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Theorem dvdsflsumcom 25692
Description: A sum commutation from Σ𝑛𝐴, Σ𝑑𝑛, 𝐵(𝑛, 𝑑) to Σ𝑑𝐴, Σ𝑚𝐴 / 𝑑, 𝐵(𝑛, 𝑑𝑚). (Contributed by Mario Carneiro, 4-May-2016.)
Hypotheses
Ref Expression
dvdsflsumcom.1 (𝑛 = (𝑑 · 𝑚) → 𝐵 = 𝐶)
dvdsflsumcom.2 (𝜑𝐴 ∈ ℝ)
dvdsflsumcom.3 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛})) → 𝐵 ∈ ℂ)
Assertion
Ref Expression
dvdsflsumcom (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
Distinct variable groups:   𝑚,𝑑,𝑛,𝑥,𝐴   𝐵,𝑚   𝐶,𝑛   𝜑,𝑑,𝑚,𝑛
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥,𝑛,𝑑)   𝐶(𝑥,𝑚,𝑑)

Proof of Theorem dvdsflsumcom
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fzfid 13329 . . 3 (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2 fzfid 13329 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (1...𝑛) ∈ Fin)
3 elfznn 12924 . . . . . 6 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
43adantl 482 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
5 dvdsssfz1 15656 . . . . 5 (𝑛 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥𝑛} ⊆ (1...𝑛))
64, 5syl 17 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ ℕ ∣ 𝑥𝑛} ⊆ (1...𝑛))
72, 6ssfid 8729 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ ℕ ∣ 𝑥𝑛} ∈ Fin)
8 nnre 11633 . . . . . . . . . . . 12 (𝑑 ∈ ℕ → 𝑑 ∈ ℝ)
98ad2antrl 724 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑑 ∈ ℝ)
104adantr 481 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑛 ∈ ℕ)
1110nnred 11641 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑛 ∈ ℝ)
12 dvdsflsumcom.2 . . . . . . . . . . . 12 (𝜑𝐴 ∈ ℝ)
1312ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝐴 ∈ ℝ)
14 nnz 11992 . . . . . . . . . . . . 13 (𝑑 ∈ ℕ → 𝑑 ∈ ℤ)
15 dvdsle 15648 . . . . . . . . . . . . 13 ((𝑑 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑑𝑛𝑑𝑛))
1614, 4, 15syl2anr 596 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ ℕ) → (𝑑𝑛𝑑𝑛))
1716impr 455 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑑𝑛)
18 fznnfl 13218 . . . . . . . . . . . . . 14 (𝐴 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝐴)))
1912, 18syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝐴)))
2019simplbda 500 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛𝐴)
2120adantr 481 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑛𝐴)
229, 11, 13, 17, 21letrd 10785 . . . . . . . . . 10 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑑𝐴)
2322ex 413 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) → 𝑑𝐴))
2423pm4.71rd 563 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ↔ (𝑑𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛))))
25 ancom 461 . . . . . . . . 9 ((𝑑𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ∧ 𝑑𝐴))
26 an32 642 . . . . . . . . 9 (((𝑑 ∈ ℕ ∧ 𝑑𝑛) ∧ 𝑑𝐴) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛))
2725, 26bitri 276 . . . . . . . 8 ((𝑑𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛))
2824, 27syl6bb 288 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛)))
29 fznnfl 13218 . . . . . . . . . 10 (𝐴 ∈ ℝ → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑𝐴)))
3012, 29syl 17 . . . . . . . . 9 (𝜑 → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑𝐴)))
3130adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑𝐴)))
3231anbi1d 629 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛)))
3328, 32bitr4d 283 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛)))
3433pm5.32da 579 . . . . 5 (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛))))
35 an12 641 . . . . 5 ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛)))
3634, 35syl6bb 288 . . . 4 (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛))))
37 breq1 5060 . . . . . 6 (𝑥 = 𝑑 → (𝑥𝑛𝑑𝑛))
3837elrab 3677 . . . . 5 (𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛} ↔ (𝑑 ∈ ℕ ∧ 𝑑𝑛))
3938anbi2i 622 . . . 4 ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)))
40 breq2 5061 . . . . . 6 (𝑥 = 𝑛 → (𝑑𝑥𝑑𝑛))
4140elrab 3677 . . . . 5 (𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥} ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛))
4241anbi2i 622 . . . 4 ((𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛)))
4336, 39, 423bitr4g 315 . . 3 (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥})))
44 dvdsflsumcom.3 . . 3 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛})) → 𝐵 ∈ ℂ)
451, 1, 7, 43, 44fsumcom2 15117 . 2 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}𝐵)
46 dvdsflsumcom.1 . . . 4 (𝑛 = (𝑑 · 𝑚) → 𝐵 = 𝐶)
47 fzfid 13329 . . . 4 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘(𝐴 / 𝑑))) ∈ Fin)
4812adantr 481 . . . . 5 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
4930simprbda 499 . . . . 5 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℕ)
50 eqid 2818 . . . . 5 (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)) = (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))
5148, 49, 50dvdsflf1o 25691 . . . 4 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)):(1...(⌊‘(𝐴 / 𝑑)))–1-1-onto→{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥})
52 oveq2 7153 . . . . . 6 (𝑦 = 𝑚 → (𝑑 · 𝑦) = (𝑑 · 𝑚))
53 ovex 7178 . . . . . 6 (𝑑 · 𝑚) ∈ V
5452, 50, 53fvmpt 6761 . . . . 5 (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑))) → ((𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))‘𝑚) = (𝑑 · 𝑚))
5554adantl 482 . . . 4 (((𝜑𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))‘𝑚) = (𝑑 · 𝑚))
5643biimpar 478 . . . . . 6 ((𝜑 ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥})) → (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}))
5756, 44syldan 591 . . . . 5 ((𝜑 ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥})) → 𝐵 ∈ ℂ)
5857anassrs 468 . . . 4 (((𝜑𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}) → 𝐵 ∈ ℂ)
5946, 47, 51, 55, 58fsumf1o 15068 . . 3 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}𝐵 = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
6059sumeq2dv 15048 . 2 (𝜑 → Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
6145, 60eqtrd 2853 1 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  {crab 3139  wss 3933   class class class wbr 5057  cmpt 5137  cfv 6348  (class class class)co 7145  cc 10523  cr 10524  1c1 10526   · cmul 10530  cle 10664   / cdiv 11285  cn 11626  cz 11969  ...cfz 12880  cfl 13148  Σcsu 15030  cdvds 15595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-fl 13150  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-sum 15031  df-dvds 15596
This theorem is referenced by:  dchrmusum2  25997  dchrvmasumlem1  25998  dchrvmasum2lem  25999  dchrisum0  26023  mudivsum  26033  mulogsum  26035  mulog2sumlem2  26038  vmalogdivsum2  26041  selberglem3  26050  selberg  26051  selberg34r  26074  pntsval2  26079
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