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Theorem chtprm 25730

Description: The Chebyshev function at a prime. (Contributed by Mario Carneiro, 22-Sep-2014.)

**Assertion**
Ref	Expression
chtprm	⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (θ‘(𝐴 + 1)) = ((θ‘𝐴) + (log‘(𝐴 + 1))))

Proof of Theorem chtprm Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2z 12024	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴 + 1) ∈ ℤ)
2	1	adantr 483	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (𝐴 + 1) ∈ ℤ)
3		zre 11986	. . . . 5 ⊢ ((𝐴 + 1) ∈ ℤ → (𝐴 + 1) ∈ ℝ)
4	2, 3	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (𝐴 + 1) ∈ ℝ)
5		chtval 25687	. . . 4 ⊢ ((𝐴 + 1) ∈ ℝ → (θ‘(𝐴 + 1)) = Σ𝑝 ∈ ((0[,](𝐴 + 1)) ∩ ℙ)(log‘𝑝))
6	4, 5	syl 17	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (θ‘(𝐴 + 1)) = Σ𝑝 ∈ ((0[,](𝐴 + 1)) ∩ ℙ)(log‘𝑝))
7		ppisval 25681	. . . . . 6 ⊢ ((𝐴 + 1) ∈ ℝ → ((0[,](𝐴 + 1)) ∩ ℙ) = ((2...(⌊‘(𝐴 + 1))) ∩ ℙ))
8	4, 7	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((0[,](𝐴 + 1)) ∩ ℙ) = ((2...(⌊‘(𝐴 + 1))) ∩ ℙ))
9		flid 13179	. . . . . . . 8 ⊢ ((𝐴 + 1) ∈ ℤ → (⌊‘(𝐴 + 1)) = (𝐴 + 1))
10	2, 9	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (⌊‘(𝐴 + 1)) = (𝐴 + 1))
11	10	oveq2d 7172	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (2...(⌊‘(𝐴 + 1))) = (2...(𝐴 + 1)))
12	11	ineq1d 4188	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((2...(⌊‘(𝐴 + 1))) ∩ ℙ) = ((2...(𝐴 + 1)) ∩ ℙ))
13	8, 12	eqtrd 2856	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((0[,](𝐴 + 1)) ∩ ℙ) = ((2...(𝐴 + 1)) ∩ ℙ))
14	13	sumeq1d 15058	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → Σ𝑝 ∈ ((0[,](𝐴 + 1)) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)(log‘𝑝))
15	6, 14	eqtrd 2856	. 2 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (θ‘(𝐴 + 1)) = Σ𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)(log‘𝑝))
16		zre 11986	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
17	16	adantr 483	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → 𝐴 ∈ ℝ)
18	17	ltp1d 11570	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → 𝐴 < (𝐴 + 1))
19	17, 4	ltnled 10787	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (𝐴 < (𝐴 + 1) ↔ ¬ (𝐴 + 1) ≤ 𝐴))
20	18, 19	mpbid 234	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ¬ (𝐴 + 1) ≤ 𝐴)
21		elinel1 4172	. . . . . 6 ⊢ ((𝐴 + 1) ∈ ((2...𝐴) ∩ ℙ) → (𝐴 + 1) ∈ (2...𝐴))
22		elfzle2 12912	. . . . . 6 ⊢ ((𝐴 + 1) ∈ (2...𝐴) → (𝐴 + 1) ≤ 𝐴)
23	21, 22	syl 17	. . . . 5 ⊢ ((𝐴 + 1) ∈ ((2...𝐴) ∩ ℙ) → (𝐴 + 1) ≤ 𝐴)
24	20, 23	nsyl 142	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ¬ (𝐴 + 1) ∈ ((2...𝐴) ∩ ℙ))
25		disjsn 4647	. . . 4 ⊢ ((((2...𝐴) ∩ ℙ) ∩ {(𝐴 + 1)}) = ∅ ↔ ¬ (𝐴 + 1) ∈ ((2...𝐴) ∩ ℙ))
26	24, 25	sylibr 236	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (((2...𝐴) ∩ ℙ) ∩ {(𝐴 + 1)}) = ∅)
27		2z 12015	. . . . . . 7 ⊢ 2 ∈ ℤ
28		zcn 11987	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
29	28	adantr 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → 𝐴 ∈ ℂ)
30		ax-1cn 10595	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
31		pncan 10892	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 1) − 1) = 𝐴)
32	29, 30, 31	sylancl 588	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((𝐴 + 1) − 1) = 𝐴)
33		prmuz2 16040	. . . . . . . . . . 11 ⊢ ((𝐴 + 1) ∈ ℙ → (𝐴 + 1) ∈ (ℤ_≥‘2))
34	33	adantl 484	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (𝐴 + 1) ∈ (ℤ_≥‘2))
35		uz2m1nn 12324	. . . . . . . . . 10 ⊢ ((𝐴 + 1) ∈ (ℤ_≥‘2) → ((𝐴 + 1) − 1) ∈ ℕ)
36	34, 35	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((𝐴 + 1) − 1) ∈ ℕ)
37	32, 36	eqeltrrd 2914	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → 𝐴 ∈ ℕ)
38		nnuz 12282	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
39		2m1e1 11764	. . . . . . . . . 10 ⊢ (2 − 1) = 1
40	39	fveq2i 6673	. . . . . . . . 9 ⊢ (ℤ_≥‘(2 − 1)) = (ℤ_≥‘1)
41	38, 40	eqtr4i 2847	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘(2 − 1))
42	37, 41	eleqtrdi 2923	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → 𝐴 ∈ (ℤ_≥‘(2 − 1)))
43		fzsuc2 12966	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ (ℤ_≥‘(2 − 1))) → (2...(𝐴 + 1)) = ((2...𝐴) ∪ {(𝐴 + 1)}))
44	27, 42, 43	sylancr 589	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (2...(𝐴 + 1)) = ((2...𝐴) ∪ {(𝐴 + 1)}))
45	44	ineq1d 4188	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((2...(𝐴 + 1)) ∩ ℙ) = (((2...𝐴) ∪ {(𝐴 + 1)}) ∩ ℙ))
46		indir 4252	. . . . 5 ⊢ (((2...𝐴) ∪ {(𝐴 + 1)}) ∩ ℙ) = (((2...𝐴) ∩ ℙ) ∪ ({(𝐴 + 1)} ∩ ℙ))
47	45, 46	syl6eq 2872	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((2...(𝐴 + 1)) ∩ ℙ) = (((2...𝐴) ∩ ℙ) ∪ ({(𝐴 + 1)} ∩ ℙ)))
48		simpr 487	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (𝐴 + 1) ∈ ℙ)
49	48	snssd 4742	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → {(𝐴 + 1)} ⊆ ℙ)
50		df-ss 3952	. . . . . 6 ⊢ ({(𝐴 + 1)} ⊆ ℙ ↔ ({(𝐴 + 1)} ∩ ℙ) = {(𝐴 + 1)})
51	49, 50	sylib 220	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ({(𝐴 + 1)} ∩ ℙ) = {(𝐴 + 1)})
52	51	uneq2d 4139	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (((2...𝐴) ∩ ℙ) ∪ ({(𝐴 + 1)} ∩ ℙ)) = (((2...𝐴) ∩ ℙ) ∪ {(𝐴 + 1)}))
53	47, 52	eqtrd 2856	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((2...(𝐴 + 1)) ∩ ℙ) = (((2...𝐴) ∩ ℙ) ∪ {(𝐴 + 1)}))
54		fzfid 13342	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (2...(𝐴 + 1)) ∈ Fin)
55		inss1 4205	. . . 4 ⊢ ((2...(𝐴 + 1)) ∩ ℙ) ⊆ (2...(𝐴 + 1))
56		ssfi 8738	. . . 4 ⊢ (((2...(𝐴 + 1)) ∈ Fin ∧ ((2...(𝐴 + 1)) ∩ ℙ) ⊆ (2...(𝐴 + 1))) → ((2...(𝐴 + 1)) ∩ ℙ) ∈ Fin)
57	54, 55, 56	sylancl 588	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((2...(𝐴 + 1)) ∩ ℙ) ∈ Fin)
58		simpr 487	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) ∧ 𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)) → 𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ))
59	58	elin2d 4176	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) ∧ 𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)) → 𝑝 ∈ ℙ)
60		prmnn 16018	. . . . . . 7 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
61	59, 60	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) ∧ 𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)) → 𝑝 ∈ ℕ)
62	61	nnrpd 12430	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) ∧ 𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
63	62	relogcld 25206	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) ∧ 𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
64	63	recnd 10669	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) ∧ 𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
65	26, 53, 57, 64	fsumsplit 15097	. 2 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → Σ𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)(log‘𝑝) = (Σ𝑝 ∈ ((2...𝐴) ∩ ℙ)(log‘𝑝) + Σ𝑝 ∈ {(𝐴 + 1)} (log‘𝑝)))
66		chtval 25687	. . . . 5 ⊢ (𝐴 ∈ ℝ → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝))
67	17, 66	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (θ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝))
68		ppisval 25681	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) = ((2...(⌊‘𝐴)) ∩ ℙ))
69	17, 68	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((0[,]𝐴) ∩ ℙ) = ((2...(⌊‘𝐴)) ∩ ℙ))
70		flid 13179	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
71	70	adantr 483	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (⌊‘𝐴) = 𝐴)
72	71	oveq2d 7172	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (2...(⌊‘𝐴)) = (2...𝐴))
73	72	ineq1d 4188	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((2...(⌊‘𝐴)) ∩ ℙ) = ((2...𝐴) ∩ ℙ))
74	69, 73	eqtrd 2856	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((0[,]𝐴) ∩ ℙ) = ((2...𝐴) ∩ ℙ))
75	74	sumeq1d 15058	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((2...𝐴) ∩ ℙ)(log‘𝑝))
76	67, 75	eqtr2d 2857	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → Σ𝑝 ∈ ((2...𝐴) ∩ ℙ)(log‘𝑝) = (θ‘𝐴))
77		prmnn 16018	. . . . 5 ⊢ ((𝐴 + 1) ∈ ℙ → (𝐴 + 1) ∈ ℕ)
78	77	adantl 484	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (𝐴 + 1) ∈ ℕ)
79	78	nnrpd 12430	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (𝐴 + 1) ∈ ℝ⁺)
80	79	relogcld 25206	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (log‘(𝐴 + 1)) ∈ ℝ)
81	80	recnd 10669	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (log‘(𝐴 + 1)) ∈ ℂ)
82		fveq2 6670	. . . . 5 ⊢ (𝑝 = (𝐴 + 1) → (log‘𝑝) = (log‘(𝐴 + 1)))
83	82	sumsn 15101	. . . 4 ⊢ (((𝐴 + 1) ∈ ℕ ∧ (log‘(𝐴 + 1)) ∈ ℂ) → Σ𝑝 ∈ {(𝐴 + 1)} (log‘𝑝) = (log‘(𝐴 + 1)))
84	78, 81, 83	syl2anc 586	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → Σ𝑝 ∈ {(𝐴 + 1)} (log‘𝑝) = (log‘(𝐴 + 1)))
85	76, 84	oveq12d 7174	. 2 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (Σ𝑝 ∈ ((2...𝐴) ∩ ℙ)(log‘𝑝) + Σ𝑝 ∈ {(𝐴 + 1)} (log‘𝑝)) = ((θ‘𝐴) + (log‘(𝐴 + 1))))
86	15, 65, 85	3eqtrd 2860	1 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (θ‘(𝐴 + 1)) = ((θ‘𝐴) + (log‘(𝐴 + 1))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 {csn 4567 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 < clt 10675 ≤ cle 10676 − cmin 10870 ℕcn 11638 2c2 11693 ℤcz 11982 ℤ_≥cuz 12244 [,]cicc 12742 ...cfz 12893 ⌊cfl 13161 Σcsu 15042 ℙcprime 16015 logclog 25138 θccht 25668

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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-dvds 15608 df-prm 16016 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-log 25140 df-cht 25674

This theorem is referenced by: cht2 25749 cht3 25750

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