aks4d1p1p4 - Mathbox for metakunt

	Mathbox for metakunt		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > aks4d1p1p4		Structured version Visualization version GIF version

Theorem aks4d1p1p4 41453

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 12231	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ)
3		2re 12290	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
4	3	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ)
5		2pos 12319	. . . . . . . . . 10 ⊢ 0 < 2
6	5	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < 2)
7	1	nngt0d 12265	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑁)
8		1red 11219	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℝ)
9		1lt2 12387	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 2
10	9	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 < 2)
11	8, 10	ltned 11354	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≠ 2)
12	11	necomd 2990	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≠ 1)
13	4, 6, 2, 7, 12	relogbcld 41354	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
14		5nn0 12496	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℕ₀
15	14	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 5 ∈ ℕ₀)
16	13, 15	reexpcld 14133	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ∈ ℝ)
17		ceilcl 13813	. . . . . . . . . . . 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 12670	. . . . . . . . . 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 2812	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∈ ℝ ↔ (⌈‘((2 log_b 𝑁)↑5)) ∈ ℝ))
23	19, 22	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
24		0red 11221	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
25		7re 12309	. . . . . . . . . . 11 ⊢ 7 ∈ ℝ
26	25	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 7 ∈ ℝ)
27		7pos 12327	. . . . . . . . . . 11 ⊢ 0 < 7
28	27	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 7)
29		aks4d1p1p4.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 3 ≤ 𝑁)
30	2, 29	3lexlogpow5ineq3 41439	. . . . . . . . . . . 12 ⊢ (𝜑 → 7 < ((2 log_b 𝑁)↑5))
31	26, 16, 30	ltled 11366	. . . . . . . . . . 11 ⊢ (𝜑 → 7 ≤ ((2 log_b 𝑁)↑5))
32		ceilge 13816	. . . . . . . . . . . . 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 5169	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ≤ 𝐵)
35	26, 16, 23, 31, 34	letrd 11375	. . . . . . . . . 10 ⊢ (𝜑 → 7 ≤ 𝐵)
36	24, 26, 23, 28, 35	ltletrd 11378	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝐵)
37	4, 6, 23, 36, 12	relogbcld 41354	. . . . . . . 8 ⊢ (𝜑 → (2 log_b 𝐵) ∈ ℝ)
38	37	flcld 13769	. . . . . . 7 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℤ)
39	24, 8	readdcld 11247	. . . . . . . . 9 ⊢ (𝜑 → (0 + 1) ∈ ℝ)
40	38	zred 12670	. . . . . . . . . 10 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℝ)
41	40, 8	readdcld 11247	. . . . . . . . 9 ⊢ (𝜑 → ((⌊‘(2 log_b 𝐵)) + 1) ∈ ℝ)
42	4, 6, 4, 6, 12	relogbcld 41354	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b 2) ∈ ℝ)
43	8	leidd 11784	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ≤ 1)
44		1cnd 11213	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℂ)
45	44	addlidd 11419	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 + 1) = 1)
46	4	recnd 11246	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ∈ ℂ)
47	24, 6	gtned 11353	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 2 ≠ 0)
48		logbid1 26655	. . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 = (2 log_b 2))
51	50	eqcomd 2732	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 log_b 2) = 1)
52	45, 51	breq12d 5154	. . . . . . . . . . 11 ⊢ (𝜑 → ((0 + 1) ≤ (2 log_b 2) ↔ 1 ≤ 1))
53	43, 52	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → (0 + 1) ≤ (2 log_b 2))
54		5re 12303	. . . . . . . . . . . . . . 15 ⊢ 5 ∈ ℝ
55	54	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 5 ∈ ℝ)
56	4, 55	readdcld 11247	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 + 5) ∈ ℝ)
57	3, 14	nn0addge1i 12524	. . . . . . . . . . . . . 14 ⊢ 2 ≤ (2 + 5)
58	57	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ≤ (2 + 5))
59	3	recni 11232	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℂ
60		5cn 12304	. . . . . . . . . . . . . . . . 17 ⊢ 5 ∈ ℂ
61	59, 60	addcomi 11409	. . . . . . . . . . . . . . . 16 ⊢ (2 + 5) = (5 + 2)
62		5p2e7 12372	. . . . . . . . . . . . . . . 16 ⊢ (5 + 2) = 7
63	61, 62	eqtri 2754	. . . . . . . . . . . . . . 15 ⊢ (2 + 5) = 7
64	63	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2 + 5) = 7)
65	26	leidd 11784	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 7 ≤ 7)
66	64, 65	eqbrtrd 5163	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 + 5) ≤ 7)
67	4, 56, 26, 58, 66	letrd 11375	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ≤ 7)
68	4, 26, 23, 67, 35	letrd 11375	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ 𝐵)
69		2z 12598	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℤ
70	69	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℤ)
71	70	uzidd 12842	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ (ℤ_≥‘2))
72		2rp 12985	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
73	72	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℝ⁺)
74	23, 36	elrpd 13019	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
75		logbleb 26670	. . . . . . . . . . . 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 11375	. . . . . . . . 9 ⊢ (𝜑 → (0 + 1) ≤ (2 log_b 𝐵))
79		fllep1 13772	. . . . . . . . . 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 11375	. . . . . . . 8 ⊢ (𝜑 → (0 + 1) ≤ ((⌊‘(2 log_b 𝐵)) + 1))
82	24, 40, 8	leadd1d 11812	. . . . . . . 8 ⊢ (𝜑 → (0 ≤ (⌊‘(2 log_b 𝐵)) ↔ (0 + 1) ≤ ((⌊‘(2 log_b 𝐵)) + 1)))
83	81, 82	mpbird 257	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (⌊‘(2 log_b 𝐵)))
84	38, 83	jca 511	. . . . . 6 ⊢ (𝜑 → ((⌊‘(2 log_b 𝐵)) ∈ ℤ ∧ 0 ≤ (⌊‘(2 log_b 𝐵))))
85		elnn0z 12575	. . . . . 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 14133	. . . 4 ⊢ (𝜑 → (𝑁↑(⌊‘(2 log_b 𝐵))) ∈ ℝ)
88		fzfid 13944	. . . . 5 ⊢ (𝜑 → (1...(⌊‘((2 log_b 𝑁)↑2))) ∈ Fin)
89	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑁 ∈ ℝ)
90		elfznn 13536	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) → 𝑘 ∈ ℕ)
91	90	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑘 ∈ ℕ)
92	91	nnnn0d 12536	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑘 ∈ ℕ₀)
93	89, 92	reexpcld 14133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑘) ∈ ℝ)
94		1red 11219	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℝ)
95	93, 94	resubcld 11646	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → ((𝑁↑𝑘) − 1) ∈ ℝ)
96	88, 95	fprodrecl 15903	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1) ∈ ℝ)
97	87, 96	remulcld 11248	. . 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 2812	. . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ ↔ ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1)) ∈ ℝ))
101	97, 100	mpbird 257	. 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 7421	. . . . . . 7 ⊢ (𝜑 → (𝑁↑_𝑐𝐸) = (𝑁↑_𝑐((2 log_b 𝑁)↑4)))
105		2cnd 12294	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℂ)
106	73	rpne0d 13027	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ≠ 0)
107	106, 12	nelprd 4654	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 2 ∈ {0, 1})
108	105, 107	eldifd 3954	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ (ℂ ∖ {0, 1}))
109	2	recnd 11246	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
110	24, 7	ltned 11354	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≠ 𝑁)
111		necom 2988	. . . . . . . . . . . . . . . 16 ⊢ (0 ≠ 𝑁 ↔ 𝑁 ≠ 0)
112	111	imbi2i 336	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 → 0 ≠ 𝑁) ↔ (𝜑 → 𝑁 ≠ 0))
113	110, 112	mpbi 229	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ≠ 0)
114	113	neneqd 2939	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑁 = 0)
115		c0ex 11212	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
116	115	elsn2 4662	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ {0} ↔ 𝑁 = 0)
117	114, 116	sylnibr 329	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑁 ∈ {0})
118	109, 117	eldifd 3954	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℂ ∖ {0}))
119		cxplogb 26673	. . . . . . . . . . 11 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 𝑁 ∈ (ℂ ∖ {0})) → (2↑_𝑐(2 log_b 𝑁)) = 𝑁)
120	108, 118, 119	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (2↑_𝑐(2 log_b 𝑁)) = 𝑁)
121	120	eqcomd 2732	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 = (2↑_𝑐(2 log_b 𝑁)))
122	121	oveq1d 7420	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑_𝑐((2 log_b 𝑁)↑4)) = ((2↑_𝑐(2 log_b 𝑁))↑_𝑐((2 log_b 𝑁)↑4)))
123		eqidd 2727	. . . . . . . 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 2766	. . . . . . 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 2766	. . . . . 6 ⊢ (𝜑 → (𝑁↑_𝑐𝐸) = ((2↑_𝑐(2 log_b 𝑁))↑_𝑐((2 log_b 𝑁)↑4)))
126	103	eqcomd 2732	. . . . . . . . 9 ⊢ (𝜑 → ((2 log_b 𝑁)↑4) = 𝐸)
127		4nn0 12495	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℕ₀
128	127	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 4 ∈ ℕ₀)
129	13, 128	reexpcld 14133	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 log_b 𝑁)↑4) ∈ ℝ)
130	103	eleq1d 2812	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 ∈ ℝ ↔ ((2 log_b 𝑁)↑4) ∈ ℝ))
131	129, 130	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℝ)
132	131	recnd 11246	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℂ)
133	126, 132	eqeltrd 2827	. . . . . . . 8 ⊢ (𝜑 → ((2 log_b 𝑁)↑4) ∈ ℂ)
134	73, 13, 133	cxpmuld 26626	. . . . . . 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 2732	. . . . . 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 2766	. . . . 5 ⊢ (𝜑 → (𝑁↑_𝑐𝐸) = (2↑_𝑐((2 log_b 𝑁) · ((2 log_b 𝑁)↑4))))
137	13	recnd 11246	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℂ)
138	137	exp1d 14111	. . . . . . . . 9 ⊢ (𝜑 → ((2 log_b 𝑁)↑1) = (2 log_b 𝑁))
139	138	eqcomd 2732	. . . . . . . 8 ⊢ (𝜑 → (2 log_b 𝑁) = ((2 log_b 𝑁)↑1))
140	139	oveq1d 7420	. . . . . . 7 ⊢ (𝜑 → ((2 log_b 𝑁) · ((2 log_b 𝑁)↑4)) = (((2 log_b 𝑁)↑1) · ((2 log_b 𝑁)↑4)))
141		1nn0 12492	. . . . . . . . . 10 ⊢ 1 ∈ ℕ₀
142	141	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℕ₀)
143	137, 128, 142	expaddd 14118	. . . . . . . 8 ⊢ (𝜑 → ((2 log_b 𝑁)↑(1 + 4)) = (((2 log_b 𝑁)↑1) · ((2 log_b 𝑁)↑4)))
144	143	eqcomd 2732	. . . . . . 7 ⊢ (𝜑 → (((2 log_b 𝑁)↑1) · ((2 log_b 𝑁)↑4)) = ((2 log_b 𝑁)↑(1 + 4)))
145	140, 144	eqtrd 2766	. . . . . 6 ⊢ (𝜑 → ((2 log_b 𝑁) · ((2 log_b 𝑁)↑4)) = ((2 log_b 𝑁)↑(1 + 4)))
146	145	oveq2d 7421	. . . . 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 2766	. . . 4 ⊢ (𝜑 → (𝑁↑_𝑐𝐸) = (2↑_𝑐((2 log_b 𝑁)↑(1 + 4))))
148		4cn 12301	. . . . . . . 8 ⊢ 4 ∈ ℂ
149		ax-1cn 11170	. . . . . . . 8 ⊢ 1 ∈ ℂ
150		4p1e5 12362	. . . . . . . 8 ⊢ (4 + 1) = 5
151	148, 149, 150	addcomli 11410	. . . . . . 7 ⊢ (1 + 4) = 5
152	151	a1i 11	. . . . . 6 ⊢ (𝜑 → (1 + 4) = 5)
153	152	oveq2d 7421	. . . . 5 ⊢ (𝜑 → ((2 log_b 𝑁)↑(1 + 4)) = ((2 log_b 𝑁)↑5))
154	153	oveq2d 7421	. . . 4 ⊢ (𝜑 → (2↑_𝑐((2 log_b 𝑁)↑(1 + 4))) = (2↑_𝑐((2 log_b 𝑁)↑5)))
155	147, 154	eqtrd 2766	. . 3 ⊢ (𝜑 → (𝑁↑_𝑐𝐸) = (2↑_𝑐((2 log_b 𝑁)↑5)))
156		3re 12296	. . . . . 6 ⊢ 3 ∈ ℝ
157	156	a1i 11	. . . . 5 ⊢ (𝜑 → 3 ∈ ℝ)
158		0le1 11741	. . . . . . 7 ⊢ 0 ≤ 1
159	158	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ≤ 1)
160		1lt3 12389	. . . . . . . 8 ⊢ 1 < 3
161	160	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 < 3)
162	8, 157, 161	ltled 11366	. . . . . 6 ⊢ (𝜑 → 1 ≤ 3)
163	24, 8, 157, 159, 162	letrd 11375	. . . . 5 ⊢ (𝜑 → 0 ≤ 3)
164	24, 157, 2, 163, 29	letrd 11375	. . . 4 ⊢ (𝜑 → 0 ≤ 𝑁)
165	2, 164, 131	recxpcld 26612	. . 3 ⊢ (𝜑 → (𝑁↑_𝑐𝐸) ∈ ℝ)
166	155, 165	eqeltrrd 2828	. 2 ⊢ (𝜑 → (2↑_𝑐((2 log_b 𝑁)↑5)) ∈ ℝ)
167	21	eleq1d 2812	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℤ ↔ (⌈‘((2 log_b 𝑁)↑5)) ∈ ℤ))
168	18, 167	mpbird 257	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℤ)
169	24, 23, 36	ltled 11366	. . . . 5 ⊢ (𝜑 → 0 ≤ 𝐵)
170	168, 169	jca 511	. . . 4 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ 0 ≤ 𝐵))
171		elnn0z 12575	. . . 4 ⊢ (𝐵 ∈ ℕ₀ ↔ (𝐵 ∈ ℤ ∧ 0 ≤ 𝐵))
172	170, 171	sylibr 233	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℕ₀)
173	4, 172	reexpcld 14133	. 2 ⊢ (𝜑 → (2↑𝐵) ∈ ℝ)
174	2, 7	elrpd 13019	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
175	16, 8	readdcld 11247	. . . . . . . . . 10 ⊢ (𝜑 → (((2 log_b 𝑁)↑5) + 1) ∈ ℝ)
176	15	nn0zd 12588	. . . . . . . . . . . . 13 ⊢ (𝜑 → 5 ∈ ℤ)
177		logb1 26656	. . . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2 log_b 1) ∈ ℝ)
180	24	leidd 11784	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ 0)
181	178	eqcomd 2732	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 = (2 log_b 1))
182	180, 181	breqtrd 5167	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (2 log_b 1))
183	8, 157, 2, 161, 29	ltletrd 11378	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 < 𝑁)
184		1rp 12984	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ⁺
185	184	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℝ⁺)
186		logblt 26671	. . . . . . . . . . . . . . . 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 11376	. . . . . . . . . . . . 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 14066	. . . . . . . . . . . 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 12058	. . . . . . . . . . . 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 11379	. . . . . . . . . 10 ⊢ (𝜑 → 0 < (((2 log_b 𝑁)↑5) + 1))
196	4, 6, 175, 195, 12	relogbcld 41354	. . . . . . . . 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 2812	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ ℝ ↔ (2 log_b (((2 log_b 𝑁)↑5) + 1)) ∈ ℝ))
200	196, 199	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
201	13	resqcld 14095	. . . . . . . . . 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 2812	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 ∈ ℝ ↔ ((2 log_b 𝑁)↑2) ∈ ℝ))
205	201, 204	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ℝ)
206	205	rehalfcld 12463	. . . . . . . 8 ⊢ (𝜑 → (𝐷 / 2) ∈ ℝ)
207	200, 206	readdcld 11247	. . . . . . 7 ⊢ (𝜑 → (𝐶 + (𝐷 / 2)) ∈ ℝ)
208	131, 4, 106	redivcld 12046	. . . . . . 7 ⊢ (𝜑 → (𝐸 / 2) ∈ ℝ)
209	207, 208	readdcld 11247	. . . . . 6 ⊢ (𝜑 → ((𝐶 + (𝐷 / 2)) + (𝐸 / 2)) ∈ ℝ)
210	174, 209	rpcxpcld 26622	. . . . 5 ⊢ (𝜑 → (𝑁↑_𝑐((𝐶 + (𝐷 / 2)) + (𝐸 / 2))) ∈ ℝ⁺)
211	210	rpred 13022	. . . 4 ⊢ (𝜑 → (𝑁↑_𝑐((𝐶 + (𝐷 / 2)) + (𝐸 / 2))) ∈ ℝ)
212	1, 98, 20, 29	aks4d1p1p2 41452	. . . . 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 7420	. . . . . . . 8 ⊢ (𝜑 → (((2 log_b 𝑁)↑4) / 2) = (𝐸 / 2))
214	213	oveq2d 7421	. . . . . . 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 2732	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b (((2 log_b 𝑁)↑5) + 1)) = 𝐶)
216	215	oveq1d 7420	. . . . . . . . 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 2732	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) = 𝐷)
218	217	oveq1d 7420	. . . . . . . . . 10 ⊢ (𝜑 → (((2 log_b 𝑁)↑2) / 2) = (𝐷 / 2))
219	218	oveq2d 7421	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 + (((2 log_b 𝑁)↑2) / 2)) = (𝐶 + (𝐷 / 2)))
220	216, 219	eqtrd 2766	. . . . . . . 8 ⊢ (𝜑 → ((2 log_b (((2 log_b 𝑁)↑5) + 1)) + (((2 log_b 𝑁)↑2) / 2)) = (𝐶 + (𝐷 / 2)))
221	220	oveq1d 7420	. . . . . . 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 2766	. . . . . 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 7421	. . . . 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 5167	. . . 4 ⊢ (𝜑 → 𝐴 < (𝑁↑_𝑐((𝐶 + (𝐷 / 2)) + (𝐸 / 2))))
225	200	recnd 11246	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ ℂ)
226	225, 105, 106	divcan3d 11999	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 · 𝐶) / 2) = 𝐶)
227	226	eqcomd 2732	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 = ((2 · 𝐶) / 2))
228	227	oveq1d 7420	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 + (𝐷 / 2)) = (((2 · 𝐶) / 2) + (𝐷 / 2)))
229	4, 200	remulcld 11248	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 · 𝐶) ∈ ℝ)
230	229	recnd 11246	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · 𝐶) ∈ ℂ)
231	205	recnd 11246	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ ℂ)
232	230, 231, 46, 106	divdird 12032	. . . . . . . . . 10 ⊢ (𝜑 → (((2 · 𝐶) + 𝐷) / 2) = (((2 · 𝐶) / 2) + (𝐷 / 2)))
233	232	eqcomd 2732	. . . . . . . . 9 ⊢ (𝜑 → (((2 · 𝐶) / 2) + (𝐷 / 2)) = (((2 · 𝐶) + 𝐷) / 2))
234	228, 233	eqtrd 2766	. . . . . . . 8 ⊢ (𝜑 → (𝐶 + (𝐷 / 2)) = (((2 · 𝐶) + 𝐷) / 2))
235		aks4d1p1p4.8	. . . . . . . . 9 ⊢ (𝜑 → ((2 · 𝐶) + 𝐷) ≤ 𝐸)
236	229, 205	readdcld 11247	. . . . . . . . . 10 ⊢ (𝜑 → ((2 · 𝐶) + 𝐷) ∈ ℝ)
237	236, 131, 73	lediv1d 13068	. . . . . . . . 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 5163	. . . . . . 7 ⊢ (𝜑 → (𝐶 + (𝐷 / 2)) ≤ (𝐸 / 2))
240	207, 208, 208, 239	leadd1dd 11832	. . . . . 6 ⊢ (𝜑 → ((𝐶 + (𝐷 / 2)) + (𝐸 / 2)) ≤ ((𝐸 / 2) + (𝐸 / 2)))
241	132	2halvesd 12462	. . . . . 6 ⊢ (𝜑 → ((𝐸 / 2) + (𝐸 / 2)) = 𝐸)
242	240, 241	breqtrd 5167	. . . . 5 ⊢ (𝜑 → ((𝐶 + (𝐷 / 2)) + (𝐸 / 2)) ≤ 𝐸)
243	2, 183, 209, 131	cxpled 26609	. . . . 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 11378	. . 3 ⊢ (𝜑 → 𝐴 < (𝑁↑_𝑐𝐸))
246	245, 155	breqtrd 5167	. 2 ⊢ (𝜑 → 𝐴 < (2↑_𝑐((2 log_b 𝑁)↑5)))
247		1le2 12425	. . . . 5 ⊢ 1 ≤ 2
248	247	a1i 11	. . . 4 ⊢ (𝜑 → 1 ≤ 2)
249	172	nn0red 12537	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
250	21	eqcomd 2732	. . . . 5 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) = 𝐵)
251	33, 250	breqtrd 5167	. . . 4 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ≤ 𝐵)
252	4, 248, 16, 249, 251	cxplead 26610	. . 3 ⊢ (𝜑 → (2↑_𝑐((2 log_b 𝑁)↑5)) ≤ (2↑_𝑐𝐵))
253		cxpexp 26557	. . . 4 ⊢ ((2 ∈ ℂ ∧ 𝐵 ∈ ℕ₀) → (2↑_𝑐𝐵) = (2↑𝐵))
254	105, 172, 253	syl2anc 583	. . 3 ⊢ (𝜑 → (2↑_𝑐𝐵) = (2↑𝐵))
255	252, 254	breqtrd 5167	. 2 ⊢ (𝜑 → (2↑_𝑐((2 log_b 𝑁)↑5)) ≤ (2↑𝐵))
256	101, 166, 173, 246, 255	ltletrd 11378	1 ⊢ (𝜑 → 𝐴 < (2↑𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∖ cdif 3940 {csn 4623 {cpr 4625 class class class wbr 5141 ‘cfv 6537 (class class class)co 7405 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 < clt 11252 ≤ cle 11253 − cmin 11448 / cdiv 11875 ℕcn 12216 2c2 12271 3c3 12272 4c4 12273 5c5 12274 7c7 12276 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 ℝ⁺crp 12980 ...cfz 13490 ⌊cfl 13761 ⌈cceil 13762 ↑cexp 14032 ∏cprod 15855 ↑_𝑐ccxp 26444 log_b clogb 26651

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-ceil 13764 df-mod 13841 df-seq 13973 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-prod 15856 df-ef 16017 df-sin 16019 df-cos 16020 df-pi 16022 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 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-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-lp 22995 df-perf 22996 df-cn 23086 df-cnp 23087 df-haus 23174 df-tx 23421 df-hmeo 23614 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-xms 24181 df-ms 24182 df-tms 24183 df-cncf 24753 df-limc 25750 df-dv 25751 df-log 26445 df-cxp 26446 df-logb 26652

This theorem is referenced by: aks4d1p1p5 41457

W3C validator