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Theorem selberg 27501
Description: Selberg's symmetry formula. The statement has many forms, and this one is equivalent to the statement that Σ𝑛 ≀ π‘₯, Ξ›(𝑛)log𝑛 + Ξ£π‘š Β· 𝑛 ≀ π‘₯, Ξ›(π‘š)Ξ›(𝑛) = 2π‘₯logπ‘₯ + 𝑂(π‘₯). Equation 10.4.10 of [Shapiro], p. 419. (Contributed by Mario Carneiro, 23-May-2016.)
Assertion
Ref Expression
selberg (π‘₯ ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘›) Β· ((logβ€˜π‘›) + (Οˆβ€˜(π‘₯ / 𝑛)))) / π‘₯) βˆ’ (2 Β· (logβ€˜π‘₯)))) ∈ 𝑂(1)
Distinct variable group:   π‘₯,𝑛

Proof of Theorem selberg
Dummy variables 𝑑 π‘š 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6902 . . . . . . . . . . . . 13 (𝑛 = 𝑑 β†’ (Ξ›β€˜π‘›) = (Ξ›β€˜π‘‘))
2 oveq2 7434 . . . . . . . . . . . . . 14 (𝑛 = 𝑑 β†’ (π‘₯ / 𝑛) = (π‘₯ / 𝑑))
32fveq2d 6906 . . . . . . . . . . . . 13 (𝑛 = 𝑑 β†’ (Οˆβ€˜(π‘₯ / 𝑛)) = (Οˆβ€˜(π‘₯ / 𝑑)))
41, 3oveq12d 7444 . . . . . . . . . . . 12 (𝑛 = 𝑑 β†’ ((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) = ((Ξ›β€˜π‘‘) Β· (Οˆβ€˜(π‘₯ / 𝑑))))
54cbvsumv 15682 . . . . . . . . . . 11 Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) = Σ𝑑 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘‘) Β· (Οˆβ€˜(π‘₯ / 𝑑)))
6 fzfid 13978 . . . . . . . . . . . . . 14 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (1...(βŒŠβ€˜(π‘₯ / 𝑑))) ∈ Fin)
7 elfznn 13570 . . . . . . . . . . . . . . . . 17 (𝑑 ∈ (1...(βŒŠβ€˜π‘₯)) β†’ 𝑑 ∈ β„•)
87adantl 480 . . . . . . . . . . . . . . . 16 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑑 ∈ β„•)
9 vmacl 27070 . . . . . . . . . . . . . . . 16 (𝑑 ∈ β„• β†’ (Ξ›β€˜π‘‘) ∈ ℝ)
108, 9syl 17 . . . . . . . . . . . . . . 15 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Ξ›β€˜π‘‘) ∈ ℝ)
1110recnd 11280 . . . . . . . . . . . . . 14 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Ξ›β€˜π‘‘) ∈ β„‚)
12 elfznn 13570 . . . . . . . . . . . . . . . . 17 (π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑))) β†’ π‘š ∈ β„•)
1312adantl 480 . . . . . . . . . . . . . . . 16 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ π‘š ∈ β„•)
14 vmacl 27070 . . . . . . . . . . . . . . . 16 (π‘š ∈ β„• β†’ (Ξ›β€˜π‘š) ∈ ℝ)
1513, 14syl 17 . . . . . . . . . . . . . . 15 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ (Ξ›β€˜π‘š) ∈ ℝ)
1615recnd 11280 . . . . . . . . . . . . . 14 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ (Ξ›β€˜π‘š) ∈ β„‚)
176, 11, 16fsummulc2 15770 . . . . . . . . . . . . 13 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((Ξ›β€˜π‘‘) Β· Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))(Ξ›β€˜π‘š)) = Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))((Ξ›β€˜π‘‘) Β· (Ξ›β€˜π‘š)))
187nnrpd 13054 . . . . . . . . . . . . . . . . 17 (𝑑 ∈ (1...(βŒŠβ€˜π‘₯)) β†’ 𝑑 ∈ ℝ+)
19 rpdivcl 13039 . . . . . . . . . . . . . . . . 17 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) β†’ (π‘₯ / 𝑑) ∈ ℝ+)
2018, 19sylan2 591 . . . . . . . . . . . . . . . 16 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘₯ / 𝑑) ∈ ℝ+)
2120rpred 13056 . . . . . . . . . . . . . . 15 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘₯ / 𝑑) ∈ ℝ)
22 chpval 27074 . . . . . . . . . . . . . . 15 ((π‘₯ / 𝑑) ∈ ℝ β†’ (Οˆβ€˜(π‘₯ / 𝑑)) = Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))(Ξ›β€˜π‘š))
2321, 22syl 17 . . . . . . . . . . . . . 14 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Οˆβ€˜(π‘₯ / 𝑑)) = Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))(Ξ›β€˜π‘š))
2423oveq2d 7442 . . . . . . . . . . . . 13 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((Ξ›β€˜π‘‘) Β· (Οˆβ€˜(π‘₯ / 𝑑))) = ((Ξ›β€˜π‘‘) Β· Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))(Ξ›β€˜π‘š)))
2513nncnd 12266 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ π‘š ∈ β„‚)
267ad2antlr 725 . . . . . . . . . . . . . . . . . 18 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ 𝑑 ∈ β„•)
2726nncnd 12266 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ 𝑑 ∈ β„‚)
2826nnne0d 12300 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ 𝑑 β‰  0)
2925, 27, 28divcan3d 12033 . . . . . . . . . . . . . . . 16 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ ((𝑑 Β· π‘š) / 𝑑) = π‘š)
3029fveq2d 6906 . . . . . . . . . . . . . . 15 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ (Ξ›β€˜((𝑑 Β· π‘š) / 𝑑)) = (Ξ›β€˜π‘š))
3130oveq2d 7442 . . . . . . . . . . . . . 14 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜((𝑑 Β· π‘š) / 𝑑))) = ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜π‘š)))
3231sumeq2dv 15689 . . . . . . . . . . . . 13 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))((Ξ›β€˜π‘‘) Β· (Ξ›β€˜((𝑑 Β· π‘š) / 𝑑))) = Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))((Ξ›β€˜π‘‘) Β· (Ξ›β€˜π‘š)))
3317, 24, 323eqtr4d 2778 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((Ξ›β€˜π‘‘) Β· (Οˆβ€˜(π‘₯ / 𝑑))) = Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))((Ξ›β€˜π‘‘) Β· (Ξ›β€˜((𝑑 Β· π‘š) / 𝑑))))
3433sumeq2dv 15689 . . . . . . . . . . 11 (π‘₯ ∈ ℝ+ β†’ Σ𝑑 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘‘) Β· (Οˆβ€˜(π‘₯ / 𝑑))) = Σ𝑑 ∈ (1...(βŒŠβ€˜π‘₯))Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))((Ξ›β€˜π‘‘) Β· (Ξ›β€˜((𝑑 Β· π‘š) / 𝑑))))
355, 34eqtrid 2780 . . . . . . . . . 10 (π‘₯ ∈ ℝ+ β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) = Σ𝑑 ∈ (1...(βŒŠβ€˜π‘₯))Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))((Ξ›β€˜π‘‘) Β· (Ξ›β€˜((𝑑 Β· π‘š) / 𝑑))))
36 fvoveq1 7449 . . . . . . . . . . . 12 (𝑛 = (𝑑 Β· π‘š) β†’ (Ξ›β€˜(𝑛 / 𝑑)) = (Ξ›β€˜((𝑑 Β· π‘š) / 𝑑)))
3736oveq2d 7442 . . . . . . . . . . 11 (𝑛 = (𝑑 Β· π‘š) β†’ ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))) = ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜((𝑑 Β· π‘š) / 𝑑))))
38 rpre 13022 . . . . . . . . . . 11 (π‘₯ ∈ ℝ+ β†’ π‘₯ ∈ ℝ)
39 ssrab2 4077 . . . . . . . . . . . . . . . . 17 {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} βŠ† β„•
40 simprr 771 . . . . . . . . . . . . . . . . 17 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})
4139, 40sselid 3980 . . . . . . . . . . . . . . . 16 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ 𝑑 ∈ β„•)
4241anassrs 466 . . . . . . . . . . . . . . 15 (((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ 𝑑 ∈ β„•)
4342, 9syl 17 . . . . . . . . . . . . . 14 (((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ (Ξ›β€˜π‘‘) ∈ ℝ)
44 elfznn 13570 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) β†’ 𝑛 ∈ β„•)
4544adantl 480 . . . . . . . . . . . . . . . . 17 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑛 ∈ β„•)
46 dvdsdivcl 16300 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ β„• ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ (𝑛 / 𝑑) ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})
4745, 46sylan 578 . . . . . . . . . . . . . . . 16 (((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ (𝑛 / 𝑑) ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})
4839, 47sselid 3980 . . . . . . . . . . . . . . 15 (((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ (𝑛 / 𝑑) ∈ β„•)
49 vmacl 27070 . . . . . . . . . . . . . . 15 ((𝑛 / 𝑑) ∈ β„• β†’ (Ξ›β€˜(𝑛 / 𝑑)) ∈ ℝ)
5048, 49syl 17 . . . . . . . . . . . . . 14 (((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ (Ξ›β€˜(𝑛 / 𝑑)) ∈ ℝ)
5143, 50remulcld 11282 . . . . . . . . . . . . 13 (((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))) ∈ ℝ)
5251recnd 11280 . . . . . . . . . . . 12 (((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))) ∈ β„‚)
5352anasss 465 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))) ∈ β„‚)
5437, 38, 53dvdsflsumcom 27140 . . . . . . . . . 10 (π‘₯ ∈ ℝ+ β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))Σ𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))) = Σ𝑑 ∈ (1...(βŒŠβ€˜π‘₯))Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))((Ξ›β€˜π‘‘) Β· (Ξ›β€˜((𝑑 Β· π‘š) / 𝑑))))
5535, 54eqtr4d 2771 . . . . . . . . 9 (π‘₯ ∈ ℝ+ β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) = Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))Σ𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))))
5655oveq1d 7441 . . . . . . . 8 (π‘₯ ∈ ℝ+ β†’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) + Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))) = (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))Σ𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))) + Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))
57 fzfid 13978 . . . . . . . . 9 (π‘₯ ∈ ℝ+ β†’ (1...(βŒŠβ€˜π‘₯)) ∈ Fin)
58 vmacl 27070 . . . . . . . . . . . 12 (𝑛 ∈ β„• β†’ (Ξ›β€˜π‘›) ∈ ℝ)
5945, 58syl 17 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Ξ›β€˜π‘›) ∈ ℝ)
6059recnd 11280 . . . . . . . . . 10 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Ξ›β€˜π‘›) ∈ β„‚)
6144nnrpd 13054 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) β†’ 𝑛 ∈ ℝ+)
62 rpdivcl 13039 . . . . . . . . . . . . . 14 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) β†’ (π‘₯ / 𝑛) ∈ ℝ+)
6361, 62sylan2 591 . . . . . . . . . . . . 13 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘₯ / 𝑛) ∈ ℝ+)
6463rpred 13056 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘₯ / 𝑛) ∈ ℝ)
65 chpcl 27076 . . . . . . . . . . . 12 ((π‘₯ / 𝑛) ∈ ℝ β†’ (Οˆβ€˜(π‘₯ / 𝑛)) ∈ ℝ)
6664, 65syl 17 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Οˆβ€˜(π‘₯ / 𝑛)) ∈ ℝ)
6766recnd 11280 . . . . . . . . . 10 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Οˆβ€˜(π‘₯ / 𝑛)) ∈ β„‚)
6860, 67mulcld 11272 . . . . . . . . 9 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) ∈ β„‚)
6945nnrpd 13054 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑛 ∈ ℝ+)
70 relogcl 26529 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ β†’ (logβ€˜π‘›) ∈ ℝ)
7169, 70syl 17 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (logβ€˜π‘›) ∈ ℝ)
7271recnd 11280 . . . . . . . . . 10 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (logβ€˜π‘›) ∈ β„‚)
7360, 72mulcld 11272 . . . . . . . . 9 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) ∈ β„‚)
7457, 68, 73fsumadd 15726 . . . . . . . 8 (π‘₯ ∈ ℝ+ β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))) = (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) + Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))
75 fzfid 13978 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (1...𝑛) ∈ Fin)
76 dvdsssfz1 16302 . . . . . . . . . . . . 13 (𝑛 ∈ β„• β†’ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} βŠ† (1...𝑛))
7745, 76syl 17 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} βŠ† (1...𝑛))
7875, 77ssfid 9298 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ∈ Fin)
7978, 51fsumrecl 15720 . . . . . . . . . 10 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ Σ𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))) ∈ ℝ)
8079recnd 11280 . . . . . . . . 9 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ Σ𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))) ∈ β„‚)
8157, 80, 73fsumadd 15726 . . . . . . . 8 (π‘₯ ∈ ℝ+ β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(Σ𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))) = (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))Σ𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))) + Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))
8256, 74, 813eqtr4d 2778 . . . . . . 7 (π‘₯ ∈ ℝ+ β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))) = Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(Σ𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))
8372, 67addcomd 11454 . . . . . . . . . 10 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((logβ€˜π‘›) + (Οˆβ€˜(π‘₯ / 𝑛))) = ((Οˆβ€˜(π‘₯ / 𝑛)) + (logβ€˜π‘›)))
8483oveq2d 7442 . . . . . . . . 9 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((Ξ›β€˜π‘›) Β· ((logβ€˜π‘›) + (Οˆβ€˜(π‘₯ / 𝑛)))) = ((Ξ›β€˜π‘›) Β· ((Οˆβ€˜(π‘₯ / 𝑛)) + (logβ€˜π‘›))))
8560, 67, 72adddid 11276 . . . . . . . . 9 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((Ξ›β€˜π‘›) Β· ((Οˆβ€˜(π‘₯ / 𝑛)) + (logβ€˜π‘›))) = (((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))
8684, 85eqtrd 2768 . . . . . . . 8 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((Ξ›β€˜π‘›) Β· ((logβ€˜π‘›) + (Οˆβ€˜(π‘₯ / 𝑛)))) = (((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))
8786sumeq2dv 15689 . . . . . . 7 (π‘₯ ∈ ℝ+ β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘›) Β· ((logβ€˜π‘›) + (Οˆβ€˜(π‘₯ / 𝑛)))) = Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))
88 logsqvma2 27496 . . . . . . . . 9 (𝑛 ∈ β„• β†’ Σ𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((ΞΌβ€˜π‘‘) Β· ((logβ€˜(𝑛 / 𝑑))↑2)) = (Σ𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))
8945, 88syl 17 . . . . . . . 8 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ Σ𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((ΞΌβ€˜π‘‘) Β· ((logβ€˜(𝑛 / 𝑑))↑2)) = (Σ𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))
9089sumeq2dv 15689 . . . . . . 7 (π‘₯ ∈ ℝ+ β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))Σ𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((ΞΌβ€˜π‘‘) Β· ((logβ€˜(𝑛 / 𝑑))↑2)) = Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))(Σ𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))) + ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›))))
9182, 87, 903eqtr4d 2778 . . . . . 6 (π‘₯ ∈ ℝ+ β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘›) Β· ((logβ€˜π‘›) + (Οˆβ€˜(π‘₯ / 𝑛)))) = Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))Σ𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((ΞΌβ€˜π‘‘) Β· ((logβ€˜(𝑛 / 𝑑))↑2)))
92 fvoveq1 7449 . . . . . . . . 9 (𝑛 = (𝑑 Β· π‘š) β†’ (logβ€˜(𝑛 / 𝑑)) = (logβ€˜((𝑑 Β· π‘š) / 𝑑)))
9392oveq1d 7441 . . . . . . . 8 (𝑛 = (𝑑 Β· π‘š) β†’ ((logβ€˜(𝑛 / 𝑑))↑2) = ((logβ€˜((𝑑 Β· π‘š) / 𝑑))↑2))
9493oveq2d 7442 . . . . . . 7 (𝑛 = (𝑑 Β· π‘š) β†’ ((ΞΌβ€˜π‘‘) Β· ((logβ€˜(𝑛 / 𝑑))↑2)) = ((ΞΌβ€˜π‘‘) Β· ((logβ€˜((𝑑 Β· π‘š) / 𝑑))↑2)))
95 mucl 27093 . . . . . . . . . 10 (𝑑 ∈ β„• β†’ (ΞΌβ€˜π‘‘) ∈ β„€)
9641, 95syl 17 . . . . . . . . 9 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ (ΞΌβ€˜π‘‘) ∈ β„€)
9796zcnd 12705 . . . . . . . 8 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ (ΞΌβ€˜π‘‘) ∈ β„‚)
9861ad2antrl 726 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ 𝑛 ∈ ℝ+)
9941nnrpd 13054 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ 𝑑 ∈ ℝ+)
10098, 99rpdivcld 13073 . . . . . . . . . 10 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ (𝑛 / 𝑑) ∈ ℝ+)
101 relogcl 26529 . . . . . . . . . . 11 ((𝑛 / 𝑑) ∈ ℝ+ β†’ (logβ€˜(𝑛 / 𝑑)) ∈ ℝ)
102101recnd 11280 . . . . . . . . . 10 ((𝑛 / 𝑑) ∈ ℝ+ β†’ (logβ€˜(𝑛 / 𝑑)) ∈ β„‚)
103100, 102syl 17 . . . . . . . . 9 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ (logβ€˜(𝑛 / 𝑑)) ∈ β„‚)
104103sqcld 14148 . . . . . . . 8 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ ((logβ€˜(𝑛 / 𝑑))↑2) ∈ β„‚)
10597, 104mulcld 11272 . . . . . . 7 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ ((ΞΌβ€˜π‘‘) Β· ((logβ€˜(𝑛 / 𝑑))↑2)) ∈ β„‚)
10694, 38, 105dvdsflsumcom 27140 . . . . . 6 (π‘₯ ∈ ℝ+ β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))Σ𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((ΞΌβ€˜π‘‘) Β· ((logβ€˜(𝑛 / 𝑑))↑2)) = Σ𝑑 ∈ (1...(βŒŠβ€˜π‘₯))Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))((ΞΌβ€˜π‘‘) Β· ((logβ€˜((𝑑 Β· π‘š) / 𝑑))↑2)))
10729fveq2d 6906 . . . . . . . . . 10 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ (logβ€˜((𝑑 Β· π‘š) / 𝑑)) = (logβ€˜π‘š))
108107oveq1d 7441 . . . . . . . . 9 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ ((logβ€˜((𝑑 Β· π‘š) / 𝑑))↑2) = ((logβ€˜π‘š)↑2))
109108oveq2d 7442 . . . . . . . 8 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ ((ΞΌβ€˜π‘‘) Β· ((logβ€˜((𝑑 Β· π‘š) / 𝑑))↑2)) = ((ΞΌβ€˜π‘‘) Β· ((logβ€˜π‘š)↑2)))
110109sumeq2dv 15689 . . . . . . 7 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))((ΞΌβ€˜π‘‘) Β· ((logβ€˜((𝑑 Β· π‘š) / 𝑑))↑2)) = Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))((ΞΌβ€˜π‘‘) Β· ((logβ€˜π‘š)↑2)))
111110sumeq2dv 15689 . . . . . 6 (π‘₯ ∈ ℝ+ β†’ Σ𝑑 ∈ (1...(βŒŠβ€˜π‘₯))Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))((ΞΌβ€˜π‘‘) Β· ((logβ€˜((𝑑 Β· π‘š) / 𝑑))↑2)) = Σ𝑑 ∈ (1...(βŒŠβ€˜π‘₯))Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))((ΞΌβ€˜π‘‘) Β· ((logβ€˜π‘š)↑2)))
11291, 106, 1113eqtrd 2772 . . . . 5 (π‘₯ ∈ ℝ+ β†’ Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘›) Β· ((logβ€˜π‘›) + (Οˆβ€˜(π‘₯ / 𝑛)))) = Σ𝑑 ∈ (1...(βŒŠβ€˜π‘₯))Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))((ΞΌβ€˜π‘‘) Β· ((logβ€˜π‘š)↑2)))
113112oveq1d 7441 . . . 4 (π‘₯ ∈ ℝ+ β†’ (Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘›) Β· ((logβ€˜π‘›) + (Οˆβ€˜(π‘₯ / 𝑛)))) / π‘₯) = (Σ𝑑 ∈ (1...(βŒŠβ€˜π‘₯))Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))((ΞΌβ€˜π‘‘) Β· ((logβ€˜π‘š)↑2)) / π‘₯))
114113oveq1d 7441 . . 3 (π‘₯ ∈ ℝ+ β†’ ((Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘›) Β· ((logβ€˜π‘›) + (Οˆβ€˜(π‘₯ / 𝑛)))) / π‘₯) βˆ’ (2 Β· (logβ€˜π‘₯))) = ((Σ𝑑 ∈ (1...(βŒŠβ€˜π‘₯))Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))((ΞΌβ€˜π‘‘) Β· ((logβ€˜π‘š)↑2)) / π‘₯) βˆ’ (2 Β· (logβ€˜π‘₯))))
115114mpteq2ia 5255 . 2 (π‘₯ ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘›) Β· ((logβ€˜π‘›) + (Οˆβ€˜(π‘₯ / 𝑛)))) / π‘₯) βˆ’ (2 Β· (logβ€˜π‘₯)))) = (π‘₯ ∈ ℝ+ ↦ ((Σ𝑑 ∈ (1...(βŒŠβ€˜π‘₯))Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))((ΞΌβ€˜π‘‘) Β· ((logβ€˜π‘š)↑2)) / π‘₯) βˆ’ (2 Β· (logβ€˜π‘₯))))
116 eqid 2728 . . 3 ((((logβ€˜(π‘₯ / 𝑑))↑2) + (2 βˆ’ (2 Β· (logβ€˜(π‘₯ / 𝑑))))) / 𝑑) = ((((logβ€˜(π‘₯ / 𝑑))↑2) + (2 βˆ’ (2 Β· (logβ€˜(π‘₯ / 𝑑))))) / 𝑑)
117116selberglem2 27499 . 2 (π‘₯ ∈ ℝ+ ↦ ((Σ𝑑 ∈ (1...(βŒŠβ€˜π‘₯))Ξ£π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))((ΞΌβ€˜π‘‘) Β· ((logβ€˜π‘š)↑2)) / π‘₯) βˆ’ (2 Β· (logβ€˜π‘₯)))) ∈ 𝑂(1)
118115, 117eqeltri 2825 1 (π‘₯ ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(βŒŠβ€˜π‘₯))((Ξ›β€˜π‘›) Β· ((logβ€˜π‘›) + (Οˆβ€˜(π‘₯ / 𝑛)))) / π‘₯) βˆ’ (2 Β· (logβ€˜π‘₯)))) ∈ 𝑂(1)
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {crab 3430   βŠ† wss 3949   class class class wbr 5152   ↦ cmpt 5235  β€˜cfv 6553  (class class class)co 7426  β„‚cc 11144  β„cr 11145  1c1 11147   + caddc 11149   Β· cmul 11151   βˆ’ cmin 11482   / cdiv 11909  β„•cn 12250  2c2 12305  β„€cz 12596  β„+crp 13014  ...cfz 13524  βŒŠcfl 13795  β†‘cexp 14066  π‘‚(1)co1 15470  Ξ£csu 15672   βˆ₯ cdvds 16238  logclog 26508  Ξ›cvma 27044  Οˆcchp 27045  ΞΌcmu 27047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224  ax-addf 11225
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-disj 5118  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-of 7691  df-om 7877  df-1st 7999  df-2nd 8000  df-supp 8172  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-2o 8494  df-oadd 8497  df-er 8731  df-map 8853  df-pm 8854  df-ixp 8923  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-fsupp 9394  df-fi 9442  df-sup 9473  df-inf 9474  df-oi 9541  df-dju 9932  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-xnn0 12583  df-z 12597  df-dec 12716  df-uz 12861  df-q 12971  df-rp 13015  df-xneg 13132  df-xadd 13133  df-xmul 13134  df-ioo 13368  df-ioc 13369  df-ico 13370  df-icc 13371  df-fz 13525  df-fzo 13668  df-fl 13797  df-mod 13875  df-seq 14007  df-exp 14067  df-fac 14273  df-bc 14302  df-hash 14330  df-shft 15054  df-cj 15086  df-re 15087  df-im 15088  df-sqrt 15222  df-abs 15223  df-limsup 15455  df-clim 15472  df-rlim 15473  df-o1 15474  df-lo1 15475  df-sum 15673  df-ef 16051  df-e 16052  df-sin 16053  df-cos 16054  df-tan 16055  df-pi 16056  df-dvds 16239  df-gcd 16477  df-prm 16650  df-pc 16813  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-mulr 17254  df-starv 17255  df-sca 17256  df-vsca 17257  df-ip 17258  df-tset 17259  df-ple 17260  df-ds 17262  df-unif 17263  df-hom 17264  df-cco 17265  df-rest 17411  df-topn 17412  df-0g 17430  df-gsum 17431  df-topgen 17432  df-pt 17433  df-prds 17436  df-xrs 17491  df-qtop 17496  df-imas 17497  df-xps 17499  df-mre 17573  df-mrc 17574  df-acs 17576  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-submnd 18748  df-mulg 19031  df-cntz 19275  df-cmn 19744  df-psmet 21278  df-xmet 21279  df-met 21280  df-bl 21281  df-mopn 21282  df-fbas 21283  df-fg 21284  df-cnfld 21287  df-top 22816  df-topon 22833  df-topsp 22855  df-bases 22869  df-cld 22943  df-ntr 22944  df-cls 22945  df-nei 23022  df-lp 23060  df-perf 23061  df-cn 23151  df-cnp 23152  df-haus 23239  df-cmp 23311  df-tx 23486  df-hmeo 23679  df-fil 23770  df-fm 23862  df-flim 23863  df-flf 23864  df-xms 24246  df-ms 24247  df-tms 24248  df-cncf 24818  df-limc 25815  df-dv 25816  df-ulm 26333  df-log 26510  df-cxp 26511  df-atan 26819  df-em 26945  df-vma 27050  df-chp 27051  df-mu 27053
This theorem is referenced by:  selbergb  27502  selberg2  27504  selbergs  27527
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