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Theorem dvdsflsumcom 25219
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 12985 . . 3 (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2 fzfid 12985 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (1...𝑛) ∈ Fin)
3 elfznn 12582 . . . . . 6 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
43adantl 473 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
5 dvdsssfz1 15339 . . . . 5 (𝑛 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥𝑛} ⊆ (1...𝑛))
64, 5syl 17 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ ℕ ∣ 𝑥𝑛} ⊆ (1...𝑛))
7 ssfi 8391 . . . 4 (((1...𝑛) ∈ Fin ∧ {𝑥 ∈ ℕ ∣ 𝑥𝑛} ⊆ (1...𝑛)) → {𝑥 ∈ ℕ ∣ 𝑥𝑛} ∈ Fin)
82, 6, 7syl2anc 579 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ ℕ ∣ 𝑥𝑛} ∈ Fin)
9 nnre 11286 . . . . . . . . . . . 12 (𝑑 ∈ ℕ → 𝑑 ∈ ℝ)
109ad2antrl 719 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑑 ∈ ℝ)
114adantr 472 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑛 ∈ ℕ)
1211nnred 11295 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑛 ∈ ℝ)
13 dvdsflsumcom.2 . . . . . . . . . . . 12 (𝜑𝐴 ∈ ℝ)
1413ad2antrr 717 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝐴 ∈ ℝ)
15 nnz 11651 . . . . . . . . . . . . 13 (𝑑 ∈ ℕ → 𝑑 ∈ ℤ)
16 dvdsle 15331 . . . . . . . . . . . . 13 ((𝑑 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑑𝑛𝑑𝑛))
1715, 4, 16syl2anr 590 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ ℕ) → (𝑑𝑛𝑑𝑛))
1817impr 446 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑑𝑛)
19 fznnfl 12874 . . . . . . . . . . . . . 14 (𝐴 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝐴)))
2013, 19syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝐴)))
2120simplbda 493 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛𝐴)
2221adantr 472 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑛𝐴)
2310, 12, 14, 18, 22letrd 10452 . . . . . . . . . 10 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑑𝐴)
2423ex 401 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) → 𝑑𝐴))
2524pm4.71rd 558 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ↔ (𝑑𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛))))
26 ancom 452 . . . . . . . . 9 ((𝑑𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ∧ 𝑑𝐴))
27 an32 636 . . . . . . . . 9 (((𝑑 ∈ ℕ ∧ 𝑑𝑛) ∧ 𝑑𝐴) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛))
2826, 27bitri 266 . . . . . . . 8 ((𝑑𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛))
2925, 28syl6bb 278 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛)))
30 fznnfl 12874 . . . . . . . . . 10 (𝐴 ∈ ℝ → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑𝐴)))
3113, 30syl 17 . . . . . . . . 9 (𝜑 → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑𝐴)))
3231adantr 472 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑𝐴)))
3332anbi1d 623 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛)))
3429, 33bitr4d 273 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛)))
3534pm5.32da 574 . . . . 5 (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛))))
36 an12 635 . . . . 5 ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛)))
3735, 36syl6bb 278 . . . 4 (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛))))
38 breq1 4814 . . . . . 6 (𝑥 = 𝑑 → (𝑥𝑛𝑑𝑛))
3938elrab 3521 . . . . 5 (𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛} ↔ (𝑑 ∈ ℕ ∧ 𝑑𝑛))
4039anbi2i 616 . . . 4 ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)))
41 breq2 4815 . . . . . 6 (𝑥 = 𝑛 → (𝑑𝑥𝑑𝑛))
4241elrab 3521 . . . . 5 (𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥} ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛))
4342anbi2i 616 . . . 4 ((𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛)))
4437, 40, 433bitr4g 305 . . 3 (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥})))
45 dvdsflsumcom.3 . . 3 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛})) → 𝐵 ∈ ℂ)
461, 1, 8, 44, 45fsumcom2 14804 . 2 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}𝐵)
47 dvdsflsumcom.1 . . . 4 (𝑛 = (𝑑 · 𝑚) → 𝐵 = 𝐶)
48 fzfid 12985 . . . 4 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘(𝐴 / 𝑑))) ∈ Fin)
4913adantr 472 . . . . 5 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
5031simprbda 492 . . . . 5 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℕ)
51 eqid 2765 . . . . 5 (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)) = (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))
5249, 50, 51dvdsflf1o 25218 . . . 4 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)):(1...(⌊‘(𝐴 / 𝑑)))–1-1-onto→{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥})
53 oveq2 6854 . . . . . 6 (𝑦 = 𝑚 → (𝑑 · 𝑦) = (𝑑 · 𝑚))
54 ovex 6878 . . . . . 6 (𝑑 · 𝑚) ∈ V
5553, 51, 54fvmpt 6475 . . . . 5 (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑))) → ((𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))‘𝑚) = (𝑑 · 𝑚))
5655adantl 473 . . . 4 (((𝜑𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))‘𝑚) = (𝑑 · 𝑚))
5744biimpar 469 . . . . . 6 ((𝜑 ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥})) → (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}))
5857, 45syldan 585 . . . . 5 ((𝜑 ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥})) → 𝐵 ∈ ℂ)
5958anassrs 459 . . . 4 (((𝜑𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}) → 𝐵 ∈ ℂ)
6047, 48, 52, 56, 59fsumf1o 14753 . . 3 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}𝐵 = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
6160sumeq2dv 14732 . 2 (𝜑 → Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
6246, 61eqtrd 2799 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 197  wa 384   = wceq 1652  wcel 2155  {crab 3059  wss 3734   class class class wbr 4811  cmpt 4890  cfv 6070  (class class class)co 6846  Fincfn 8164  cc 10191  cr 10192  1c1 10194   · cmul 10198  cle 10333   / cdiv 10942  cn 11278  cz 11628  ...cfz 12538  cfl 12804  Σcsu 14715  cdvds 15279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-oadd 7772  df-er 7951  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-sup 8559  df-inf 8560  df-oi 8626  df-card 9020  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-div 10943  df-nn 11279  df-2 11339  df-3 11340  df-n0 11543  df-z 11629  df-uz 11892  df-rp 12034  df-fz 12539  df-fzo 12679  df-fl 12806  df-seq 13014  df-exp 13073  df-hash 13327  df-cj 14138  df-re 14139  df-im 14140  df-sqrt 14274  df-abs 14275  df-clim 14518  df-sum 14716  df-dvds 15280
This theorem is referenced by:  dchrmusum2  25488  dchrvmasumlem1  25489  dchrvmasum2lem  25490  dchrisum0  25514  mudivsum  25524  mulogsum  25526  mulog2sumlem2  25529  vmalogdivsum2  25532  selberglem3  25541  selberg  25542  selberg34r  25565  pntsval2  25570
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