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Theorem chtprm 25738

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 12011	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴 + 1) ∈ ℤ)
2	1	adantr 484	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (𝐴 + 1) ∈ ℤ)
3		zre 11973	. . . . 5 ⊢ ((𝐴 + 1) ∈ ℤ → (𝐴 + 1) ∈ ℝ)
4	2, 3	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (𝐴 + 1) ∈ ℝ)
5		chtval 25695	. . . 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 25689	. . . . . 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 13173	. . . . . . . 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 7151	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (2...(⌊‘(𝐴 + 1))) = (2...(𝐴 + 1)))
12	11	ineq1d 4138	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((2...(⌊‘(𝐴 + 1))) ∩ ℙ) = ((2...(𝐴 + 1)) ∩ ℙ))
13	8, 12	eqtrd 2833	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((0[,](𝐴 + 1)) ∩ ℙ) = ((2...(𝐴 + 1)) ∩ ℙ))
14	13	sumeq1d 15050	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → Σ𝑝 ∈ ((0[,](𝐴 + 1)) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)(log‘𝑝))
15	6, 14	eqtrd 2833	. 2 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (θ‘(𝐴 + 1)) = Σ𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)(log‘𝑝))
16		zre 11973	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
17	16	adantr 484	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → 𝐴 ∈ ℝ)
18	17	ltp1d 11559	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → 𝐴 < (𝐴 + 1))
19	17, 4	ltnled 10776	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (𝐴 < (𝐴 + 1) ↔ ¬ (𝐴 + 1) ≤ 𝐴))
20	18, 19	mpbid 235	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ¬ (𝐴 + 1) ≤ 𝐴)
21		elinel1 4122	. . . . . 6 ⊢ ((𝐴 + 1) ∈ ((2...𝐴) ∩ ℙ) → (𝐴 + 1) ∈ (2...𝐴))
22		elfzle2 12906	. . . . . 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 4607	. . . 4 ⊢ ((((2...𝐴) ∩ ℙ) ∩ {(𝐴 + 1)}) = ∅ ↔ ¬ (𝐴 + 1) ∈ ((2...𝐴) ∩ ℙ))
26	24, 25	sylibr 237	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (((2...𝐴) ∩ ℙ) ∩ {(𝐴 + 1)}) = ∅)
27		2z 12002	. . . . . . 7 ⊢ 2 ∈ ℤ
28		zcn 11974	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
29	28	adantr 484	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → 𝐴 ∈ ℂ)
30		ax-1cn 10584	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
31		pncan 10881	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 1) − 1) = 𝐴)
32	29, 30, 31	sylancl 589	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((𝐴 + 1) − 1) = 𝐴)
33		prmuz2 16030	. . . . . . . . . . 11 ⊢ ((𝐴 + 1) ∈ ℙ → (𝐴 + 1) ∈ (ℤ_≥‘2))
34	33	adantl 485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (𝐴 + 1) ∈ (ℤ_≥‘2))
35		uz2m1nn 12311	. . . . . . . . . 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 2891	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → 𝐴 ∈ ℕ)
38		nnuz 12269	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
39		2m1e1 11751	. . . . . . . . . 10 ⊢ (2 − 1) = 1
40	39	fveq2i 6648	. . . . . . . . 9 ⊢ (ℤ_≥‘(2 − 1)) = (ℤ_≥‘1)
41	38, 40	eqtr4i 2824	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘(2 − 1))
42	37, 41	eleqtrdi 2900	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → 𝐴 ∈ (ℤ_≥‘(2 − 1)))
43		fzsuc2 12960	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ (ℤ_≥‘(2 − 1))) → (2...(𝐴 + 1)) = ((2...𝐴) ∪ {(𝐴 + 1)}))
44	27, 42, 43	sylancr 590	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (2...(𝐴 + 1)) = ((2...𝐴) ∪ {(𝐴 + 1)}))
45	44	ineq1d 4138	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((2...(𝐴 + 1)) ∩ ℙ) = (((2...𝐴) ∪ {(𝐴 + 1)}) ∩ ℙ))
46		indir 4202	. . . . 5 ⊢ (((2...𝐴) ∪ {(𝐴 + 1)}) ∩ ℙ) = (((2...𝐴) ∩ ℙ) ∪ ({(𝐴 + 1)} ∩ ℙ))
47	45, 46	eqtrdi 2849	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((2...(𝐴 + 1)) ∩ ℙ) = (((2...𝐴) ∩ ℙ) ∪ ({(𝐴 + 1)} ∩ ℙ)))
48		simpr 488	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (𝐴 + 1) ∈ ℙ)
49	48	snssd 4702	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → {(𝐴 + 1)} ⊆ ℙ)
50		df-ss 3898	. . . . . 6 ⊢ ({(𝐴 + 1)} ⊆ ℙ ↔ ({(𝐴 + 1)} ∩ ℙ) = {(𝐴 + 1)})
51	49, 50	sylib 221	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ({(𝐴 + 1)} ∩ ℙ) = {(𝐴 + 1)})
52	51	uneq2d 4090	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (((2...𝐴) ∩ ℙ) ∪ ({(𝐴 + 1)} ∩ ℙ)) = (((2...𝐴) ∩ ℙ) ∪ {(𝐴 + 1)}))
53	47, 52	eqtrd 2833	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((2...(𝐴 + 1)) ∩ ℙ) = (((2...𝐴) ∩ ℙ) ∪ {(𝐴 + 1)}))
54		fzfid 13336	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (2...(𝐴 + 1)) ∈ Fin)
55		inss1 4155	. . . 4 ⊢ ((2...(𝐴 + 1)) ∩ ℙ) ⊆ (2...(𝐴 + 1))
56		ssfi 8722	. . . 4 ⊢ (((2...(𝐴 + 1)) ∈ Fin ∧ ((2...(𝐴 + 1)) ∩ ℙ) ⊆ (2...(𝐴 + 1))) → ((2...(𝐴 + 1)) ∩ ℙ) ∈ Fin)
57	54, 55, 56	sylancl 589	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((2...(𝐴 + 1)) ∩ ℙ) ∈ Fin)
58		simpr 488	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) ∧ 𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)) → 𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ))
59	58	elin2d 4126	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) ∧ 𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)) → 𝑝 ∈ ℙ)
60		prmnn 16008	. . . . . . 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 12417	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) ∧ 𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
63	62	relogcld 25214	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) ∧ 𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
64	63	recnd 10658	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) ∧ 𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
65	26, 53, 57, 64	fsumsplit 15089	. 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 25695	. . . . 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 25689	. . . . . . 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 13173	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
71	70	adantr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (⌊‘𝐴) = 𝐴)
72	71	oveq2d 7151	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (2...(⌊‘𝐴)) = (2...𝐴))
73	72	ineq1d 4138	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((2...(⌊‘𝐴)) ∩ ℙ) = ((2...𝐴) ∩ ℙ))
74	69, 73	eqtrd 2833	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → ((0[,]𝐴) ∩ ℙ) = ((2...𝐴) ∩ ℙ))
75	74	sumeq1d 15050	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((2...𝐴) ∩ ℙ)(log‘𝑝))
76	67, 75	eqtr2d 2834	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → Σ𝑝 ∈ ((2...𝐴) ∩ ℙ)(log‘𝑝) = (θ‘𝐴))
77		prmnn 16008	. . . . 5 ⊢ ((𝐴 + 1) ∈ ℙ → (𝐴 + 1) ∈ ℕ)
78	77	adantl 485	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (𝐴 + 1) ∈ ℕ)
79	78	nnrpd 12417	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (𝐴 + 1) ∈ ℝ⁺)
80	79	relogcld 25214	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (log‘(𝐴 + 1)) ∈ ℝ)
81	80	recnd 10658	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → (log‘(𝐴 + 1)) ∈ ℂ)
82		fveq2 6645	. . . . 5 ⊢ (𝑝 = (𝐴 + 1) → (log‘𝑝) = (log‘(𝐴 + 1)))
83	82	sumsn 15093	. . . 4 ⊢ (((𝐴 + 1) ∈ ℕ ∧ (log‘(𝐴 + 1)) ∈ ℂ) → Σ𝑝 ∈ {(𝐴 + 1)} (log‘𝑝) = (log‘(𝐴 + 1)))
84	78, 81, 83	syl2anc 587	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) → Σ𝑝 ∈ {(𝐴 + 1)} (log‘𝑝) = (log‘(𝐴 + 1)))
85	76, 84	oveq12d 7153	. 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 2837	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 399 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {csn 4525 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 < clt 10664 ≤ cle 10665 − cmin 10859 ℕcn 11625 2c2 11680 ℤcz 11969 ℤ_≥cuz 12231 [,]cicc 12729 ...cfz 12885 ⌊cfl 13155 Σcsu 15034 ℙcprime 16005 logclog 25146 θccht 25676

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-pi 15418 df-dvds 15600 df-prm 16006 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 df-log 25148 df-cht 25682

This theorem is referenced by: cht2 25757 cht3 25758

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