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Theorem selberg 26912
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 6847 . . . . . . . . . . . . 13 (𝑛 = 𝑑 β†’ (Ξ›β€˜π‘›) = (Ξ›β€˜π‘‘))
2 oveq2 7370 . . . . . . . . . . . . . 14 (𝑛 = 𝑑 β†’ (π‘₯ / 𝑛) = (π‘₯ / 𝑑))
32fveq2d 6851 . . . . . . . . . . . . 13 (𝑛 = 𝑑 β†’ (Οˆβ€˜(π‘₯ / 𝑛)) = (Οˆβ€˜(π‘₯ / 𝑑)))
41, 3oveq12d 7380 . . . . . . . . . . . 12 (𝑛 = 𝑑 β†’ ((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) = ((Ξ›β€˜π‘‘) Β· (Οˆβ€˜(π‘₯ / 𝑑))))
54cbvsumv 15588 . . . . . . . . . . 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 13885 . . . . . . . . . . . . . 14 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (1...(βŒŠβ€˜(π‘₯ / 𝑑))) ∈ Fin)
7 elfznn 13477 . . . . . . . . . . . . . . . . 17 (𝑑 ∈ (1...(βŒŠβ€˜π‘₯)) β†’ 𝑑 ∈ β„•)
87adantl 483 . . . . . . . . . . . . . . . 16 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑑 ∈ β„•)
9 vmacl 26483 . . . . . . . . . . . . . . . 16 (𝑑 ∈ β„• β†’ (Ξ›β€˜π‘‘) ∈ ℝ)
108, 9syl 17 . . . . . . . . . . . . . . 15 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Ξ›β€˜π‘‘) ∈ ℝ)
1110recnd 11190 . . . . . . . . . . . . . 14 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Ξ›β€˜π‘‘) ∈ β„‚)
12 elfznn 13477 . . . . . . . . . . . . . . . . 17 (π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑))) β†’ π‘š ∈ β„•)
1312adantl 483 . . . . . . . . . . . . . . . 16 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ π‘š ∈ β„•)
14 vmacl 26483 . . . . . . . . . . . . . . . 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 11190 . . . . . . . . . . . . . 14 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ (Ξ›β€˜π‘š) ∈ β„‚)
176, 11, 16fsummulc2 15676 . . . . . . . . . . . . 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 12962 . . . . . . . . . . . . . . . . 17 (𝑑 ∈ (1...(βŒŠβ€˜π‘₯)) β†’ 𝑑 ∈ ℝ+)
19 rpdivcl 12947 . . . . . . . . . . . . . . . . 17 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ ℝ+) β†’ (π‘₯ / 𝑑) ∈ ℝ+)
2018, 19sylan2 594 . . . . . . . . . . . . . . . 16 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘₯ / 𝑑) ∈ ℝ+)
2120rpred 12964 . . . . . . . . . . . . . . 15 ((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘₯ / 𝑑) ∈ ℝ)
22 chpval 26487 . . . . . . . . . . . . . . 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 7378 . . . . . . . . . . . . 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 12176 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ π‘š ∈ β„‚)
267ad2antlr 726 . . . . . . . . . . . . . . . . . 18 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ 𝑑 ∈ β„•)
2726nncnd 12176 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ 𝑑 ∈ β„‚)
2826nnne0d 12210 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ 𝑑 β‰  0)
2925, 27, 28divcan3d 11943 . . . . . . . . . . . . . . . 16 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ ((𝑑 Β· π‘š) / 𝑑) = π‘š)
3029fveq2d 6851 . . . . . . . . . . . . . . 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 7378 . . . . . . . . . . . . . 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 15595 . . . . . . . . . . . . 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 2787 . . . . . . . . . . . 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 15595 . . . . . . . . . . 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 2789 . . . . . . . . . 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 7385 . . . . . . . . . . . 12 (𝑛 = (𝑑 Β· π‘š) β†’ (Ξ›β€˜(𝑛 / 𝑑)) = (Ξ›β€˜((𝑑 Β· π‘š) / 𝑑)))
3736oveq2d 7378 . . . . . . . . . . 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 12930 . . . . . . . . . . 11 (π‘₯ ∈ ℝ+ β†’ π‘₯ ∈ ℝ)
39 ssrab2 4042 . . . . . . . . . . . . . . . . 17 {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} βŠ† β„•
40 simprr 772 . . . . . . . . . . . . . . . . 17 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})
4139, 40sselid 3947 . . . . . . . . . . . . . . . 16 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ 𝑑 ∈ β„•)
4241anassrs 469 . . . . . . . . . . . . . . 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 13477 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) β†’ 𝑛 ∈ β„•)
4544adantl 483 . . . . . . . . . . . . . . . . 17 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑛 ∈ β„•)
46 dvdsdivcl 16205 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ β„• ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ (𝑛 / 𝑑) ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})
4745, 46sylan 581 . . . . . . . . . . . . . . . 16 (((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ (𝑛 / 𝑑) ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})
4839, 47sselid 3947 . . . . . . . . . . . . . . 15 (((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ (𝑛 / 𝑑) ∈ β„•)
49 vmacl 26483 . . . . . . . . . . . . . . 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 11192 . . . . . . . . . . . . 13 (((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))) ∈ ℝ)
5251recnd 11190 . . . . . . . . . . . 12 (((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛}) β†’ ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))) ∈ β„‚)
5352anasss 468 . . . . . . . . . . 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 26553 . . . . . . . . . 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 2780 . . . . . . . . 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 7377 . . . . . . . 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 13885 . . . . . . . . 9 (π‘₯ ∈ ℝ+ β†’ (1...(βŒŠβ€˜π‘₯)) ∈ Fin)
58 vmacl 26483 . . . . . . . . . . . 12 (𝑛 ∈ β„• β†’ (Ξ›β€˜π‘›) ∈ ℝ)
5945, 58syl 17 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Ξ›β€˜π‘›) ∈ ℝ)
6059recnd 11190 . . . . . . . . . 10 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Ξ›β€˜π‘›) ∈ β„‚)
6144nnrpd 12962 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) β†’ 𝑛 ∈ ℝ+)
62 rpdivcl 12947 . . . . . . . . . . . . . 14 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) β†’ (π‘₯ / 𝑛) ∈ ℝ+)
6361, 62sylan2 594 . . . . . . . . . . . . 13 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘₯ / 𝑛) ∈ ℝ+)
6463rpred 12964 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (π‘₯ / 𝑛) ∈ ℝ)
65 chpcl 26489 . . . . . . . . . . . 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 11190 . . . . . . . . . 10 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (Οˆβ€˜(π‘₯ / 𝑛)) ∈ β„‚)
6860, 67mulcld 11182 . . . . . . . . 9 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((Ξ›β€˜π‘›) Β· (Οˆβ€˜(π‘₯ / 𝑛))) ∈ β„‚)
6945nnrpd 12962 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ 𝑛 ∈ ℝ+)
70 relogcl 25947 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ β†’ (logβ€˜π‘›) ∈ ℝ)
7169, 70syl 17 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (logβ€˜π‘›) ∈ ℝ)
7271recnd 11190 . . . . . . . . . 10 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (logβ€˜π‘›) ∈ β„‚)
7360, 72mulcld 11182 . . . . . . . . 9 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((Ξ›β€˜π‘›) Β· (logβ€˜π‘›)) ∈ β„‚)
7457, 68, 73fsumadd 15632 . . . . . . . 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 13885 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ (1...𝑛) ∈ Fin)
76 dvdsssfz1 16207 . . . . . . . . . . . . 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 9218 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ∈ Fin)
7978, 51fsumrecl 15626 . . . . . . . . . 10 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ Σ𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛} ((Ξ›β€˜π‘‘) Β· (Ξ›β€˜(𝑛 / 𝑑))) ∈ ℝ)
8079recnd 11190 . . . . . . . . 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 15632 . . . . . . . 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 2787 . . . . . . 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 11364 . . . . . . . . . 10 ((π‘₯ ∈ ℝ+ ∧ 𝑛 ∈ (1...(βŒŠβ€˜π‘₯))) β†’ ((logβ€˜π‘›) + (Οˆβ€˜(π‘₯ / 𝑛))) = ((Οˆβ€˜(π‘₯ / 𝑛)) + (logβ€˜π‘›)))
8483oveq2d 7378 . . . . . . . . 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 11186 . . . . . . . . 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 2777 . . . . . . . 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 15595 . . . . . . 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 26907 . . . . . . . . 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 15595 . . . . . . 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 2787 . . . . . 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 7385 . . . . . . . . 9 (𝑛 = (𝑑 Β· π‘š) β†’ (logβ€˜(𝑛 / 𝑑)) = (logβ€˜((𝑑 Β· π‘š) / 𝑑)))
9392oveq1d 7377 . . . . . . . 8 (𝑛 = (𝑑 Β· π‘š) β†’ ((logβ€˜(𝑛 / 𝑑))↑2) = ((logβ€˜((𝑑 Β· π‘š) / 𝑑))↑2))
9493oveq2d 7378 . . . . . . 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 26506 . . . . . . . . . 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 12615 . . . . . . . 8 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ (ΞΌβ€˜π‘‘) ∈ β„‚)
9861ad2antrl 727 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ 𝑛 ∈ ℝ+)
9941nnrpd 12962 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ 𝑑 ∈ ℝ+)
10098, 99rpdivcld 12981 . . . . . . . . . 10 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ (𝑛 / 𝑑) ∈ ℝ+)
101 relogcl 25947 . . . . . . . . . . 11 ((𝑛 / 𝑑) ∈ ℝ+ β†’ (logβ€˜(𝑛 / 𝑑)) ∈ ℝ)
102101recnd 11190 . . . . . . . . . 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 14056 . . . . . . . 8 ((π‘₯ ∈ ℝ+ ∧ (𝑛 ∈ (1...(βŒŠβ€˜π‘₯)) ∧ 𝑑 ∈ {𝑦 ∈ β„• ∣ 𝑦 βˆ₯ 𝑛})) β†’ ((logβ€˜(𝑛 / 𝑑))↑2) ∈ β„‚)
10597, 104mulcld 11182 . . . . . . 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 26553 . . . . . 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 6851 . . . . . . . . . 10 (((π‘₯ ∈ ℝ+ ∧ 𝑑 ∈ (1...(βŒŠβ€˜π‘₯))) ∧ π‘š ∈ (1...(βŒŠβ€˜(π‘₯ / 𝑑)))) β†’ (logβ€˜((𝑑 Β· π‘š) / 𝑑)) = (logβ€˜π‘š))
108107oveq1d 7377 . . . . . . . . 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 7378 . . . . . . . 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 15595 . . . . . . 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 15595 . . . . . 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 2781 . . . . 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 7377 . . . 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 7377 . . 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 5213 . 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 2737 . . 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 26910 . 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 2834 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 397   = wceq 1542   ∈ wcel 2107  {crab 3410   βŠ† wss 3915   class class class wbr 5110   ↦ cmpt 5193  β€˜cfv 6501  (class class class)co 7362  β„‚cc 11056  β„cr 11057  1c1 11059   + caddc 11061   Β· cmul 11063   βˆ’ cmin 11392   / cdiv 11819  β„•cn 12160  2c2 12215  β„€cz 12506  β„+crp 12922  ...cfz 13431  βŒŠcfl 13702  β†‘cexp 13974  π‘‚(1)co1 15375  Ξ£csu 15577   βˆ₯ cdvds 16143  logclog 25926  Ξ›cvma 26457  Οˆcchp 26458  ΞΌcmu 26460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136  ax-addf 11137  ax-mulf 11138
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-disj 5076  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622  df-om 7808  df-1st 7926  df-2nd 7927  df-supp 8098  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-er 8655  df-map 8774  df-pm 8775  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9313  df-fi 9354  df-sup 9385  df-inf 9386  df-oi 9453  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-xnn0 12493  df-z 12507  df-dec 12626  df-uz 12771  df-q 12881  df-rp 12923  df-xneg 13040  df-xadd 13041  df-xmul 13042  df-ioo 13275  df-ioc 13276  df-ico 13277  df-icc 13278  df-fz 13432  df-fzo 13575  df-fl 13704  df-mod 13782  df-seq 13914  df-exp 13975  df-fac 14181  df-bc 14210  df-hash 14238  df-shft 14959  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-limsup 15360  df-clim 15377  df-rlim 15378  df-o1 15379  df-lo1 15380  df-sum 15578  df-ef 15957  df-e 15958  df-sin 15959  df-cos 15960  df-tan 15961  df-pi 15962  df-dvds 16144  df-gcd 16382  df-prm 16555  df-pc 16716  df-struct 17026  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-mulr 17154  df-starv 17155  df-sca 17156  df-vsca 17157  df-ip 17158  df-tset 17159  df-ple 17160  df-ds 17162  df-unif 17163  df-hom 17164  df-cco 17165  df-rest 17311  df-topn 17312  df-0g 17330  df-gsum 17331  df-topgen 17332  df-pt 17333  df-prds 17336  df-xrs 17391  df-qtop 17396  df-imas 17397  df-xps 17399  df-mre 17473  df-mrc 17474  df-acs 17476  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-submnd 18609  df-mulg 18880  df-cntz 19104  df-cmn 19571  df-psmet 20804  df-xmet 20805  df-met 20806  df-bl 20807  df-mopn 20808  df-fbas 20809  df-fg 20810  df-cnfld 20813  df-top 22259  df-topon 22276  df-topsp 22298  df-bases 22312  df-cld 22386  df-ntr 22387  df-cls 22388  df-nei 22465  df-lp 22503  df-perf 22504  df-cn 22594  df-cnp 22595  df-haus 22682  df-cmp 22754  df-tx 22929  df-hmeo 23122  df-fil 23213  df-fm 23305  df-flim 23306  df-flf 23307  df-xms 23689  df-ms 23690  df-tms 23691  df-cncf 24257  df-limc 25246  df-dv 25247  df-ulm 25752  df-log 25928  df-cxp 25929  df-atan 26233  df-em 26358  df-vma 26463  df-chp 26464  df-mu 26466
This theorem is referenced by:  selbergb  26913  selberg2  26915  selbergs  26938
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