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Theorem pclogsum 27196

Description: The logarithmic analogue of pcprod 16857. The sum of the logarithms of the primes dividing 𝐴 multiplied by their powers yields the logarithm of 𝐴. (Contributed by Mario Carneiro, 15-Apr-2016.)

**Assertion**
Ref	Expression
pclogsum	⊢ (𝐴 ∈ ℕ → Σ𝑝 ∈ ((1...𝐴) ∩ ℙ)((𝑝 pCnt 𝐴) · (log‘𝑝)) = (log‘𝐴))

Distinct variable group: 𝐴,𝑝

Proof of Theorem pclogsum Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3899	. . . . . 6 ⊢ (𝑝 ∈ ((1...𝐴) ∩ ℙ) ↔ (𝑝 ∈ (1...𝐴) ∧ 𝑝 ∈ ℙ))
2	1	baib 540	. . . . 5 ⊢ (𝑝 ∈ (1...𝐴) → (𝑝 ∈ ((1...𝐴) ∩ ℙ) ↔ 𝑝 ∈ ℙ))
3	2	ifbid 4478	. . . 4 ⊢ (𝑝 ∈ (1...𝐴) → if(𝑝 ∈ ((1...𝐴) ∩ ℙ), (log‘(𝑝↑(𝑝 pCnt 𝐴))), 0) = if(𝑝 ∈ ℙ, (log‘(𝑝↑(𝑝 pCnt 𝐴))), 0))
4		fvif 6843	. . . . 5 ⊢ (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)) = if(𝑝 ∈ ℙ, (log‘(𝑝↑(𝑝 pCnt 𝐴))), (log‘1))
5		log1 26567	. . . . . 6 ⊢ (log‘1) = 0
6		ifeq2 4459	. . . . . 6 ⊢ ((log‘1) = 0 → if(𝑝 ∈ ℙ, (log‘(𝑝↑(𝑝 pCnt 𝐴))), (log‘1)) = if(𝑝 ∈ ℙ, (log‘(𝑝↑(𝑝 pCnt 𝐴))), 0))
7	5, 6	ax-mp 5	. . . . 5 ⊢ if(𝑝 ∈ ℙ, (log‘(𝑝↑(𝑝 pCnt 𝐴))), (log‘1)) = if(𝑝 ∈ ℙ, (log‘(𝑝↑(𝑝 pCnt 𝐴))), 0)
8	4, 7	eqtri 2762	. . . 4 ⊢ (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)) = if(𝑝 ∈ ℙ, (log‘(𝑝↑(𝑝 pCnt 𝐴))), 0)
9	3, 8	eqtr4di 2792	. . 3 ⊢ (𝑝 ∈ (1...𝐴) → if(𝑝 ∈ ((1...𝐴) ∩ ℙ), (log‘(𝑝↑(𝑝 pCnt 𝐴))), 0) = (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)))
10	9	sumeq2i 15651	. 2 ⊢ Σ𝑝 ∈ (1...𝐴)if(𝑝 ∈ ((1...𝐴) ∩ ℙ), (log‘(𝑝↑(𝑝 pCnt 𝐴))), 0) = Σ𝑝 ∈ (1...𝐴)(log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1))
11		inss1 4165	. . . 4 ⊢ ((1...𝐴) ∩ ℙ) ⊆ (1...𝐴)
12		simpr 485	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝑝 ∈ ((1...𝐴) ∩ ℙ))
13	12	elin1d 4133	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝑝 ∈ (1...𝐴))
14		elfznn 13498	. . . . . . . . . 10 ⊢ (𝑝 ∈ (1...𝐴) → 𝑝 ∈ ℕ)
15	13, 14	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
16	12	elin2d 4134	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
17		simpl 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝐴 ∈ ℕ)
18	16, 17	pccld 16812	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (𝑝 pCnt 𝐴) ∈ ℕ₀)
19	15, 18	nnexpcld 14198	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (𝑝↑(𝑝 pCnt 𝐴)) ∈ ℕ)
20	19	nnrpd 12975	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (𝑝↑(𝑝 pCnt 𝐴)) ∈ ℝ⁺)
21	20	relogcld 26605	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (log‘(𝑝↑(𝑝 pCnt 𝐴))) ∈ ℝ)
22	21	recnd 11164	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (log‘(𝑝↑(𝑝 pCnt 𝐴))) ∈ ℂ)
23	22	ralrimiva 3131	. . . 4 ⊢ (𝐴 ∈ ℕ → ∀𝑝 ∈ ((1...𝐴) ∩ ℙ)(log‘(𝑝↑(𝑝 pCnt 𝐴))) ∈ ℂ)
24		fzfi 13925	. . . . . 6 ⊢ (1...𝐴) ∈ Fin
25	24	olci 872	. . . . 5 ⊢ ((1...𝐴) ⊆ (ℤ_≥‘1) ∨ (1...𝐴) ∈ Fin)
26		sumss2 15679	. . . . 5 ⊢ (((((1...𝐴) ∩ ℙ) ⊆ (1...𝐴) ∧ ∀𝑝 ∈ ((1...𝐴) ∩ ℙ)(log‘(𝑝↑(𝑝 pCnt 𝐴))) ∈ ℂ) ∧ ((1...𝐴) ⊆ (ℤ_≥‘1) ∨ (1...𝐴) ∈ Fin)) → Σ𝑝 ∈ ((1...𝐴) ∩ ℙ)(log‘(𝑝↑(𝑝 pCnt 𝐴))) = Σ𝑝 ∈ (1...𝐴)if(𝑝 ∈ ((1...𝐴) ∩ ℙ), (log‘(𝑝↑(𝑝 pCnt 𝐴))), 0))
27	25, 26	mpan2 697	. . . 4 ⊢ ((((1...𝐴) ∩ ℙ) ⊆ (1...𝐴) ∧ ∀𝑝 ∈ ((1...𝐴) ∩ ℙ)(log‘(𝑝↑(𝑝 pCnt 𝐴))) ∈ ℂ) → Σ𝑝 ∈ ((1...𝐴) ∩ ℙ)(log‘(𝑝↑(𝑝 pCnt 𝐴))) = Σ𝑝 ∈ (1...𝐴)if(𝑝 ∈ ((1...𝐴) ∩ ℙ), (log‘(𝑝↑(𝑝 pCnt 𝐴))), 0))
28	11, 23, 27	sylancr 593	. . 3 ⊢ (𝐴 ∈ ℕ → Σ𝑝 ∈ ((1...𝐴) ∩ ℙ)(log‘(𝑝↑(𝑝 pCnt 𝐴))) = Σ𝑝 ∈ (1...𝐴)if(𝑝 ∈ ((1...𝐴) ∩ ℙ), (log‘(𝑝↑(𝑝 pCnt 𝐴))), 0))
29	15	nnrpd 12975	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
30	18	nn0zd 12540	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (𝑝 pCnt 𝐴) ∈ ℤ)
31		relogexp 26578	. . . . 5 ⊢ ((𝑝 ∈ ℝ⁺ ∧ (𝑝 pCnt 𝐴) ∈ ℤ) → (log‘(𝑝↑(𝑝 pCnt 𝐴))) = ((𝑝 pCnt 𝐴) · (log‘𝑝)))
32	29, 30, 31	syl2anc 590	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (log‘(𝑝↑(𝑝 pCnt 𝐴))) = ((𝑝 pCnt 𝐴) · (log‘𝑝)))
33	32	sumeq2dv 15655	. . 3 ⊢ (𝐴 ∈ ℕ → Σ𝑝 ∈ ((1...𝐴) ∩ ℙ)(log‘(𝑝↑(𝑝 pCnt 𝐴))) = Σ𝑝 ∈ ((1...𝐴) ∩ ℙ)((𝑝 pCnt 𝐴) · (log‘𝑝)))
34	28, 33	eqtr3d 2776	. 2 ⊢ (𝐴 ∈ ℕ → Σ𝑝 ∈ (1...𝐴)if(𝑝 ∈ ((1...𝐴) ∩ ℙ), (log‘(𝑝↑(𝑝 pCnt 𝐴))), 0) = Σ𝑝 ∈ ((1...𝐴) ∩ ℙ)((𝑝 pCnt 𝐴) · (log‘𝑝)))
35	14	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → 𝑝 ∈ ℕ)
36		eleq1w 2822	. . . . . . . 8 ⊢ (𝑛 = 𝑝 → (𝑛 ∈ ℙ ↔ 𝑝 ∈ ℙ))
37		id 22	. . . . . . . . 9 ⊢ (𝑛 = 𝑝 → 𝑛 = 𝑝)
38		oveq1 7363	. . . . . . . . 9 ⊢ (𝑛 = 𝑝 → (𝑛 pCnt 𝐴) = (𝑝 pCnt 𝐴))
39	37, 38	oveq12d 7374	. . . . . . . 8 ⊢ (𝑛 = 𝑝 → (𝑛↑(𝑛 pCnt 𝐴)) = (𝑝↑(𝑝 pCnt 𝐴)))
40	36, 39	ifbieq1d 4479	. . . . . . 7 ⊢ (𝑛 = 𝑝 → if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1) = if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1))
41	40	fveq2d 6831	. . . . . 6 ⊢ (𝑛 = 𝑝 → (log‘if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)) = (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)))
42		eqid 2739	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (log‘if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1))) = (𝑛 ∈ ℕ ↦ (log‘if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)))
43		fvex 6840	. . . . . 6 ⊢ (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)) ∈ V
44	41, 42, 43	fvmpt 6935	. . . . 5 ⊢ (𝑝 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (log‘if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)))‘𝑝) = (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)))
45	35, 44	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → ((𝑛 ∈ ℕ ↦ (log‘if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)))‘𝑝) = (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)))
46		elnnuz 12819	. . . . 5 ⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (ℤ_≥‘1))
47	46	biimpi 217	. . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ (ℤ_≥‘1))
48	35	adantr 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℕ)
49		simpr 485	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
50		simpll 772	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℕ)
51	49, 50	pccld 16812	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℕ₀)
52	48, 51	nnexpcld 14198	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt 𝐴)) ∈ ℕ)
53		1nn 12176	. . . . . . . . 9 ⊢ 1 ∈ ℕ
54	53	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ ¬ 𝑝 ∈ ℙ) → 1 ∈ ℕ)
55	52, 54	ifclda 4490	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1) ∈ ℕ)
56	55	nnrpd 12975	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1) ∈ ℝ⁺)
57	56	relogcld 26605	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)) ∈ ℝ)
58	57	recnd 11164	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)) ∈ ℂ)
59	45, 47, 58	fsumser 15683	. . 3 ⊢ (𝐴 ∈ ℕ → Σ𝑝 ∈ (1...𝐴)(log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)) = (seq1( + , (𝑛 ∈ ℕ ↦ (log‘if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1))))‘𝐴))
60		rpmulcl 12958	. . . . 5 ⊢ ((𝑝 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (𝑝 · 𝑚) ∈ ℝ⁺)
61	60	adantl 482	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ (𝑝 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺)) → (𝑝 · 𝑚) ∈ ℝ⁺)
62		eqid 2739	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1))
63		ovex 7389	. . . . . . . 8 ⊢ (𝑝↑(𝑝 pCnt 𝐴)) ∈ V
64		1ex 11131	. . . . . . . 8 ⊢ 1 ∈ V
65	63, 64	ifex 4505	. . . . . . 7 ⊢ if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1) ∈ V
66	40, 62, 65	fvmpt 6935	. . . . . 6 ⊢ (𝑝 ∈ ℕ → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1))‘𝑝) = if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1))
67	35, 66	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1))‘𝑝) = if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1))
68	67, 56	eqeltrd 2839	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1))‘𝑝) ∈ ℝ⁺)
69		relogmul 26574	. . . . 5 ⊢ ((𝑝 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (log‘(𝑝 · 𝑚)) = ((log‘𝑝) + (log‘𝑚)))
70	69	adantl 482	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ (𝑝 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺)) → (log‘(𝑝 · 𝑚)) = ((log‘𝑝) + (log‘𝑚)))
71	67	fveq2d 6831	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → (log‘((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1))‘𝑝)) = (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)))
72	71, 45	eqtr4d 2777	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → (log‘((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1))‘𝑝)) = ((𝑛 ∈ ℕ ↦ (log‘if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)))‘𝑝))
73	61, 68, 47, 70, 72	seqhomo 14002	. . 3 ⊢ (𝐴 ∈ ℕ → (log‘(seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)))‘𝐴)) = (seq1( + , (𝑛 ∈ ℕ ↦ (log‘if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1))))‘𝐴))
74	62	pcprod 16857	. . . 4 ⊢ (𝐴 ∈ ℕ → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)))‘𝐴) = 𝐴)
75	74	fveq2d 6831	. . 3 ⊢ (𝐴 ∈ ℕ → (log‘(seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)))‘𝐴)) = (log‘𝐴))
76	59, 73, 75	3eqtr2d 2780	. 2 ⊢ (𝐴 ∈ ℕ → Σ𝑝 ∈ (1...𝐴)(log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)) = (log‘𝐴))
77	10, 34, 76	3eqtr3a 2798	1 ⊢ (𝐴 ∈ ℕ → Σ𝑝 ∈ ((1...𝐴) ∩ ℙ)((𝑝 pCnt 𝐴) · (log‘𝑝)) = (log‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 853 = wceq 1547 ∈ wcel 2119 ∀wral 3053 ∩ cin 3882 ⊆ wss 3883 ifcif 4454 ↦ cmpt 5153 ‘cfv 6485 (class class class)co 7356 Fincfn 8883 ℂcc 11027 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 ℕcn 12165 ℤcz 12515 ℤ_≥cuz 12779 ℝ⁺crp 12933 ...cfz 13452 seqcseq 13954 ↑cexp 14014 Σcsu 15639 ℙcprime 16631 pCnt cpc 16798 logclog 26536

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-ef 16023 df-sin 16025 df-cos 16026 df-pi 16028 df-dvds 16213 df-gcd 16455 df-prm 16632 df-pc 16799 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-lp 23119 df-perf 23120 df-cn 23210 df-cnp 23211 df-haus 23298 df-tx 23545 df-hmeo 23738 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-xms 24303 df-ms 24304 df-tms 24305 df-cncf 24863 df-limc 25851 df-dv 25852 df-log 26538

This theorem is referenced by: vmasum 27197 chebbnd1lem1 27450

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