1p4.5	⊢ 𝐶 = (2 log_b (((2 log_b 𝑁)↑5) + 1))
aks4d1p1p4.6	⊢ 𝐷 = ((2 log_b 𝑁)↑2)
aks4d1p1p4.7	⊢ 𝐸 = ((2 log_b 𝑁)↑4)
aks4d1p1p4.8	⊢ (𝜑 → ((2 · 𝐶) + 𝐷) ≤ 𝐸)

**Assertion**
Ref	Expression
aks4d1p1p4	⊢ (𝜑 → 𝐴 < (2↑𝐵))

Distinct variable groups: 𝑘,𝑁 𝜑,𝑘 Allowed substitution hints: 𝐴(𝑘) 𝐵(𝑘) 𝐶(𝑘) 𝐷(𝑘) 𝐸(𝑘)

**Proof of Theorem aks4d1p1p4**
Step	Hyp	Ref	Expression
1		aks4d1p1p4.1	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2	1	nnred 12255	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ)
3		2re 12314	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
4	3	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ)
5		2pos 12343	. . . . . . . . . 10 ⊢ 0 < 2
6	5	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < 2)
7	1	nngt0d 12289	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑁)
8		1red 11243	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℝ)
9		1lt2 12411	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 2
10	9	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 < 2)
11	8, 10	ltned 11378	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≠ 2)
12	11	necomd 2986	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≠ 1)
13	4, 6, 2, 7, 12	relogbcld 41498	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
14		5nn0 12520	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℕ₀
15	14	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 5 ∈ ℕ₀)
16	13, 15	reexpcld 14157	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ∈ ℝ)
17		ceilcl 13837	. . . . . . . . . . . 12 ⊢ (((2 log_b 𝑁)↑5) ∈ ℝ → (⌈‘((2 log_b 𝑁)↑5)) ∈ ℤ)
18	16, 17	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) ∈ ℤ)
19	18	zred 12694	. . . . . . . . . 10 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) ∈ ℝ)
20		aks4d1p1p4.3	. . . . . . . . . . . 12 ⊢ 𝐵 = (⌈‘((2 log_b 𝑁)↑5))
21	20	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (⌈‘((2 log_b 𝑁)↑5)))
22	21	eleq1d 2810	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∈ ℝ ↔ (⌈‘((2 log_b 𝑁)↑5)) ∈ ℝ))
23	19, 22	mpbird 256	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
24		0red 11245	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
25		7re 12333	. . . . . . . . . . 11 ⊢ 7 ∈ ℝ
26	25	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 7 ∈ ℝ)
27		7pos 12351	. . . . . . . . . . 11 ⊢ 0 < 7
28	27	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 7)
29		aks4d1p1p4.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 3 ≤ 𝑁)
30	2, 29	3lexlogpow5ineq3 41583	. . . . . . . . . . . 12 ⊢ (𝜑 → 7 < ((2 log_b 𝑁)↑5))
31	26, 16, 30	ltled 11390	. . . . . . . . . . 11 ⊢ (𝜑 → 7 ≤ ((2 log_b 𝑁)↑5))
32		ceilge 13840	. . . . . . . . . . . . 13 ⊢ (((2 log_b 𝑁)↑5) ∈ ℝ → ((2 log_b 𝑁)↑5) ≤ (⌈‘((2 log_b 𝑁)↑5)))
33	16, 32	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ≤ (⌈‘((2 log_b 𝑁)↑5)))
34	33, 21	breqtrrd 5171	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ≤ 𝐵)
35	26, 16, 23, 31, 34	letrd 11399	. . . . . . . . . 10 ⊢ (𝜑 → 7 ≤ 𝐵)
36	24, 26, 23, 28, 35	ltletrd 11402	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝐵)
37	4, 6, 23, 36, 12	relogbcld 41498	. . . . . . . 8 ⊢ (𝜑 → (2 log_b 𝐵) ∈ ℝ)
38	37	flcld 13793	. . . . . . 7 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℤ)
39	24, 8	readdcld 11271	. . . . . . . . 9 ⊢ (𝜑 → (0 + 1) ∈ ℝ)
40	38	zred 12694	. . . . . . . . . 10 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℝ)
41	40, 8	readdcld 11271	. . . . . . . . 9 ⊢ (𝜑 → ((⌊‘(2 log_b 𝐵)) + 1) ∈ ℝ)
42	4, 6, 4, 6, 12	relogbcld 41498	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b 2) ∈ ℝ)
43	8	leidd 11808	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ≤ 1)
44		1cnd 11237	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℂ)
45	44	addlidd 11443	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 + 1) = 1)
46	4	recnd 11270	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℂ)
47	24, 6	gtned 11377	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ≠ 0)
48		logbid1 26716	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 log_b 2) = 1)
49	46, 47, 12, 48	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2 log_b 2) = 1)
50	49	eqcomd 2731	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 = (2 log_b 2))
51	50	eqcomd 2731	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 log_b 2) = 1)
52	45, 51	breq12d 5156	. . . . . . . . . . 11 ⊢ (𝜑 → ((0 + 1) ≤ (2 log_b 2) ↔ 1 ≤ 1))
53	43, 52	mpbird 256	. . . . . . . . . 10 ⊢ (𝜑 → (0 + 1) ≤ (2 log_b 2))
54		5re 12327	. . . . . . . . . . . . . . 15 ⊢ 5 ∈ ℝ
55	54	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 5 ∈ ℝ)
56	4, 55	readdcld 11271	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 + 5) ∈ ℝ)
57	3, 14	nn0addge1i 12548	. . . . . . . . . . . . . 14 ⊢ 2 ≤ (2 + 5)
58	57	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ≤ (2 + 5))
59	3	recni 11256	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℂ
60		5cn 12328	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℂ
61	59, 60	addcomi 11433	. . . . . . . . . . . . . . . 16 ⊢ (2 + 5) = (5 + 2)
62		5p2e7 12396	. . . . . . . . . . . . . . . 16 ⊢ (5 + 2) = 7
63	61, 62	eqtri 2753	. . . . . . . . . . . . . . 15 ⊢ (2 + 5) = 7
64	63	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2 + 5) = 7)
65	26	leidd 11808	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 7 ≤ 7)
66	64, 65	eqbrtrd 5165	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 + 5) ≤ 7)
67	4, 56, 26, 58, 66	letrd 11399	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ≤ 7)
68	4, 26, 23, 67, 35	letrd 11399	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ 𝐵)
69		2z 12622	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℤ
70	69	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℤ)
71	70	uzidd 12866	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ (ℤ_≥‘2))
72		2rp 13009	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
73	72	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℝ⁺)
74	23, 36	elrpd 13043	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
75		logbleb 26731	. . . . . . . . . . . 12 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 2 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) → (2 ≤ 𝐵 ↔ (2 log_b 2) ≤ (2 log_b 𝐵)))
76	71, 73, 74, 75	syl3anc 1368	. . . . . . . . . . 11 ⊢ (𝜑 → (2 ≤ 𝐵 ↔ (2 log_b 2) ≤ (2 log_b 𝐵)))
77	68, 76	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b 2) ≤ (2 log_b 𝐵))
78	39, 42, 37, 53, 77	letrd 11399	. . . . . . . . 9 ⊢ (𝜑 → (0 + 1) ≤ (2 log_b 𝐵))
79		fllep1 13796	. . . . . . . . . 10 ⊢ ((2 log_b 𝐵) ∈ ℝ → (2 log_b 𝐵) ≤ ((⌊‘(2 log_b 𝐵)) + 1))
80	37, 79	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (2 log_b 𝐵) ≤ ((⌊‘(2 log_b 𝐵)) + 1))
81	39, 37, 41, 78, 80	letrd 11399	. . . . . . . 8 ⊢ (𝜑 → (0 + 1) ≤ ((⌊‘(2 log_b 𝐵)) + 1))
82	24, 40, 8	leadd1d 11836	. . . . . . . 8 ⊢ (𝜑 → (0 ≤ (⌊‘(2 log_b 𝐵)) ↔ (0 + 1) ≤ ((⌊‘(2 log_b 𝐵)) + 1)))
83	81, 82	mpbird 256	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (⌊‘(2 log_b 𝐵)))
84	38, 83	jca 510	. . . . . 6 ⊢ (𝜑 → ((⌊‘(2 log_b 𝐵)) ∈ ℤ ∧ 0 ≤ (⌊‘(2 log_b 𝐵))))
85		elnn0z 12599	. . . . . 6 ⊢ ((⌊‘(2 log_b 𝐵)) ∈ ℕ₀ ↔ ((⌊‘(2 log_b 𝐵)) ∈ ℤ ∧ 0 ≤ (⌊‘(2 log_b 𝐵))))
86	84, 85	sylibr 233	. . . . 5 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℕ₀)
87	2, 86	reexpcld 14157	. . . 4 ⊢ (𝜑 → (𝑁↑(⌊‘(2 log_b 𝐵))) ∈ ℝ)
88		fzfid 13968	. . . . 5 ⊢ (𝜑 → (1...(⌊‘((2 log_b 𝑁)↑2))) ∈ Fin)
89	2	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑁 ∈ ℝ)
90		elfznn 13560	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) → 𝑘 ∈ ℕ)
91	90	adantl 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑘 ∈ ℕ)
92	91	nnnn0d 12560	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑘 ∈ ℕ₀)
93	89, 92	reexpcld 14157	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑘) ∈ ℝ)
94		1red 11243	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℝ)
95	93, 94	resubcld 11670	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → ((𝑁↑𝑘) − 1) ∈ ℝ)
96	88, 95	fprodrecl 15927	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1) ∈ ℝ)
97	87, 96	remulcld 11272	. . 3 ⊢ (𝜑 → ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1)) ∈ ℝ)
98		aks4d1p1p4.2	. . . . 5 ⊢ 𝐴 = ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1))
99	98	a1i 11	. . . 4 ⊢ (𝜑 → 𝐴 = ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1)))
100	99	eleq1d 2810	. . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ ↔ ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1)) ∈ ℝ))
101	97, 100	mpbird 256	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
102		aks4d1p1p4.7	. . . . . . . . 9 ⊢ 𝐸 = ((2 log_b 𝑁)↑4)
103	102	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐸 = ((2 log_b 𝑁)↑4))
104	103	oveq2d 7431	. . . . . . 7 ⊢ (𝜑 → (𝑁↑_𝑐𝐸) = (𝑁↑_𝑐((2 log_b 𝑁)↑4)))
105		2cnd 12318	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℂ)
106	73	rpne0d 13051	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ≠ 0)
107	106, 12	nelprd 4655	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 2 ∈ {0, 1})
108	105, 107	eldifd 3951	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ (ℂ ∖ {0, 1}))
109	2	recnd 11270	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
110	24, 7	ltned 11378	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≠ 𝑁)
111		necom 2984	. . . . . . . . . . . . . . . 16 ⊢ (0 ≠ 𝑁 ↔ 𝑁 ≠ 0)
112	111	imbi2i 335	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 → 0 ≠ 𝑁) ↔ (𝜑 → 𝑁 ≠ 0))
113	110, 112	mpbi 229	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ≠ 0)
114	113	neneqd 2935	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑁 = 0)
115		c0ex 11236	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
116	115	elsn2 4663	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ {0} ↔ 𝑁 = 0)
117	114, 116	sylnibr 328	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑁 ∈ {0})
118	109, 117	eldifd 3951	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℂ ∖ {0}))
119		cxplogb 26734	. . . . . . . . . . 11 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 𝑁 ∈ (ℂ ∖ {0})) → (2↑_𝑐(2 log_b 𝑁)) = 𝑁)
120	108, 118, 119	syl2anc 582	. . . . . . . . . 10 ⊢ (𝜑 → (2↑_𝑐(2 log_b 𝑁)) = 𝑁)
121	120	eqcomd 2731	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 = (2↑_𝑐(2 log_b 𝑁)))
122	121	oveq1d 7430	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑_𝑐((2 log_b 𝑁)↑4)) = ((2↑_𝑐(2 log_b 𝑁))↑_𝑐((2 log_b 𝑁)↑4)))
123		eqidd 2726	. . . . . . . 8 ⊢ (𝜑 → ((2↑_𝑐(2 log_b 𝑁))↑_𝑐((2 log_b 𝑁)↑4)) = ((2↑_𝑐(2 log_b 𝑁))↑_𝑐((2 log_b 𝑁)↑4)))
124	122, 123	eqtrd 2765	. . . . . . 7 ⊢ (𝜑 → (𝑁↑_𝑐((2 log_b 𝑁)↑4)) = ((2↑_𝑐(2 log_b 𝑁))↑_𝑐((2 log_b 𝑁)↑4)))
125	104, 124	eqtrd 2765	. . . . . 6 ⊢ (𝜑 → (𝑁↑_𝑐𝐸) = ((2↑_𝑐(2 log_b 𝑁))↑_𝑐((2 log_b 𝑁)↑4)))
126	103	eqcomd 2731	. . . . . . . . 9 ⊢ (𝜑 → ((2 log_b 𝑁)↑4) = 𝐸)
127		4nn0 12519	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℕ₀
128	127	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 4 ∈ ℕ₀)
129	13, 128	reexpcld 14157	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 log_b 𝑁)↑4) ∈ ℝ)
130	103	eleq1d 2810	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 ∈ ℝ ↔ ((2 log_b 𝑁)↑4) ∈ ℝ))
131	129, 130	mpbird 256	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℝ)
132	131	recnd 11270	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℂ)
133	126, 132	eqeltrd 2825	. . . . . . . 8 ⊢ (𝜑 → ((2 log_b 𝑁)↑4) ∈ ℂ)
134	73, 13, 133	cxpmuld 26687	. . . . . . 7 ⊢ (𝜑 → (2↑_𝑐((2 log_b 𝑁) · ((2 log_b 𝑁)↑4))) = ((2↑_𝑐(2 log_b 𝑁))↑_𝑐((2 log_b 𝑁)↑4)))
135	134	eqcomd 2731	. . . . . 6 ⊢ (𝜑 → ((2↑_𝑐(2 log_b 𝑁))↑_𝑐((2 log_b 𝑁)↑4)) = (2↑_𝑐((2 log_b 𝑁) · ((2 log_b 𝑁)↑4))))
136	125, 135	eqtrd 2765	. . . . 5 ⊢ (𝜑 → (𝑁↑_𝑐𝐸) = (2↑_𝑐((2 log_b 𝑁) · ((2 log_b 𝑁)↑4))))
137	13	recnd 11270	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℂ)
138	137	exp1d 14135	. . . . . . . . 9 ⊢ (𝜑 → ((2 log_b 𝑁)↑1) = (2 log_b 𝑁))
139	138	eqcomd 2731	. . . . . . . 8 ⊢ (𝜑 → (2 log_b 𝑁) = ((2 log_b 𝑁)↑1))
140	139	oveq1d 7430	. . . . . . 7 ⊢ (𝜑 → ((2 log_b 𝑁) · ((2 log_b 𝑁)↑4)) = (((2 log_b 𝑁)↑1) · ((2 log_b 𝑁)↑4)))
141		1nn0 12516	. . . . . . . . . 10 ⊢ 1 ∈ ℕ₀
142	141	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℕ₀)
143	137, 128, 142	expaddd 14142	. . . . . . . 8 ⊢ (𝜑 → ((2 log_b 𝑁)↑(1 + 4)) = (((2 log_b 𝑁)↑1) · ((2 log_b 𝑁)↑4)))
144	143	eqcomd 2731	. . . . . . 7 ⊢ (𝜑 → (((2 log_b 𝑁)↑1) · ((2 log_b 𝑁)↑4)) = ((2 log_b 𝑁)↑(1 + 4)))
145	140, 144	eqtrd 2765	. . . . . 6 ⊢ (𝜑 → ((2 log_b 𝑁) · ((2 log_b 𝑁)↑4)) = ((2 log_b 𝑁)↑(1 + 4)))
146	145	oveq2d 7431	. . . . 5 ⊢ (𝜑 → (2↑_𝑐((2 log_b 𝑁) · ((2 log_b 𝑁)↑4))) = (2↑_𝑐((2 log_b 𝑁)↑(1 + 4))))
147	136, 146	eqtrd 2765	. . . 4 ⊢ (𝜑 → (𝑁↑_𝑐𝐸) = (2↑_𝑐((2 log_b 𝑁)↑(1 + 4))))
148		4cn 12325	. . . . . . . 8 ⊢ 4 ∈ ℂ
149		ax-1cn 11194	. . . . . . . 8 ⊢ 1 ∈ ℂ
150		4p1e5 12386	. . . . . . . 8 ⊢ (4 + 1) = 5
151	148, 149, 150	addcomli 11434	. . . . . . 7 ⊢ (1 + 4) = 5
152	151	a1i 11	. . . . . 6 ⊢ (𝜑 → (1 + 4) = 5)
153	152	oveq2d 7431	. . . . 5 ⊢ (𝜑 → ((2 log_b 𝑁)↑(1 + 4)) = ((2 log_b 𝑁)↑5))
154	153	oveq2d 7431	. . . 4 ⊢ (𝜑 → (2↑_𝑐((2 log_b 𝑁)↑(1 + 4))) = (2↑_𝑐((2 log_b 𝑁)↑5)))
155	147, 154	eqtrd 2765	. . 3 ⊢ (𝜑 → (𝑁↑_𝑐𝐸) = (2↑_𝑐((2 log_b 𝑁)↑5)))
156		3re 12320	. . . . . 6 ⊢ 3 ∈ ℝ
157	156	a1i 11	. . . . 5 ⊢ (𝜑 → 3 ∈ ℝ)
158		0le1 11765	. . . . . . 7 ⊢ 0 ≤ 1
159	158	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ≤ 1)
160		1lt3 12413	. . . . . . . 8 ⊢ 1 < 3
161	160	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 < 3)
162	8, 157, 161	ltled 11390	. . . . . 6 ⊢ (𝜑 → 1 ≤ 3)
163	24, 8, 157, 159, 162	letrd 11399	. . . . 5 ⊢ (𝜑 → 0 ≤ 3)
164	24, 157, 2, 163, 29	letrd 11399	. . . 4 ⊢ (𝜑 → 0 ≤ 𝑁)
165	2, 164, 131	recxpcld 26673	. . 3 ⊢ (𝜑 → (𝑁↑_𝑐𝐸) ∈ ℝ)
166	155, 165	eqeltrrd 2826	. 2 ⊢ (𝜑 → (2↑_𝑐((2 log_b 𝑁)↑5)) ∈ ℝ)
167	21	eleq1d 2810	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℤ ↔ (⌈‘((2 log_b 𝑁)↑5)) ∈ ℤ))
168	18, 167	mpbird 256	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℤ)
169	24, 23, 36	ltled 11390	. . . . 5 ⊢ (𝜑 → 0 ≤ 𝐵)
170	168, 169	jca 510	. . . 4 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ 0 ≤ 𝐵))
171		elnn0z 12599	. . . 4 ⊢ (𝐵 ∈ ℕ₀ ↔ (𝐵 ∈ ℤ ∧ 0 ≤ 𝐵))
172	170, 171	sylibr 233	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℕ₀)
173	4, 172	reexpcld 14157	. 2 ⊢ (𝜑 → (2↑𝐵) ∈ ℝ)
174	2, 7	elrpd 13043	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
175	16, 8	readdcld 11271	. . . . . . . . . 10 ⊢ (𝜑 → (((2 log_b 𝑁)↑5) + 1) ∈ ℝ)
176	15	nn0zd 12612	. . . . . . . . . . . . 13 ⊢ (𝜑 → 5 ∈ ℤ)
177		logb1 26717	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 log_b 1) = 0)
178	46, 47, 12, 177	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 1) = 0)
179	178, 24	eqeltrd 2825	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2 log_b 1) ∈ ℝ)
180	24	leidd 11808	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ 0)
181	178	eqcomd 2731	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 = (2 log_b 1))
182	180, 181	breqtrd 5169	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (2 log_b 1))
183	8, 157, 2, 161, 29	ltletrd 11402	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 < 𝑁)
184		1rp 13008	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ⁺
185	184	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℝ⁺)
186		logblt 26732	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 1 ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → (1 < 𝑁 ↔ (2 log_b 1) < (2 log_b 𝑁)))
187	71, 185, 174, 186	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1 < 𝑁 ↔ (2 log_b 1) < (2 log_b 𝑁)))
188	183, 187	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2 log_b 1) < (2 log_b 𝑁))
189	24, 179, 13, 182, 188	lelttrd 11400	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < (2 log_b 𝑁))
190	13, 176, 189	3jca 1125	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2 log_b 𝑁) ∈ ℝ ∧ 5 ∈ ℤ ∧ 0 < (2 log_b 𝑁)))
191		expgt0 14090	. . . . . . . . . . . 12 ⊢ (((2 log_b 𝑁) ∈ ℝ ∧ 5 ∈ ℤ ∧ 0 < (2 log_b 𝑁)) → 0 < ((2 log_b 𝑁)↑5))
192	190, 191	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < ((2 log_b 𝑁)↑5))
193		ltp1 12082	. . . . . . . . . . . 12 ⊢ (((2 log_b 𝑁)↑5) ∈ ℝ → ((2 log_b 𝑁)↑5) < (((2 log_b 𝑁)↑5) + 1))
194	16, 193	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) < (((2 log_b 𝑁)↑5) + 1))
195	24, 16, 175, 192, 194	lttrd 11403	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (((2 log_b 𝑁)↑5) + 1))
196	4, 6, 175, 195, 12	relogbcld 41498	. . . . . . . . 9 ⊢ (𝜑 → (2 log_b (((2 log_b 𝑁)↑5) + 1)) ∈ ℝ)
197		aks4d1p1p4.5	. . . . . . . . . . 11 ⊢ 𝐶 = (2 log_b (((2 log_b 𝑁)↑5) + 1))
198	197	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 = (2 log_b (((2 log_b 𝑁)↑5) + 1)))
199	198	eleq1d 2810	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ ℝ ↔ (2 log_b (((2 log_b 𝑁)↑5) + 1)) ∈ ℝ))
200	196, 199	mpbird 256	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
201	13	resqcld 14119	. . . . . . . . . 10 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) ∈ ℝ)
202		aks4d1p1p4.6	. . . . . . . . . . . 12 ⊢ 𝐷 = ((2 log_b 𝑁)↑2)
203	202	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 = ((2 log_b 𝑁)↑2))
204	203	eleq1d 2810	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 ∈ ℝ ↔ ((2 log_b 𝑁)↑2) ∈ ℝ))
205	201, 204	mpbird 256	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℝ)
206	205	rehalfcld 12487	. . . . . . . 8 ⊢ (𝜑 → (𝐷 / 2) ∈ ℝ)
207	200, 206	readdcld 11271	. . . . . . 7 ⊢ (𝜑 → (𝐶 + (𝐷 / 2)) ∈ ℝ)
208	131, 4, 106	redivcld 12070	. . . . . . 7 ⊢ (𝜑 → (𝐸 / 2) ∈ ℝ)
209	207, 208	readdcld 11271	. . . . . 6 ⊢ (𝜑 → ((𝐶 + (𝐷 / 2)) + (𝐸 / 2)) ∈ ℝ)
210	174, 209	rpcxpcld 26683	. . . . 5 ⊢ (𝜑 → (𝑁↑_𝑐((𝐶 + (𝐷 / 2)) + (𝐸 / 2))) ∈ ℝ⁺)
211	210	rpred 13046	. . . 4 ⊢ (𝜑 → (𝑁↑_𝑐((𝐶 + (𝐷 / 2)) + (𝐸 / 2))) ∈ ℝ)
212	1, 98, 20, 29	aks4d1p1p2 41596	. . . . 5 ⊢ (𝜑 → 𝐴 < (𝑁↑_𝑐(((2 log_b (((2 log_b 𝑁)↑5) + 1)) + (((2 log_b 𝑁)↑2) / 2)) + (((2 log_b 𝑁)↑4) / 2))))
213	126	oveq1d 7430	. . . . . . . 8 ⊢ (𝜑 → (((2 log_b 𝑁)↑4) / 2) = (𝐸 / 2))
214	213	oveq2d 7431	. . . . . . 7 ⊢ (𝜑 → (((2 log_b (((2 log_b 𝑁)↑5) + 1)) + (((2 log_b 𝑁)↑2) / 2)) + (((2 log_b 𝑁)↑4) / 2)) = (((2 log_b (((2 log_b 𝑁)↑5) + 1)) + (((2 log_b 𝑁)↑2) / 2)) + (𝐸 / 2)))
215	198	eqcomd 2731	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b (((2 log_b 𝑁)↑5) + 1)) = 𝐶)
216	215	oveq1d 7430	. . . . . . . . 9 ⊢ (𝜑 → ((2 log_b (((2 log_b 𝑁)↑5) + 1)) + (((2 log_b 𝑁)↑2) / 2)) = (𝐶 + (((2 log_b 𝑁)↑2) / 2)))
217	203	eqcomd 2731	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) = 𝐷)
218	217	oveq1d 7430	. . . . . . . . . 10 ⊢ (𝜑 → (((2 log_b 𝑁)↑2) / 2) = (𝐷 / 2))
219	218	oveq2d 7431	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 + (((2 log_b 𝑁)↑2) / 2)) = (𝐶 + (𝐷 / 2)))
220	216, 219	eqtrd 2765	. . . . . . . 8 ⊢ (𝜑 → ((2 log_b (((2 log_b 𝑁)↑5) + 1)) + (((2 log_b 𝑁)↑2) / 2)) = (𝐶 + (𝐷 / 2)))
221	220	oveq1d 7430	. . . . . . 7 ⊢ (𝜑 → (((2 log_b (((2 log_b 𝑁)↑5) + 1)) + (((2 log_b 𝑁)↑2) / 2)) + (𝐸 / 2)) = ((𝐶 + (𝐷 / 2)) + (𝐸 / 2)))
222	214, 221	eqtrd 2765	. . . . . 6 ⊢ (𝜑 → (((2 log_b (((2 log_b 𝑁)↑5) + 1)) + (((2 log_b 𝑁)↑2) / 2)) + (((2 log_b 𝑁)↑4) / 2)) = ((𝐶 + (𝐷 / 2)) + (𝐸 / 2)))
223	222	oveq2d 7431	. . . . 5 ⊢ (𝜑 → (𝑁↑_𝑐(((2 log_b (((2 log_b 𝑁)↑5) + 1)) + (((2 log_b 𝑁)↑2) / 2)) + (((2 log_b 𝑁)↑4) / 2))) = (𝑁↑_𝑐((𝐶 + (𝐷 / 2)) + (𝐸 / 2))))
224	212, 223	breqtrd 5169	. . . 4 ⊢ (𝜑 → 𝐴 < (𝑁↑_𝑐((𝐶 + (𝐷 / 2)) + (𝐸 / 2))))
225	200	recnd 11270	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ ℂ)
226	225, 105, 106	divcan3d 12023	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 · 𝐶) / 2) = 𝐶)
227	226	eqcomd 2731	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 = ((2 · 𝐶) / 2))
228	227	oveq1d 7430	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 + (𝐷 / 2)) = (((2 · 𝐶) / 2) + (𝐷 / 2)))
229	4, 200	remulcld 11272	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 · 𝐶) ∈ ℝ)
230	229	recnd 11270	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · 𝐶) ∈ ℂ)
231	205	recnd 11270	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ ℂ)
232	230, 231, 46, 106	divdird 12056	. . . . . . . . . 10 ⊢ (𝜑 → (((2 · 𝐶) + 𝐷) / 2) = (((2 · 𝐶) / 2) + (𝐷 / 2)))
233	232	eqcomd 2731	. . . . . . . . 9 ⊢ (𝜑 → (((2 · 𝐶) / 2) + (𝐷 / 2)) = (((2 · 𝐶) + 𝐷) / 2))
234	228, 233	eqtrd 2765	. . . . . . . 8 ⊢ (𝜑 → (𝐶 + (𝐷 / 2)) = (((2 · 𝐶) + 𝐷) / 2))
235		aks4d1p1p4.8	. . . . . . . . 9 ⊢ (𝜑 → ((2 · 𝐶) + 𝐷) ≤ 𝐸)
236	229, 205	readdcld 11271	. . . . . . . . . 10 ⊢ (𝜑 → ((2 · 𝐶) + 𝐷) ∈ ℝ)
237	236, 131, 73	lediv1d 13092	. . . . . . . . 9 ⊢ (𝜑 → (((2 · 𝐶) + 𝐷) ≤ 𝐸 ↔ (((2 · 𝐶) + 𝐷) / 2) ≤ (𝐸 / 2)))
238	235, 237	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → (((2 · 𝐶) + 𝐷) / 2) ≤ (𝐸 / 2))
239	234, 238	eqbrtrd 5165	. . . . . . 7 ⊢ (𝜑 → (𝐶 + (𝐷 / 2)) ≤ (𝐸 / 2))
240	207, 208, 208, 239	leadd1dd 11856	. . . . . 6 ⊢ (𝜑 → ((𝐶 + (𝐷 / 2)) + (𝐸 / 2)) ≤ ((𝐸 / 2) + (𝐸 / 2)))
241	132	2halvesd 12486	. . . . . 6 ⊢ (𝜑 → ((𝐸 / 2) + (𝐸 / 2)) = 𝐸)
242	240, 241	breqtrd 5169	. . . . 5 ⊢ (𝜑 → ((𝐶 + (𝐷 / 2)) + (𝐸 / 2)) ≤ 𝐸)
243	2, 183, 209, 131	cxpled 26670	. . . . 5 ⊢ (𝜑 → (((𝐶 + (𝐷 / 2)) + (𝐸 / 2)) ≤ 𝐸 ↔ (𝑁↑_𝑐((𝐶 + (𝐷 / 2)) + (𝐸 / 2))) ≤ (𝑁↑_𝑐𝐸)))
244	242, 243	mpbid 231	. . . 4 ⊢ (𝜑 → (𝑁↑_𝑐((𝐶 + (𝐷 / 2)) + (𝐸 / 2))) ≤ (𝑁↑_𝑐𝐸))
245	101, 211, 165, 224, 244	ltletrd 11402	. . 3 ⊢ (𝜑 → 𝐴 < (𝑁↑_𝑐𝐸))
246	245, 155	breqtrd 5169	. 2 ⊢ (𝜑 → 𝐴 < (2↑_𝑐((2 log_b 𝑁)↑5)))
247		1le2 12449	. . . . 5 ⊢ 1 ≤ 2
248	247	a1i 11	. . . 4 ⊢ (𝜑 → 1 ≤ 2)
249	172	nn0red 12561	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
250	21	eqcomd 2731	. . . . 5 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) = 𝐵)
251	33, 250	breqtrd 5169	. . . 4 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ≤ 𝐵)
252	4, 248, 16, 249, 251	cxplead 26671	. . 3 ⊢ (𝜑 → (2↑_𝑐((2 log_b 𝑁)↑5)) ≤ (2↑_𝑐𝐵))
253		cxpexp 26618	. . . 4 ⊢ ((2 ∈ ℂ ∧ 𝐵 ∈ ℕ₀) → (2↑_𝑐𝐵) = (2↑𝐵))
254	105, 172, 253	syl2anc 582	. . 3 ⊢ (𝜑 → (2↑_𝑐𝐵) = (2↑𝐵))
255	252, 254	breqtrd 5169	. 2 ⊢ (𝜑 → (2↑_𝑐((2 log_b 𝑁)↑5)) ≤ (2↑𝐵))
256	101, 166, 173, 246, 255	ltletrd 11402	1 ⊢ (𝜑 → 𝐴 < (2↑𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∖ cdif 3937 {csn 4624 {cpr 4626 class class class wbr 5143 ‘cfv 6542 (class class class)co 7415 ℂcc 11134 ℝcr 11135 0cc0 11136 1c1 11137 + caddc 11139 · cmul 11141 < clt 11276 ≤ cle 11277 − cmin 11472 / cdiv 11899 ℕcn 12240 2c2 12295 3c3 12296 4c4 12297 5c5 12298 7c7 12300 ℕ₀cn0 12500 ℤcz 12586 ℤ_≥cuz 12850 ℝ⁺crp 13004 ...cfz 13514 ⌊cfl 13785 ⌈cceil 13786 ↑cexp 14056 ∏cprod 15879 ↑_𝑐ccxp 26505 log_b clogb 26712

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 ax-addf 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-fi 9432 df-sup 9463 df-inf 9464 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-q 12961 df-rp 13005 df-xneg 13122 df-xadd 13123 df-xmul 13124 df-ioo 13358 df-ioc 13359 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13658 df-fl 13787 df-ceil 13788 df-mod 13865 df-seq 13997 df-exp 14057 df-fac 14263 df-bc 14292 df-hash 14320 df-shft 15044 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-limsup 15445 df-clim 15462 df-rlim 15463 df-sum 15663 df-prod 15880 df-ef 16041 df-sin 16043 df-cos 16044 df-pi 16046 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-hom 17254 df-cco 17255 df-rest 17401 df-topn 17402 df-0g 17420 df-gsum 17421 df-topgen 17422 df-pt 17423 df-prds 17426 df-xrs 17481 df-qtop 17486 df-imas 17487 df-xps 17489 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-mulg 19026 df-cntz 19270 df-cmn 19739 df-psmet 21273 df-xmet 21274 df-met 21275 df-bl 21276 df-mopn 21277 df-fbas 21278 df-fg 21279 df-cnfld 21282 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22865 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24242 df-ms 24243 df-tms 24244 df-cncf 24814 df-limc 25811 df-dv 25812 df-log 26506 df-cxp 26507 df-logb 26713

This theorem is referenced by: aks4d1p1p5 41601

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