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Theorem pclogsum 24654
Description: The logarithmic analogue of pcprod 15380. 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.
StepHypRef Expression
1 elin 3754 . . . . . 6 (𝑝 ∈ ((1...𝐴) ∩ ℙ) ↔ (𝑝 ∈ (1...𝐴) ∧ 𝑝 ∈ ℙ))
21baib 941 . . . . 5 (𝑝 ∈ (1...𝐴) → (𝑝 ∈ ((1...𝐴) ∩ ℙ) ↔ 𝑝 ∈ ℙ))
32ifbid 4054 . . . 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 6096 . . . . 5 (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)) = if(𝑝 ∈ ℙ, (log‘(𝑝↑(𝑝 pCnt 𝐴))), (log‘1))
5 log1 24050 . . . . . 6 (log‘1) = 0
6 ifeq2 4037 . . . . . 6 ((log‘1) = 0 → if(𝑝 ∈ ℙ, (log‘(𝑝↑(𝑝 pCnt 𝐴))), (log‘1)) = if(𝑝 ∈ ℙ, (log‘(𝑝↑(𝑝 pCnt 𝐴))), 0))
75, 6ax-mp 5 . . . . 5 if(𝑝 ∈ ℙ, (log‘(𝑝↑(𝑝 pCnt 𝐴))), (log‘1)) = if(𝑝 ∈ ℙ, (log‘(𝑝↑(𝑝 pCnt 𝐴))), 0)
84, 7eqtri 2628 . . . 4 (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)) = if(𝑝 ∈ ℙ, (log‘(𝑝↑(𝑝 pCnt 𝐴))), 0)
93, 8syl6eqr 2658 . . 3 (𝑝 ∈ (1...𝐴) → if(𝑝 ∈ ((1...𝐴) ∩ ℙ), (log‘(𝑝↑(𝑝 pCnt 𝐴))), 0) = (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)))
109sumeq2i 14220 . 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 3791 . . . 4 ((1...𝐴) ∩ ℙ) ⊆ (1...𝐴)
12 simpr 475 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝑝 ∈ ((1...𝐴) ∩ ℙ))
1311, 12sseldi 3562 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝑝 ∈ (1...𝐴))
14 elfznn 12193 . . . . . . . . . 10 (𝑝 ∈ (1...𝐴) → 𝑝 ∈ ℕ)
1513, 14syl 17 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
16 inss2 3792 . . . . . . . . . . 11 ((1...𝐴) ∩ ℙ) ⊆ ℙ
1716, 12sseldi 3562 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
18 simpl 471 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝐴 ∈ ℕ)
1917, 18pccld 15336 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (𝑝 pCnt 𝐴) ∈ ℕ0)
2015, 19nnexpcld 12844 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (𝑝↑(𝑝 pCnt 𝐴)) ∈ ℕ)
2120nnrpd 11699 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (𝑝↑(𝑝 pCnt 𝐴)) ∈ ℝ+)
2221relogcld 24087 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (log‘(𝑝↑(𝑝 pCnt 𝐴))) ∈ ℝ)
2322recnd 9921 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (log‘(𝑝↑(𝑝 pCnt 𝐴))) ∈ ℂ)
2423ralrimiva 2945 . . . 4 (𝐴 ∈ ℕ → ∀𝑝 ∈ ((1...𝐴) ∩ ℙ)(log‘(𝑝↑(𝑝 pCnt 𝐴))) ∈ ℂ)
25 fzfi 12585 . . . . . 6 (1...𝐴) ∈ Fin
2625olci 404 . . . . 5 ((1...𝐴) ⊆ (ℤ‘1) ∨ (1...𝐴) ∈ Fin)
27 sumss2 14247 . . . . 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))
2826, 27mpan2 702 . . . 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))
2911, 24, 28sylancr 693 . . 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))
3015nnrpd 11699 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ+)
3119nn0zd 11309 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (𝑝 pCnt 𝐴) ∈ ℤ)
32 relogexp 24060 . . . . 5 ((𝑝 ∈ ℝ+ ∧ (𝑝 pCnt 𝐴) ∈ ℤ) → (log‘(𝑝↑(𝑝 pCnt 𝐴))) = ((𝑝 pCnt 𝐴) · (log‘𝑝)))
3330, 31, 32syl2anc 690 . . . 4 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ((1...𝐴) ∩ ℙ)) → (log‘(𝑝↑(𝑝 pCnt 𝐴))) = ((𝑝 pCnt 𝐴) · (log‘𝑝)))
3433sumeq2dv 14224 . . 3 (𝐴 ∈ ℕ → Σ𝑝 ∈ ((1...𝐴) ∩ ℙ)(log‘(𝑝↑(𝑝 pCnt 𝐴))) = Σ𝑝 ∈ ((1...𝐴) ∩ ℙ)((𝑝 pCnt 𝐴) · (log‘𝑝)))
3529, 34eqtr3d 2642 . 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)))
3614adantl 480 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → 𝑝 ∈ ℕ)
37 eleq1 2672 . . . . . . . 8 (𝑛 = 𝑝 → (𝑛 ∈ ℙ ↔ 𝑝 ∈ ℙ))
38 id 22 . . . . . . . . 9 (𝑛 = 𝑝𝑛 = 𝑝)
39 oveq1 6531 . . . . . . . . 9 (𝑛 = 𝑝 → (𝑛 pCnt 𝐴) = (𝑝 pCnt 𝐴))
4038, 39oveq12d 6542 . . . . . . . 8 (𝑛 = 𝑝 → (𝑛↑(𝑛 pCnt 𝐴)) = (𝑝↑(𝑝 pCnt 𝐴)))
4137, 40ifbieq1d 4055 . . . . . . 7 (𝑛 = 𝑝 → if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1) = if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1))
4241fveq2d 6089 . . . . . 6 (𝑛 = 𝑝 → (log‘if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)) = (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)))
43 eqid 2606 . . . . . 6 (𝑛 ∈ ℕ ↦ (log‘if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1))) = (𝑛 ∈ ℕ ↦ (log‘if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)))
44 fvex 6095 . . . . . 6 (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)) ∈ V
4542, 43, 44fvmpt 6173 . . . . 5 (𝑝 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (log‘if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)))‘𝑝) = (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)))
4636, 45syl 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)))
47 elnnuz 11553 . . . . 5 (𝐴 ∈ ℕ ↔ 𝐴 ∈ (ℤ‘1))
4847biimpi 204 . . . 4 (𝐴 ∈ ℕ → 𝐴 ∈ (ℤ‘1))
4936adantr 479 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℕ)
50 simpr 475 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
51 simpll 785 . . . . . . . . . 10 (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℕ)
5250, 51pccld 15336 . . . . . . . . 9 (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℕ0)
5349, 52nnexpcld 12844 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt 𝐴)) ∈ ℕ)
54 1nn 10875 . . . . . . . . 9 1 ∈ ℕ
5554a1i 11 . . . . . . . 8 (((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) ∧ ¬ 𝑝 ∈ ℙ) → 1 ∈ ℕ)
5653, 55ifclda 4066 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1) ∈ ℕ)
5756nnrpd 11699 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1) ∈ ℝ+)
5857relogcld 24087 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)) ∈ ℝ)
5958recnd 9921 . . . 4 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → (log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)) ∈ ℂ)
6046, 48, 59fsumser 14251 . . 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))
61 rpmulcl 11684 . . . . 5 ((𝑝 ∈ ℝ+𝑚 ∈ ℝ+) → (𝑝 · 𝑚) ∈ ℝ+)
6261adantl 480 . . . 4 ((𝐴 ∈ ℕ ∧ (𝑝 ∈ ℝ+𝑚 ∈ ℝ+)) → (𝑝 · 𝑚) ∈ ℝ+)
63 eqid 2606 . . . . . . 7 (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1))
64 ovex 6552 . . . . . . . 8 (𝑝↑(𝑝 pCnt 𝐴)) ∈ V
65 1ex 9888 . . . . . . . 8 1 ∈ V
6664, 65ifex 4102 . . . . . . 7 if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1) ∈ V
6741, 63, 66fvmpt 6173 . . . . . 6 (𝑝 ∈ ℕ → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1))‘𝑝) = if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1))
6836, 67syl 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))
6968, 57eqeltrd 2684 . . . 4 ((𝐴 ∈ ℕ ∧ 𝑝 ∈ (1...𝐴)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1))‘𝑝) ∈ ℝ+)
70 relogmul 24056 . . . . 5 ((𝑝 ∈ ℝ+𝑚 ∈ ℝ+) → (log‘(𝑝 · 𝑚)) = ((log‘𝑝) + (log‘𝑚)))
7170adantl 480 . . . 4 ((𝐴 ∈ ℕ ∧ (𝑝 ∈ ℝ+𝑚 ∈ ℝ+)) → (log‘(𝑝 · 𝑚)) = ((log‘𝑝) + (log‘𝑚)))
7268fveq2d 6089 . . . . 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)))
7372, 46eqtr4d 2643 . . . 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))
7462, 69, 48, 71, 73seqhomo 12662 . . 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))
7563pcprod 15380 . . . 4 (𝐴 ∈ ℕ → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)))‘𝐴) = 𝐴)
7675fveq2d 6089 . . 3 (𝐴 ∈ ℕ → (log‘(seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt 𝐴)), 1)))‘𝐴)) = (log‘𝐴))
7760, 74, 763eqtr2d 2646 . 2 (𝐴 ∈ ℕ → Σ𝑝 ∈ (1...𝐴)(log‘if(𝑝 ∈ ℙ, (𝑝↑(𝑝 pCnt 𝐴)), 1)) = (log‘𝐴))
7810, 35, 773eqtr3a 2664 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  wo 381  wa 382   = wceq 1474  wcel 1976  wral 2892  cin 3535  wss 3536  ifcif 4032  cmpt 4634  cfv 5787  (class class class)co 6524  Fincfn 7815  cc 9787  0cc0 9789  1c1 9790   + caddc 9792   · cmul 9794  cn 10864  cz 11207  cuz 11516  +crp 11661  ...cfz 12149  seqcseq 12615  cexp 12674  Σcsu 14207  cprime 15166   pCnt cpc 15322  logclog 24019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-inf2 8395  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866  ax-pre-sup 9867  ax-addf 9868  ax-mulf 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-iin 4449  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-se 4985  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-isom 5796  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-of 6769  df-om 6932  df-1st 7033  df-2nd 7034  df-supp 7157  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-2o 7422  df-oadd 7425  df-er 7603  df-map 7720  df-pm 7721  df-ixp 7769  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-fsupp 8133  df-fi 8174  df-sup 8205  df-inf 8206  df-oi 8272  df-card 8622  df-cda 8847  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-div 10531  df-nn 10865  df-2 10923  df-3 10924  df-4 10925  df-5 10926  df-6 10927  df-7 10928  df-8 10929  df-9 10930  df-n0 11137  df-z 11208  df-dec 11323  df-uz 11517  df-q 11618  df-rp 11662  df-xneg 11775  df-xadd 11776  df-xmul 11777  df-ioo 12003  df-ioc 12004  df-ico 12005  df-icc 12006  df-fz 12150  df-fzo 12287  df-fl 12407  df-mod 12483  df-seq 12616  df-exp 12675  df-fac 12875  df-bc 12904  df-hash 12932  df-shft 13598  df-cj 13630  df-re 13631  df-im 13632  df-sqrt 13766  df-abs 13767  df-limsup 13993  df-clim 14010  df-rlim 14011  df-sum 14208  df-ef 14580  df-sin 14582  df-cos 14583  df-pi 14585  df-dvds 14765  df-gcd 14998  df-prm 15167  df-pc 15323  df-struct 15640  df-ndx 15641  df-slot 15642  df-base 15643  df-sets 15644  df-ress 15645  df-plusg 15724  df-mulr 15725  df-starv 15726  df-sca 15727  df-vsca 15728  df-ip 15729  df-tset 15730  df-ple 15731  df-ds 15734  df-unif 15735  df-hom 15736  df-cco 15737  df-rest 15849  df-topn 15850  df-0g 15868  df-gsum 15869  df-topgen 15870  df-pt 15871  df-prds 15874  df-xrs 15928  df-qtop 15933  df-imas 15934  df-xps 15936  df-mre 16012  df-mrc 16013  df-acs 16015  df-mgm 17008  df-sgrp 17050  df-mnd 17061  df-submnd 17102  df-mulg 17307  df-cntz 17516  df-cmn 17961  df-psmet 19502  df-xmet 19503  df-met 19504  df-bl 19505  df-mopn 19506  df-fbas 19507  df-fg 19508  df-cnfld 19511  df-top 20460  df-bases 20461  df-topon 20462  df-topsp 20463  df-cld 20572  df-ntr 20573  df-cls 20574  df-nei 20651  df-lp 20689  df-perf 20690  df-cn 20780  df-cnp 20781  df-haus 20868  df-tx 21114  df-hmeo 21307  df-fil 21399  df-fm 21491  df-flim 21492  df-flf 21493  df-xms 21873  df-ms 21874  df-tms 21875  df-cncf 22417  df-limc 23350  df-dv 23351  df-log 24021
This theorem is referenced by:  vmasum  24655  chebbnd1lem1  24872
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