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Theorem dvdsflsumcom 27170
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 13979 . . 3 (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2 fzfid 13979 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (1...𝑛) ∈ Fin)
3 elfznn 13570 . . . . . 6 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
43adantl 480 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
5 dvdsssfz1 16303 . . . . 5 (𝑛 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥𝑛} ⊆ (1...𝑛))
64, 5syl 17 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ ℕ ∣ 𝑥𝑛} ⊆ (1...𝑛))
72, 6ssfid 9295 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ ℕ ∣ 𝑥𝑛} ∈ Fin)
8 nnre 12257 . . . . . . . . . . . 12 (𝑑 ∈ ℕ → 𝑑 ∈ ℝ)
98ad2antrl 726 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑑 ∈ ℝ)
104adantr 479 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑛 ∈ ℕ)
1110nnred 12265 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑛 ∈ ℝ)
12 dvdsflsumcom.2 . . . . . . . . . . . 12 (𝜑𝐴 ∈ ℝ)
1312ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝐴 ∈ ℝ)
14 nnz 12617 . . . . . . . . . . . . 13 (𝑑 ∈ ℕ → 𝑑 ∈ ℤ)
15 dvdsle 16295 . . . . . . . . . . . . 13 ((𝑑 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑑𝑛𝑑𝑛))
1614, 4, 15syl2anr 595 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ ℕ) → (𝑑𝑛𝑑𝑛))
1716impr 453 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑑𝑛)
18 fznnfl 13868 . . . . . . . . . . . . . 14 (𝐴 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝐴)))
1912, 18syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝐴)))
2019simplbda 498 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛𝐴)
2120adantr 479 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑛𝐴)
229, 11, 13, 17, 21letrd 11408 . . . . . . . . . 10 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑑𝐴)
2322ex 411 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) → 𝑑𝐴))
2423pm4.71rd 561 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ↔ (𝑑𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛))))
25 ancom 459 . . . . . . . . 9 ((𝑑𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ∧ 𝑑𝐴))
26 an32 644 . . . . . . . . 9 (((𝑑 ∈ ℕ ∧ 𝑑𝑛) ∧ 𝑑𝐴) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛))
2725, 26bitri 274 . . . . . . . 8 ((𝑑𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛))
2824, 27bitrdi 286 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛)))
29 fznnfl 13868 . . . . . . . . . 10 (𝐴 ∈ ℝ → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑𝐴)))
3012, 29syl 17 . . . . . . . . 9 (𝜑 → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑𝐴)))
3130adantr 479 . . . . . . . 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 281 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛)))
3433pm5.32da 577 . . . . 5 (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛))))
35 an12 643 . . . . 5 ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛)))
3634, 35bitrdi 286 . . . 4 (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛))))
37 breq1 5152 . . . . . 6 (𝑥 = 𝑑 → (𝑥𝑛𝑑𝑛))
3837elrab 3679 . . . . 5 (𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛} ↔ (𝑑 ∈ ℕ ∧ 𝑑𝑛))
3938anbi2i 621 . . . 4 ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)))
40 breq2 5153 . . . . . 6 (𝑥 = 𝑛 → (𝑑𝑥𝑑𝑛))
4140elrab 3679 . . . . 5 (𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥} ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛))
4241anbi2i 621 . . . 4 ((𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛)))
4336, 39, 423bitr4g 313 . . 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 15761 . 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 13979 . . . 4 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘(𝐴 / 𝑑))) ∈ Fin)
4812adantr 479 . . . . 5 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
4930simprbda 497 . . . . 5 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℕ)
50 eqid 2725 . . . . 5 (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)) = (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))
5148, 49, 50dvdsflf1o 27169 . . . 4 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)):(1...(⌊‘(𝐴 / 𝑑)))–1-1-onto→{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥})
52 oveq2 7427 . . . . . 6 (𝑦 = 𝑚 → (𝑑 · 𝑦) = (𝑑 · 𝑚))
53 ovex 7452 . . . . . 6 (𝑑 · 𝑚) ∈ V
5452, 50, 53fvmpt 7004 . . . . 5 (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑))) → ((𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))‘𝑚) = (𝑑 · 𝑚))
5554adantl 480 . . . 4 (((𝜑𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))‘𝑚) = (𝑑 · 𝑚))
5643biimpar 476 . . . . . 6 ((𝜑 ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥})) → (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}))
5756, 44syldan 589 . . . . 5 ((𝜑 ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥})) → 𝐵 ∈ ℂ)
5857anassrs 466 . . . 4 (((𝜑𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}) → 𝐵 ∈ ℂ)
5946, 47, 51, 55, 58fsumf1o 15710 . . 3 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}𝐵 = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
6059sumeq2dv 15690 . 2 (𝜑 → Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
6145, 60eqtrd 2765 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 205  wa 394   = wceq 1533  wcel 2098  {crab 3418  wss 3944   class class class wbr 5149  cmpt 5232  cfv 6549  (class class class)co 7419  cc 11143  cr 11144  1c1 11146   · cmul 11150  cle 11286   / cdiv 11908  cn 12250  cz 12596  ...cfz 13524  cfl 13796  Σcsu 15673  cdvds 16239
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 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9671  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222  ax-pre-sup 11223
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9472  df-inf 9473  df-oi 9540  df-card 9969  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-div 11909  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-z 12597  df-uz 12861  df-rp 13015  df-fz 13525  df-fzo 13668  df-fl 13798  df-seq 14008  df-exp 14068  df-hash 14331  df-cj 15087  df-re 15088  df-im 15089  df-sqrt 15223  df-abs 15224  df-clim 15473  df-sum 15674  df-dvds 16240
This theorem is referenced by:  dchrmusum2  27477  dchrvmasumlem1  27478  dchrvmasum2lem  27479  dchrisum0  27503  mudivsum  27513  mulogsum  27515  mulog2sumlem2  27518  vmalogdivsum2  27521  selberglem3  27530  selberg  27531  selberg34r  27554  pntsval2  27559
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