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Theorem pclogsum 27268

Description: The logarithmic analogue of pcprod 16937. 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 3986	. . . . . 6 ⊢ (𝑝 ∈ ((1...𝐴) ∩ ℙ) ↔ (𝑝 ∈ (1...𝐴) ∧ 𝑝 ∈ ℙ))
2	1	baib 535	. . . . 5 ⊢ (𝑝 ∈ (1...𝐴) → (𝑝 ∈ ((1...𝐴) ∩ ℙ) ↔ 𝑝 ∈ ℙ))
3	2	ifbid 4571	. . . 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 6935	. . . . 5 ⊢ (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)) = if(𝑝 ∈ ℙ, (log‘(𝑝↑(𝑝 pCnt 𝐴))), (log‘1))
5		log1 26636	. . . . . 6 ⊢ (log‘1) = 0
6		ifeq2 4553	. . . . . 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 15742	. 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 4252	. . . 4 ⊢ ((1...𝐴) ∩ ℙ) ⊆ (1...𝐴)
12		simpr 484	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝑝 ∈ ((1...𝐴) ∩ ℙ))
13	12	elin1d 4221	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝑝 ∈ (1...𝐴))
14		elfznn 13609	. . . . . . . . . 10 ⊢ (𝑝 ∈ (1...𝐴) → 𝑝 ∈ ℕ)
15	13, 14	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
16	12	elin2d 4222	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
17		simpl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝐴 ∈ ℕ)
18	16, 17	pccld 16892	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (𝑝 pCnt 𝐴) ∈ ℕ₀)
19	15, 18	nnexpcld 14290	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (𝑝↑(𝑝 pCnt 𝐴)) ∈ ℕ)
20	19	nnrpd 13093	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (𝑝↑(𝑝 pCnt 𝐴)) ∈ ℝ⁺)
21	20	relogcld 26674	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (log‘(𝑝↑(𝑝 pCnt 𝐴))) ∈ ℝ)
22	21	recnd 11314	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (log‘(𝑝↑(𝑝 pCnt 𝐴))) ∈ ℂ)
23	22	ralrimiva 3148	. . . 4 ⊢ (𝐴 ∈ ℕ → ∀𝑝 ∈ ((1...𝐴) ∩ ℙ)(log‘(𝑝↑(𝑝 pCnt 𝐴))) ∈ ℂ)
24		fzfi 14019	. . . . . 6 ⊢ (1...𝐴) ∈ Fin
25	24	olci 865	. . . . 5 ⊢ ((1...𝐴) ⊆ (ℤ_≥‘1) ∨ (1...𝐴) ∈ Fin)
26		sumss2 15770	. . . . 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 690	. . . 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 586	. . 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 13093	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
30	18	nn0zd 12661	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (𝑝 pCnt 𝐴) ∈ ℤ)
31		relogexp 26647	. . . . 5 ⊢ ((𝑝 ∈ ℝ⁺ ∧ (𝑝 pCnt 𝐴) ∈ ℤ) → (log‘(𝑝↑(𝑝 pCnt 𝐴))) = ((𝑝 pCnt 𝐴) · (log‘𝑝)))
32	29, 30, 31	syl2anc 583	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (log‘(𝑝↑(𝑝 pCnt 𝐴))) = ((𝑝 pCnt 𝐴) · (log‘𝑝)))
33	32	sumeq2dv 15746	. . 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 481	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → 𝑝 ∈ ℕ)
36		eleq1w 2821	. . . . . . . 8 ⊢ (𝑛 = 𝑝 → (𝑛 ∈ ℙ ↔ 𝑝 ∈ ℙ))
37		id 22	. . . . . . . . 9 ⊢ (𝑛 = 𝑝 → 𝑛 = 𝑝)
38		oveq1 7452	. . . . . . . . 9 ⊢ (𝑛 = 𝑝 → (𝑛 pCnt 𝐴) = (𝑝 pCnt 𝐴))
39	37, 38	oveq12d 7463	. . . . . . . 8 ⊢ (𝑛 = 𝑝 → (𝑛↑(𝑛 pCnt 𝐴)) = (𝑝↑(𝑝 pCnt 𝐴)))
40	36, 39	ifbieq1d 4572	. . . . . . 7 ⊢ (𝑛 = 𝑝 → if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1) = if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1))
41	40	fveq2d 6923	. . . . . 6 ⊢ (𝑛 = 𝑝 → (log‘if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)) = (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)))
42		eqid 2734	. . . . . 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 6932	. . . . . 6 ⊢ (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)) ∈ V
44	41, 42, 43	fvmpt 7027	. . . . 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 12943	. . . . 5 ⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (ℤ_≥‘1))
47	46	biimpi 216	. . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ (ℤ_≥‘1))
48	35	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℕ)
49		simpr 484	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
50		simpll 766	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℕ)
51	49, 50	pccld 16892	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℕ₀)
52	48, 51	nnexpcld 14290	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt 𝐴)) ∈ ℕ)
53		1nn 12300	. . . . . . . . 9 ⊢ 1 ∈ ℕ
54	53	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ ¬ 𝑝 ∈ ℙ) → 1 ∈ ℕ)
55	52, 54	ifclda 4583	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1) ∈ ℕ)
56	55	nnrpd 13093	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1) ∈ ℝ⁺)
57	56	relogcld 26674	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)) ∈ ℝ)
58	57	recnd 11314	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)) ∈ ℂ)
59	45, 47, 58	fsumser 15774	. . 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 13076	. . . . 5 ⊢ ((𝑝 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (𝑝 · 𝑚) ∈ ℝ⁺)
61	60	adantl 481	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ (𝑝 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺)) → (𝑝 · 𝑚) ∈ ℝ⁺)
62		eqid 2734	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1))
63		ovex 7478	. . . . . . . 8 ⊢ (𝑝↑(𝑝 pCnt 𝐴)) ∈ V
64		1ex 11282	. . . . . . . 8 ⊢ 1 ∈ V
65	63, 64	ifex 4598	. . . . . . 7 ⊢ if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1) ∈ V
66	40, 62, 65	fvmpt 7027	. . . . . 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 2838	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1))‘𝑝) ∈ ℝ⁺)
69		relogmul 26643	. . . . 5 ⊢ ((𝑝 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (log‘(𝑝 · 𝑚)) = ((log‘𝑝) + (log‘𝑚)))
70	69	adantl 481	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ (𝑝 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺)) → (log‘(𝑝 · 𝑚)) = ((log‘𝑝) + (log‘𝑚)))
71	67	fveq2d 6923	. . . . 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 14096	. . 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 16937	. . . 4 ⊢ (𝐴 ∈ ℕ → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)))‘𝐴) = 𝐴)
75	74	fveq2d 6923	. . 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 395 ∨ wo 846 = wceq 1537 ∈ wcel 2103 ∀wral 3063 ∩ cin 3969 ⊆ wss 3970 ifcif 4548 ↦ cmpt 5252 ‘cfv 6572 (class class class)co 7445 Fincfn 8999 ℂcc 11178 0cc0 11180 1c1 11181 + caddc 11183 · cmul 11185 ℕcn 12289 ℤcz 12635 ℤ_≥cuz 12899 ℝ⁺crp 13053 ...cfz 13563 seqcseq 14048 ↑cexp 14108 Σcsu 15730 ℙcprime 16712 pCnt cpc 16878 logclog 26605

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-inf2 9706 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 ax-addf 11259

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-iin 5022 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-om 7900 df-1st 8026 df-2nd 8027 df-supp 8198 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-er 8759 df-map 8882 df-pm 8883 df-ixp 8952 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-fsupp 9428 df-fi 9476 df-sup 9507 df-inf 9508 df-oi 9575 df-card 10004 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-q 13010 df-rp 13054 df-xneg 13171 df-xadd 13172 df-xmul 13173 df-ioo 13407 df-ioc 13408 df-ico 13409 df-icc 13410 df-fz 13564 df-fzo 13708 df-fl 13839 df-mod 13917 df-seq 14049 df-exp 14109 df-fac 14319 df-bc 14348 df-hash 14376 df-shft 15112 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-limsup 15513 df-clim 15530 df-rlim 15531 df-sum 15731 df-ef 16109 df-sin 16111 df-cos 16112 df-pi 16114 df-dvds 16297 df-gcd 16535 df-prm 16713 df-pc 16879 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-hom 17330 df-cco 17331 df-rest 17477 df-topn 17478 df-0g 17496 df-gsum 17497 df-topgen 17498 df-pt 17499 df-prds 17502 df-xrs 17557 df-qtop 17562 df-imas 17563 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-submnd 18814 df-mulg 19103 df-cntz 19352 df-cmn 19819 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22914 df-topon 22931 df-topsp 22953 df-bases 22967 df-cld 23041 df-ntr 23042 df-cls 23043 df-nei 23120 df-lp 23158 df-perf 23159 df-cn 23249 df-cnp 23250 df-haus 23337 df-tx 23584 df-hmeo 23777 df-fil 23868 df-fm 23960 df-flim 23961 df-flf 23962 df-xms 24344 df-ms 24345 df-tms 24346 df-cncf 24916 df-limc 25913 df-dv 25914 df-log 26607

This theorem is referenced by: vmasum 27269 chebbnd1lem1 27522

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