aks4d1p6 - Mathbox for metakunt

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Theorem aks4d1p6 41253

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 41251	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴))
9	8	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (1...𝐵))
10		elfznn 13535	. . . . . . . 8 ⊢ (𝑅 ∈ (1...𝐵) → 𝑅 ∈ ℕ)
11	9, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℕ)
12	3, 11	pccld 16788	. . . . . 6 ⊢ (𝜑 → (𝑃 pCnt 𝑅) ∈ ℕ₀)
13	2, 12	eqeltrd 2832	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
14	13	nn0zd 12589	. . . 4 ⊢ (𝜑 → 𝐾 ∈ ℤ)
15	14	zred 12671	. . 3 ⊢ (𝜑 → 𝐾 ∈ ℝ)
16		prmnn 16616	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
17	3, 16	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ)
18	17	nnred 12232	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℝ)
19	17	nngt0d 12266	. . . 4 ⊢ (𝜑 → 0 < 𝑃)
20	6	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (⌈‘((2 log_b 𝑁)↑5)))
21		2re 12291	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
22	21	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
23		2pos 12320	. . . . . . . . . . . 12 ⊢ 0 < 2
24	23	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 2)
25		eluzelz 12837	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
26	4, 25	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
27	26	zred 12671	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ)
28		0red 11222	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℝ)
29		3re 12297	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ
30	29	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 3 ∈ ℝ)
31		3pos 12322	. . . . . . . . . . . . 13 ⊢ 0 < 3
32	31	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 3)
33		eluzle 12840	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
34	4, 33	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 3 ≤ 𝑁)
35	28, 30, 27, 32, 34	ltletrd 11379	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝑁)
36		1red 11220	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℝ)
37		1lt2 12388	. . . . . . . . . . . . . 14 ⊢ 1 < 2
38	37	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < 2)
39	36, 38	ltned 11355	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≠ 2)
40	39	necomd 2995	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≠ 1)
41	22, 24, 27, 35, 40	relogbcld 41145	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
42		5nn0 12497	. . . . . . . . . . 11 ⊢ 5 ∈ ℕ₀
43	42	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 5 ∈ ℕ₀)
44	41, 43	reexpcld 14133	. . . . . . . . 9 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ∈ ℝ)
45		ceilcl 13812	. . . . . . . . 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 2832	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℤ)
48	47	zred 12671	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
49		9re 12316	. . . . . . . . . 10 ⊢ 9 ∈ ℝ
50	49	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 9 ∈ ℝ)
51		9pos 12330	. . . . . . . . . 10 ⊢ 0 < 9
52	51	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < 9)
53	27, 34	3lexlogpow5ineq4 41228	. . . . . . . . 9 ⊢ (𝜑 → 9 < ((2 log_b 𝑁)↑5))
54	28, 50, 44, 52, 53	lttrd 11380	. . . . . . . 8 ⊢ (𝜑 → 0 < ((2 log_b 𝑁)↑5))
55		ceilge 13815	. . . . . . . . . 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 11379	. . . . . . 7 ⊢ (𝜑 → 0 < 𝐵)
59	47, 58	jca 511	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ 0 < 𝐵))
60		elnnz 12573	. . . . . 6 ⊢ (𝐵 ∈ ℕ ↔ (𝐵 ∈ ℤ ∧ 0 < 𝐵))
61	59, 60	sylibr 233	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℕ)
62	61	nnred 12232	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
63	61	nngt0d 12266	. . . 4 ⊢ (𝜑 → 0 < 𝐵)
64		2z 12599	. . . . . . . . 9 ⊢ 2 ∈ ℤ
65	64	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℤ)
66	65	zred 12671	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℝ)
67		prmuz2 16638	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
68	3, 67	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘2))
69		eluzle 12840	. . . . . . . 8 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 2 ≤ 𝑃)
70	68, 69	syl 17	. . . . . . 7 ⊢ (𝜑 → 2 ≤ 𝑃)
71	36, 66, 18, 38, 70	ltletrd 11379	. . . . . 6 ⊢ (𝜑 → 1 < 𝑃)
72	36, 71	ltned 11355	. . . . 5 ⊢ (𝜑 → 1 ≠ 𝑃)
73	72	necomd 2995	. . . 4 ⊢ (𝜑 → 𝑃 ≠ 1)
74	18, 19, 62, 63, 73	relogbcld 41145	. . 3 ⊢ (𝜑 → (𝑃 log_b 𝐵) ∈ ℝ)
75	66, 24, 62, 63, 40	relogbcld 41145	. . 3 ⊢ (𝜑 → (2 log_b 𝐵) ∈ ℝ)
76	17	nnrpd 13019	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
77	76	rpcnd 13023	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℂ)
78	76	rpne0d 13026	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 0)
79	77, 78, 14	cxpexpzd 26456	. . . . 5 ⊢ (𝜑 → (𝑃↑_𝑐𝐾) = (𝑃↑𝐾))
80	18, 13	reexpcld 14133	. . . . . . 7 ⊢ (𝜑 → (𝑃↑𝐾) ∈ ℝ)
81	11	nnred 12232	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℝ)
82	2	oveq2d 7428	. . . . . . . 8 ⊢ (𝜑 → (𝑃↑𝐾) = (𝑃↑(𝑃 pCnt 𝑅)))
83		pcdvds 16802	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑅 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝑅)) ∥ 𝑅)
84	3, 11, 83	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∥ 𝑅)
85	17	nnzd 12590	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℤ)
86		zexpcl 14047	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℤ ∧ (𝑃 pCnt 𝑅) ∈ ℕ₀) → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℤ)
87	85, 12, 86	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) ∈ ℤ)
88		dvdsle 16258	. . . . . . . . . 10 ⊢ (((𝑃↑(𝑃 pCnt 𝑅)) ∈ ℤ ∧ 𝑅 ∈ ℕ) → ((𝑃↑(𝑃 pCnt 𝑅)) ∥ 𝑅 → (𝑃↑(𝑃 pCnt 𝑅)) ≤ 𝑅))
89	87, 11, 88	syl2anc 583	. . . . . . . . 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 13510	. . . . . . . 8 ⊢ (𝑅 ∈ (1...𝐵) → 𝑅 ≤ 𝐵)
93	9, 92	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ≤ 𝐵)
94	80, 81, 62, 91, 93	letrd 11376	. . . . . 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 11247	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
98	28, 63	ltned 11355	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≠ 𝐵)
99	98	necomd 2995	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≠ 0)
100	99	neneqd 2944	. . . . . . . . 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 325	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐵 ∈ {0})
104	97, 103	eldifd 3959	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (ℂ ∖ {0}))
105		cxplogb 26528	. . . . . . 7 ⊢ ((𝑃 ∈ (ℂ ∖ {0, 1}) ∧ 𝐵 ∈ (ℂ ∖ {0})) → (𝑃↑_𝑐(𝑃 log_b 𝐵)) = 𝐵)
106	96, 104, 105	syl2anc 583	. . . . . 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 13021	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℝ)
110	36, 66, 109, 38, 70	ltletrd 11379	. . . . 5 ⊢ (𝜑 → 1 < 𝑃)
111	109, 110, 15, 74	cxpled 26465	. . . 4 ⊢ (𝜑 → (𝐾 ≤ (𝑃 log_b 𝐵) ↔ (𝑃↑_𝑐𝐾) ≤ (𝑃↑_𝑐(𝑃 log_b 𝐵))))
112	108, 111	mpbird 257	. . 3 ⊢ (𝜑 → 𝐾 ≤ (𝑃 log_b 𝐵))
113	22, 38	rplogcld 26374	. . . . 5 ⊢ (𝜑 → (log‘2) ∈ ℝ⁺)
114	109, 110	rplogcld 26374	. . . . 5 ⊢ (𝜑 → (log‘𝑃) ∈ ℝ⁺)
115	61	nnrpd 13019	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
116	115	relogcld 26368	. . . . 5 ⊢ (𝜑 → (log‘𝐵) ∈ ℝ)
117	61	nnge1d 12265	. . . . . 6 ⊢ (𝜑 → 1 ≤ 𝐵)
118	62, 117	logge0d 26375	. . . . 5 ⊢ (𝜑 → 0 ≤ (log‘𝐵))
119		2rp 12984	. . . . . . . 8 ⊢ 2 ∈ ℝ⁺
120	119	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℝ⁺)
121	120, 76	logled 26372	. . . . . 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 13043	. . . 4 ⊢ (𝜑 → ((log‘𝐵) / (log‘𝑃)) ≤ ((log‘𝐵) / (log‘2)))
124		relogbval 26514	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 𝐵 ∈ ℝ⁺) → (𝑃 log_b 𝐵) = ((log‘𝐵) / (log‘𝑃)))
125	68, 115, 124	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑃 log_b 𝐵) = ((log‘𝐵) / (log‘𝑃)))
126	125	eqcomd 2737	. . . 4 ⊢ (𝜑 → ((log‘𝐵) / (log‘𝑃)) = (𝑃 log_b 𝐵))
127	65	uzidd 12843	. . . . . 6 ⊢ (𝜑 → 2 ∈ (ℤ_≥‘2))
128		relogbval 26514	. . . . . 6 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝐵 ∈ ℝ⁺) → (2 log_b 𝐵) = ((log‘𝐵) / (log‘2)))
129	127, 115, 128	syl2anc 583	. . . . 5 ⊢ (𝜑 → (2 log_b 𝐵) = ((log‘𝐵) / (log‘2)))
130	129	eqcomd 2737	. . . 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 11376	. 2 ⊢ (𝜑 → 𝐾 ≤ (2 log_b 𝐵))
133		flge 13775	. . 3 ⊢ (((2 log_b 𝐵) ∈ ℝ ∧ 𝐾 ∈ ℤ) → (𝐾 ≤ (2 log_b 𝐵) ↔ 𝐾 ≤ (⌊‘(2 log_b 𝐵))))
134	75, 14, 133	syl2anc 583	. 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 395 = wceq 1540 ∈ wcel 2105 {crab 3431 ∖ cdif 3945 {csn 4628 {cpr 4630 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 infcinf 9440 ℂcc 11112 ℝcr 11113 0cc0 11114 1c1 11115 · cmul 11119 < clt 11253 ≤ cle 11254 − cmin 11449 / cdiv 11876 ℕcn 12217 2c2 12272 3c3 12273 5c5 12275 9c9 12279 ℕ₀cn0 12477 ℤcz 12563 ℤ_≥cuz 12827 ℝ⁺crp 12979 ...cfz 13489 ⌊cfl 13760 ⌈cceil 13761 ↑cexp 14032 ∏cprod 15854 ∥ cdvds 16202 ℙcprime 16613 pCnt cpc 16774 logclog 26300 ↑_𝑐ccxp 26301 log_b clogb 26506

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cc 10434 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-ofr 7675 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-oadd 8474 df-omul 8475 df-er 8707 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-dju 9900 df-card 9938 df-acn 9941 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ioc 13334 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-ceil 13763 df-mod 13840 df-seq 13972 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15019 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-limsup 15420 df-clim 15437 df-rlim 15438 df-sum 15638 df-prod 15855 df-ef 16016 df-e 16017 df-sin 16018 df-cos 16019 df-pi 16021 df-dvds 16203 df-gcd 16441 df-lcm 16532 df-lcmf 16533 df-prm 16614 df-pc 16775 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-lp 22861 df-perf 22862 df-cn 22952 df-cnp 22953 df-haus 23040 df-cmp 23112 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-xms 24047 df-ms 24048 df-tms 24049 df-cncf 24619 df-ovol 25214 df-vol 25215 df-mbf 25369 df-itg1 25370 df-itg2 25371 df-ibl 25372 df-itg 25373 df-0p 25420 df-limc 25616 df-dv 25617 df-log 26302 df-cxp 26303 df-logb 26507

This theorem is referenced by: aks4d1p7d1 41254

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