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Theorem dvdsflsumcom 24821
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 12715 . . 3 (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2 fzfid 12715 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (1...𝑛) ∈ Fin)
3 elfznn 12315 . . . . . 6 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
43adantl 482 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
5 dvdsssfz1 14967 . . . . 5 (𝑛 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥𝑛} ⊆ (1...𝑛))
64, 5syl 17 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ ℕ ∣ 𝑥𝑛} ⊆ (1...𝑛))
7 ssfi 8127 . . . 4 (((1...𝑛) ∈ Fin ∧ {𝑥 ∈ ℕ ∣ 𝑥𝑛} ⊆ (1...𝑛)) → {𝑥 ∈ ℕ ∣ 𝑥𝑛} ∈ Fin)
82, 6, 7syl2anc 692 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ ℕ ∣ 𝑥𝑛} ∈ Fin)
9 nnre 10974 . . . . . . . . . . . 12 (𝑑 ∈ ℕ → 𝑑 ∈ ℝ)
109ad2antrl 763 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑑 ∈ ℝ)
114adantr 481 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑛 ∈ ℕ)
1211nnred 10982 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑛 ∈ ℝ)
13 dvdsflsumcom.2 . . . . . . . . . . . 12 (𝜑𝐴 ∈ ℝ)
1413ad2antrr 761 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝐴 ∈ ℝ)
15 nnz 11346 . . . . . . . . . . . . 13 (𝑑 ∈ ℕ → 𝑑 ∈ ℤ)
16 dvdsle 14959 . . . . . . . . . . . . 13 ((𝑑 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑑𝑛𝑑𝑛))
1715, 4, 16syl2anr 495 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ ℕ) → (𝑑𝑛𝑑𝑛))
1817impr 648 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑑𝑛)
19 fznnfl 12604 . . . . . . . . . . . . . 14 (𝐴 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝐴)))
2013, 19syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝐴)))
2120simplbda 653 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛𝐴)
2221adantr 481 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑛𝐴)
2310, 12, 14, 18, 22letrd 10141 . . . . . . . . . 10 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) → 𝑑𝐴)
2423ex 450 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) → 𝑑𝐴))
2524pm4.71rd 666 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ↔ (𝑑𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛))))
26 ancom 466 . . . . . . . . 9 ((𝑑𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ∧ 𝑑𝐴))
27 an32 838 . . . . . . . . 9 (((𝑑 ∈ ℕ ∧ 𝑑𝑛) ∧ 𝑑𝐴) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛))
2826, 27bitri 264 . . . . . . . 8 ((𝑑𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛))
2925, 28syl6bb 276 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛)))
30 fznnfl 12604 . . . . . . . . . 10 (𝐴 ∈ ℝ → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑𝐴)))
3113, 30syl 17 . . . . . . . . 9 (𝜑 → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑𝐴)))
3231adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑𝐴)))
3332anbi1d 740 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛) ↔ ((𝑑 ∈ ℕ ∧ 𝑑𝐴) ∧ 𝑑𝑛)))
3429, 33bitr4d 271 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑𝑛) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛)))
3534pm5.32da 672 . . . . 5 (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛))))
36 an12 837 . . . . 5 ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛)))
3735, 36syl6bb 276 . . . 4 (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛))))
38 breq1 4618 . . . . . 6 (𝑥 = 𝑑 → (𝑥𝑛𝑑𝑛))
3938elrab 3347 . . . . 5 (𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛} ↔ (𝑑 ∈ ℕ ∧ 𝑑𝑛))
4039anbi2i 729 . . . 4 ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑𝑛)))
41 breq2 4619 . . . . . 6 (𝑥 = 𝑛 → (𝑑𝑥𝑑𝑛))
4241elrab 3347 . . . . 5 (𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥} ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛))
4342anbi2i 729 . . . 4 ((𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑𝑛)))
4437, 40, 433bitr4g 303 . . 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 14436 . 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 12715 . . . 4 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘(𝐴 / 𝑑))) ∈ Fin)
4913adantr 481 . . . . 5 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
5031simprbda 652 . . . . 5 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℕ)
51 eqid 2621 . . . . 5 (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)) = (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))
5249, 50, 51dvdsflf1o 24820 . . . 4 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)):(1...(⌊‘(𝐴 / 𝑑)))–1-1-onto→{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥})
53 oveq2 6615 . . . . . 6 (𝑦 = 𝑚 → (𝑑 · 𝑦) = (𝑑 · 𝑚))
54 ovex 6635 . . . . . 6 (𝑑 · 𝑚) ∈ V
5553, 51, 54fvmpt 6241 . . . . 5 (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑))) → ((𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))‘𝑚) = (𝑑 · 𝑚))
5655adantl 482 . . . 4 (((𝜑𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))‘𝑚) = (𝑑 · 𝑚))
5744biimpar 502 . . . . . 6 ((𝜑 ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥})) → (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}))
5857, 45syldan 487 . . . . 5 ((𝜑 ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥})) → 𝐵 ∈ ℂ)
5958anassrs 679 . . . 4 (((𝜑𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}) → 𝐵 ∈ ℂ)
6047, 48, 52, 56, 59fsumf1o 14390 . . 3 ((𝜑𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}𝐵 = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
6160sumeq2dv 14370 . 2 (𝜑 → Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑𝑥}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶)
6246, 61eqtrd 2655 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 196  wa 384   = wceq 1480  wcel 1987  {crab 2911  wss 3556   class class class wbr 4615  cmpt 4675  cfv 5849  (class class class)co 6607  Fincfn 7902  cc 9881  cr 9882  1c1 9884   · cmul 9888  cle 10022   / cdiv 10631  cn 10967  cz 11324  ...cfz 12271  cfl 12534  Σcsu 14353  cdvds 14910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-inf2 8485  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960  ax-pre-sup 9961
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-oadd 7512  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-sup 8295  df-inf 8296  df-oi 8362  df-card 8712  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-div 10632  df-nn 10968  df-2 11026  df-3 11027  df-n0 11240  df-z 11325  df-uz 11635  df-rp 11780  df-fz 12272  df-fzo 12410  df-fl 12536  df-seq 12745  df-exp 12804  df-hash 13061  df-cj 13776  df-re 13777  df-im 13778  df-sqrt 13912  df-abs 13913  df-clim 14156  df-sum 14354  df-dvds 14911
This theorem is referenced by:  dchrmusum2  25090  dchrvmasumlem1  25091  dchrvmasum2lem  25092  dchrisum0  25116  mudivsum  25126  mulogsum  25128  mulog2sumlem2  25131  vmalogdivsum2  25134  selberglem3  25143  selberg  25144  selberg34r  25167  pntsval2  25172
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