aks4d1p6 - Mathbox for metakunt

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Theorem aks4d1p6 41032

Description: The maximal prime power exponent is smaller than the binary logarithm floor of 𝐵. (Contributed by metakunt, 30-Oct-2024.)

**Hypotheses**
Ref	Expression
aks4d1p6.1	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
aks4d1p6.2	⊢ 𝐴 = ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1))
aks4d1p6.3	⊢ 𝐵 = (⌈‘((2 log_b 𝑁)↑5))
aks4d1p6.4	⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )
aks4d1p6.5	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks4d1p6.6	⊢ (𝜑 → 𝑃 ∥ 𝑅)
aks4d1p6.7	⊢ 𝐾 = (𝑃 pCnt 𝑅)

**Assertion**
Ref	Expression
aks4d1p6	⊢ (𝜑 → 𝐾 ≤ (⌊‘(2 log_b 𝐵)))

Distinct variable groups: 𝐴,𝑟 𝐵,𝑟 𝑘,𝑁 𝜑,𝑘 𝑅,𝑟 Allowed substitution hints: 𝜑(𝑟) 𝐴(𝑘) 𝐵(𝑘) 𝑃(𝑘,𝑟) 𝑅(𝑘) 𝐾(𝑘,𝑟) 𝑁(𝑟)

**Proof of Theorem aks4d1p6**
Step	Hyp	Ref	Expression
1		aks4d1p6.7	. . . . . . 7 ⊢ 𝐾 = (𝑃 pCnt 𝑅)
2	1	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐾 = (𝑃 pCnt 𝑅))
3		aks4d1p6.5	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℙ)
4		aks4d1p6.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
5		aks4d1p6.2	. . . . . . . . . 10 ⊢ 𝐴 = ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1))
6		aks4d1p6.3	. . . . . . . . . 10 ⊢ 𝐵 = (⌈‘((2 log_b 𝑁)↑5))
7		aks4d1p6.4	. . . . . . . . . 10 ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )
8	4, 5, 6, 7	aks4d1p4 41030	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴))
9	8	simpld 495	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (1...𝐵))
10		elfznn 13532	. . . . . . . 8 ⊢ (𝑅 ∈ (1...𝐵) → 𝑅 ∈ ℕ)
11	9, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℕ)
12	3, 11	pccld 16785	. . . . . 6 ⊢ (𝜑 → (𝑃 pCnt 𝑅) ∈ ℕ₀)
13	2, 12	eqeltrd 2833	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
14	13	nn0zd 12586	. . . 4 ⊢ (𝜑 → 𝐾 ∈ ℤ)
15	14	zred 12668	. . 3 ⊢ (𝜑 → 𝐾 ∈ ℝ)
16		prmnn 16613	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
17	3, 16	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ)
18	17	nnred 12229	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℝ)
19	17	nngt0d 12263	. . . 4 ⊢ (𝜑 → 0 < 𝑃)
20	6	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (⌈‘((2 log_b 𝑁)↑5)))
21		2re 12288	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
22	21	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
23		2pos 12317	. . . . . . . . . . . 12 ⊢ 0 < 2
24	23	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 2)
25		eluzelz 12834	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
26	4, 25	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
27	26	zred 12668	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ)
28		0red 11219	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℝ)
29		3re 12294	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ
30	29	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 3 ∈ ℝ)
31		3pos 12319	. . . . . . . . . . . . 13 ⊢ 0 < 3
32	31	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 3)
33		eluzle 12837	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
34	4, 33	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 3 ≤ 𝑁)
35	28, 30, 27, 32, 34	ltletrd 11376	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝑁)
36		1red 11217	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ)
37		1lt2 12385	. . . . . . . . . . . . . 14 ⊢ 1 < 2
38	37	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < 2)
39	36, 38	ltned 11352	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≠ 2)
40	39	necomd 2996	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≠ 1)
41	22, 24, 27, 35, 40	relogbcld 40924	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
42		5nn0 12494	. . . . . . . . . . 11 ⊢ 5 ∈ ℕ₀
43	42	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 5 ∈ ℕ₀)
44	41, 43	reexpcld 14130	. . . . . . . . 9 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ∈ ℝ)
45		ceilcl 13809	. . . . . . . . 9 ⊢ (((2 log_b 𝑁)↑5) ∈ ℝ → (⌈‘((2 log_b 𝑁)↑5)) ∈ ℤ)
46	44, 45	syl 17	. . . . . . . 8 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) ∈ ℤ)
47	20, 46	eqeltrd 2833	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℤ)
48	47	zred 12668	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
49		9re 12313	. . . . . . . . . 10 ⊢ 9 ∈ ℝ
50	49	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 9 ∈ ℝ)
51		9pos 12327	. . . . . . . . . 10 ⊢ 0 < 9
52	51	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < 9)
53	27, 34	3lexlogpow5ineq4 41007	. . . . . . . . 9 ⊢ (𝜑 → 9 < ((2 log_b 𝑁)↑5))
54	28, 50, 44, 52, 53	lttrd 11377	. . . . . . . 8 ⊢ (𝜑 → 0 < ((2 log_b 𝑁)↑5))
55		ceilge 13812	. . . . . . . . . 10 ⊢ (((2 log_b 𝑁)↑5) ∈ ℝ → ((2 log_b 𝑁)↑5) ≤ (⌈‘((2 log_b 𝑁)↑5)))
56	44, 55	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ≤ (⌈‘((2 log_b 𝑁)↑5)))
57	56, 20	breqtrrd 5176	. . . . . . . 8 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ≤ 𝐵)
58	28, 44, 48, 54, 57	ltletrd 11376	. . . . . . 7 ⊢ (𝜑 → 0 < 𝐵)
59	47, 58	jca 512	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ 0 < 𝐵))
60		elnnz 12570	. . . . . 6 ⊢ (𝐵 ∈ ℕ ↔ (𝐵 ∈ ℤ ∧ 0 < 𝐵))
61	59, 60	sylibr 233	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℕ)
62	61	nnred 12229	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
63	61	nngt0d 12263	. . . 4 ⊢ (𝜑 → 0 < 𝐵)
64		2z 12596	. . . . . . . . 9 ⊢ 2 ∈ ℤ
65	64	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℤ)
66	65	zred 12668	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℝ)
67		prmuz2 16635	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
68	3, 67	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘2))
69		eluzle 12837	. . . . . . . 8 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 2 ≤ 𝑃)
70	68, 69	syl 17	. . . . . . 7 ⊢ (𝜑 → 2 ≤ 𝑃)
71	36, 66, 18, 38, 70	ltletrd 11376	. . . . . 6 ⊢ (𝜑 → 1 < 𝑃)
72	36, 71	ltned 11352	. . . . 5 ⊢ (𝜑 → 1 ≠ 𝑃)
73	72	necomd 2996	. . . 4 ⊢ (𝜑 → 𝑃 ≠ 1)
74	18, 19, 62, 63, 73	relogbcld 40924	. . 3 ⊢ (𝜑 → (𝑃 log_b 𝐵) ∈ ℝ)
75	66, 24, 62, 63, 40	relogbcld 40924	. . 3 ⊢ (𝜑 → (2 log_b 𝐵) ∈ ℝ)
76	17	nnrpd 13016	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
77	76	rpcnd 13020	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℂ)
78	76	rpne0d 13023	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 0)
79	77, 78, 14	cxpexpzd 26226	. . . . 5 ⊢ (𝜑 → (𝑃↑_𝑐𝐾) = (𝑃↑𝐾))
80	18, 13	reexpcld 14130	. . . . . . 7 ⊢ (𝜑 → (𝑃↑𝐾) ∈ ℝ)
81	11	nnred 12229	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℝ)
82	2	oveq2d 7427	. . . . . . . 8 ⊢ (𝜑 → (𝑃↑𝐾) = (𝑃↑(𝑃 pCnt 𝑅)))
83		pcdvds 16799	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑅 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝑅)) ∥ 𝑅)
84	3, 11, 83	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∥ 𝑅)
85	17	nnzd 12587	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℤ)
86		zexpcl 14044	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℤ ∧ (𝑃 pCnt 𝑅) ∈ ℕ₀) → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℤ)
87	85, 12, 86	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℤ)
88		dvdsle 16255	. . . . . . . . . 10 ⊢ (((𝑃↑(𝑃 pCnt 𝑅)) ∈ ℤ ∧ 𝑅 ∈ ℕ) → ((𝑃↑(𝑃 pCnt 𝑅)) ∥ 𝑅 → (𝑃↑(𝑃 pCnt 𝑅)) ≤ 𝑅))
89	87, 11, 88	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃↑(𝑃 pCnt 𝑅)) ∥ 𝑅 → (𝑃↑(𝑃 pCnt 𝑅)) ≤ 𝑅))
90	84, 89	mpd 15	. . . . . . . 8 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ≤ 𝑅)
91	82, 90	eqbrtrd 5170	. . . . . . 7 ⊢ (𝜑 → (𝑃↑𝐾) ≤ 𝑅)
92		elfzle2 13507	. . . . . . . 8 ⊢ (𝑅 ∈ (1...𝐵) → 𝑅 ≤ 𝐵)
93	9, 92	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ≤ 𝐵)
94	80, 81, 62, 91, 93	letrd 11373	. . . . . 6 ⊢ (𝜑 → (𝑃↑𝐾) ≤ 𝐵)
95	78, 73	nelprd 4659	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑃 ∈ {0, 1})
96	77, 95	eldifd 3959	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ (ℂ ∖ {0, 1}))
97	62	recnd 11244	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
98	28, 63	ltned 11352	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≠ 𝐵)
99	98	necomd 2996	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≠ 0)
100	99	neneqd 2945	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝐵 = 0)
101		elsng 4642	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℕ → (𝐵 ∈ {0} ↔ 𝐵 = 0))
102	61, 101	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ∈ {0} ↔ 𝐵 = 0))
103	100, 102	mtbird 324	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐵 ∈ {0})
104	97, 103	eldifd 3959	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (ℂ ∖ {0}))
105		cxplogb 26298	. . . . . . 7 ⊢ ((𝑃 ∈ (ℂ ∖ {0, 1}) ∧ 𝐵 ∈ (ℂ ∖ {0})) → (𝑃↑_𝑐(𝑃 log_b 𝐵)) = 𝐵)
106	96, 104, 105	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑃↑_𝑐(𝑃 log_b 𝐵)) = 𝐵)
107	94, 106	breqtrrd 5176	. . . . 5 ⊢ (𝜑 → (𝑃↑𝐾) ≤ (𝑃↑_𝑐(𝑃 log_b 𝐵)))
108	79, 107	eqbrtrd 5170	. . . 4 ⊢ (𝜑 → (𝑃↑_𝑐𝐾) ≤ (𝑃↑_𝑐(𝑃 log_b 𝐵)))
109	76	rpred 13018	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℝ)
110	36, 66, 109, 38, 70	ltletrd 11376	. . . . 5 ⊢ (𝜑 → 1 < 𝑃)
111	109, 110, 15, 74	cxpled 26235	. . . 4 ⊢ (𝜑 → (𝐾 ≤ (𝑃 log_b 𝐵) ↔ (𝑃↑_𝑐𝐾) ≤ (𝑃↑_𝑐(𝑃 log_b 𝐵))))
112	108, 111	mpbird 256	. . 3 ⊢ (𝜑 → 𝐾 ≤ (𝑃 log_b 𝐵))
113	22, 38	rplogcld 26144	. . . . 5 ⊢ (𝜑 → (log‘2) ∈ ℝ⁺)
114	109, 110	rplogcld 26144	. . . . 5 ⊢ (𝜑 → (log‘𝑃) ∈ ℝ⁺)
115	61	nnrpd 13016	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
116	115	relogcld 26138	. . . . 5 ⊢ (𝜑 → (log‘𝐵) ∈ ℝ)
117	61	nnge1d 12262	. . . . . 6 ⊢ (𝜑 → 1 ≤ 𝐵)
118	62, 117	logge0d 26145	. . . . 5 ⊢ (𝜑 → 0 ≤ (log‘𝐵))
119		2rp 12981	. . . . . . . 8 ⊢ 2 ∈ ℝ⁺
120	119	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℝ⁺)
121	120, 76	logled 26142	. . . . . 6 ⊢ (𝜑 → (2 ≤ 𝑃 ↔ (log‘2) ≤ (log‘𝑃)))
122	70, 121	mpbid 231	. . . . 5 ⊢ (𝜑 → (log‘2) ≤ (log‘𝑃))
123	113, 114, 116, 118, 122	lediv2ad 13040	. . . 4 ⊢ (𝜑 → ((log‘𝐵) / (log‘𝑃)) ≤ ((log‘𝐵) / (log‘2)))
124		relogbval 26284	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝐵 ∈ ℝ⁺) → (𝑃 log_b 𝐵) = ((log‘𝐵) / (log‘𝑃)))
125	68, 115, 124	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑃 log_b 𝐵) = ((log‘𝐵) / (log‘𝑃)))
126	125	eqcomd 2738	. . . 4 ⊢ (𝜑 → ((log‘𝐵) / (log‘𝑃)) = (𝑃 log_b 𝐵))
127	65	uzidd 12840	. . . . . 6 ⊢ (𝜑 → 2 ∈ (ℤ_≥‘2))
128		relogbval 26284	. . . . . 6 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝐵 ∈ ℝ⁺) → (2 log_b 𝐵) = ((log‘𝐵) / (log‘2)))
129	127, 115, 128	syl2anc 584	. . . . 5 ⊢ (𝜑 → (2 log_b 𝐵) = ((log‘𝐵) / (log‘2)))
130	129	eqcomd 2738	. . . 4 ⊢ (𝜑 → ((log‘𝐵) / (log‘2)) = (2 log_b 𝐵))
131	123, 126, 130	3brtr3d 5179	. . 3 ⊢ (𝜑 → (𝑃 log_b 𝐵) ≤ (2 log_b 𝐵))
132	15, 74, 75, 112, 131	letrd 11373	. 2 ⊢ (𝜑 → 𝐾 ≤ (2 log_b 𝐵))
133		flge 13772	. . 3 ⊢ (((2 log_b 𝐵) ∈ ℝ ∧ 𝐾 ∈ ℤ) → (𝐾 ≤ (2 log_b 𝐵) ↔ 𝐾 ≤ (⌊‘(2 log_b 𝐵))))
134	75, 14, 133	syl2anc 584	. 2 ⊢ (𝜑 → (𝐾 ≤ (2 log_b 𝐵) ↔ 𝐾 ≤ (⌊‘(2 log_b 𝐵))))
135	132, 134	mpbid 231	1 ⊢ (𝜑 → 𝐾 ≤ (⌊‘(2 log_b 𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3432 ∖ cdif 3945 {csn 4628 {cpr 4630 class class class wbr 5148 ‘cfv 6543 (class class class)co 7411 infcinf 9438 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 · cmul 11117 < clt 11250 ≤ cle 11251 − cmin 11446 / cdiv 11873 ℕcn 12214 2c2 12269 3c3 12270 5c5 12272 9c9 12276 ℕ₀cn0 12474 ℤcz 12560 ℤ_≥cuz 12824 ℝ⁺crp 12976 ...cfz 13486 ⌊cfl 13757 ⌈cceil 13758 ↑cexp 14029 ∏cprod 15851 ∥ cdvds 16199 ℙcprime 16610 pCnt cpc 16771 logclog 26070 ↑_𝑐ccxp 26071 log_b clogb 26276

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cc 10432 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 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-symdif 4242 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-omul 8473 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-dju 9898 df-card 9936 df-acn 9939 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-ioc 13331 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-fl 13759 df-ceil 13760 df-mod 13837 df-seq 13969 df-exp 14030 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15016 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-limsup 15417 df-clim 15434 df-rlim 15435 df-sum 15635 df-prod 15852 df-ef 16013 df-e 16014 df-sin 16015 df-cos 16016 df-pi 16018 df-dvds 16200 df-gcd 16438 df-lcm 16529 df-lcmf 16530 df-prm 16611 df-pc 16772 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-hom 17223 df-cco 17224 df-rest 17370 df-topn 17371 df-0g 17389 df-gsum 17390 df-topgen 17391 df-pt 17392 df-prds 17395 df-xrs 17450 df-qtop 17455 df-imas 17456 df-xps 17458 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-mulg 18953 df-cntz 19183 df-cmn 19652 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-fbas 20947 df-fg 20948 df-cnfld 20951 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cld 22530 df-ntr 22531 df-cls 22532 df-nei 22609 df-lp 22647 df-perf 22648 df-cn 22738 df-cnp 22739 df-haus 22826 df-cmp 22898 df-tx 23073 df-hmeo 23266 df-fil 23357 df-fm 23449 df-flim 23450 df-flf 23451 df-xms 23833 df-ms 23834 df-tms 23835 df-cncf 24401 df-ovol 24988 df-vol 24989 df-mbf 25143 df-itg1 25144 df-itg2 25145 df-ibl 25146 df-itg 25147 df-0p 25194 df-limc 25390 df-dv 25391 df-log 26072 df-cxp 26073 df-logb 26277

This theorem is referenced by: aks4d1p7d1 41033

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