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Theorem dvdsflsumcom 25334
 Description: A sum commutation from Σ𝑛 ≤ 𝐴, Σ𝑑 ∥ 𝑛, 𝐵(𝑛, 𝑑) to Σ𝑑 ≤ 𝐴, Σ𝑚 ≤ 𝐴 / 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 13074 . . 3 (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2 fzfid 13074 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (1...𝑛) ∈ Fin)
3 elfznn 12670 . . . . . 6 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
43adantl 475 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
5 dvdsssfz1 15424 . . . . 5 (𝑛 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥𝑛} ⊆ (1...𝑛))
64, 5syl 17 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ ℕ ∣ 𝑥𝑛} ⊆ (1...𝑛))
7 ssfi 8455 . . . 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 11365 . . . . . . . . . . . 12 (𝑑 ∈ ℕ → 𝑑 ∈ ℝ)
109ad2antrl 719 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑑 ∈ ℝ)
114adantr 474 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑛 ∈ ℕ)
1211nnred 11374 . . . . . . . . . . 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 11734 . . . . . . . . . . . . 13 (𝑑 ∈ ℕ → 𝑑 ∈ ℤ)
16 dvdsle 15416 . . . . . . . . . . . . 13 ((𝑑 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑑𝑛𝑑𝑛))
1715, 4, 16syl2anr 590 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ ℕ) → (𝑑𝑛𝑑𝑛))
1817impr 448 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑑𝑛)
19 fznnfl 12963 . . . . . . . . . . . . . 14 (𝐴 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝐴)))
2013, 19syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝐴)))
2120simplbda 495 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛𝐴)
2221adantr 474 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑛𝐴)
2310, 12, 14, 18, 22letrd 10520 . . . . . . . . . 10 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑑𝐴)
2423ex 403 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) → 𝑑𝐴))
2524pm4.71rd 558 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ↔ (𝑑𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛))))
26 ancom 454 . . . . . . . . 9 ((𝑑𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ∧ 𝑑𝐴))
27 an32 636 . . . . . . . . 9 (((𝑑 ∈ ℕ ∧ 𝑑𝑛) ∧ 𝑑𝐴) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛))
2826, 27bitri 267 . . . . . . . 8 ((𝑑𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛))
2925, 28syl6bb 279 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛)))
30 fznnfl 12963 . . . . . . . . . 10 (𝐴 ∈ ℝ → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑𝐴)))
3113, 30syl 17 . . . . . . . . 9 (𝜑 → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑𝐴)))
3231adantr 474 . . . . . . . 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 274 . . . . . 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 279 . . . 4 (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛))))
38 breq1 4878 . . . . . 6 (𝑥 = 𝑑 → (𝑥𝑛𝑑𝑛))
3938elrab 3585 . . . . 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 4879 . . . . . 6 (𝑥 = 𝑛 → (𝑑𝑥𝑑𝑛))
4241elrab 3585 . . . . 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 306 . . 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 14887 . 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 13074 . . . 4 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘(𝐴 / 𝑑))) ∈ Fin)
4913adantr 474 . . . . 5 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
5031simprbda 494 . . . . 5 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℕ)
51 eqid 2825 . . . . 5 (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)) = (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))
5249, 50, 51dvdsflf1o 25333 . . . 4 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)):(1...(⌊‘(𝐴 / 𝑑)))–1-1-onto→{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥})
53 oveq2 6918 . . . . . 6 (𝑦 = 𝑚 → (𝑑 · 𝑦) = (𝑑 · 𝑚))
54 ovex 6942 . . . . . 6 (𝑑 · 𝑚) ∈ V
5553, 51, 54fvmpt 6533 . . . . 5 (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑))) → ((𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))‘𝑚) = (𝑑 · 𝑚))
5655adantl 475 . . . 4 (((𝜑𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))‘𝑚) = (𝑑 · 𝑚))
5744biimpar 471 . . . . . 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 461 . . . 4 (((𝜑𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}) → 𝐵 ∈ ℂ)
6047, 48, 52, 56, 59fsumf1o 14838 . . 3 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}𝐵 = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
6160sumeq2dv 14817 . 2 (𝜑 → Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
6246, 61eqtrd 2861 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 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  {crab 3121   ⊆ wss 3798   class class class wbr 4875   ↦ cmpt 4954  ‘cfv 6127  (class class class)co 6910  Fincfn 8228  ℂcc 10257  ℝcr 10258  1c1 10260   · cmul 10264   ≤ cle 10399   / cdiv 11016  ℕcn 11357  ℤcz 11711  ...cfz 12626  ⌊cfl 12893  Σcsu 14800   ∥ cdvds 15364 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-inf2 8822  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336  ax-pre-sup 10337 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-se 5306  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-isom 6136  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-oadd 7835  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-sup 8623  df-inf 8624  df-oi 8691  df-card 9085  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-div 11017  df-nn 11358  df-2 11421  df-3 11422  df-n0 11626  df-z 11712  df-uz 11976  df-rp 12120  df-fz 12627  df-fzo 12768  df-fl 12895  df-seq 13103  df-exp 13162  df-hash 13418  df-cj 14223  df-re 14224  df-im 14225  df-sqrt 14359  df-abs 14360  df-clim 14603  df-sum 14801  df-dvds 15365 This theorem is referenced by:  dchrmusum2  25603  dchrvmasumlem1  25604  dchrvmasum2lem  25605  dchrisum0  25629  mudivsum  25639  mulogsum  25641  mulog2sumlem2  25644  vmalogdivsum2  25647  selberglem3  25656  selberg  25657  selberg34r  25680  pntsval2  25685
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