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Theorem dvdsflf1o 25781
Description: A bijection from the numbers less than 𝑁 / 𝐴 to the multiples of 𝐴 less than 𝑁. Useful for some sum manipulations. (Contributed by Mario Carneiro, 3-May-2016.)
Hypotheses
Ref Expression
dvdsflf1o.1 (𝜑𝐴 ∈ ℝ)
dvdsflf1o.2 (𝜑𝑁 ∈ ℕ)
dvdsflf1o.f 𝐹 = (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ↦ (𝑁 · 𝑛))
Assertion
Ref Expression
dvdsflf1o (𝜑𝐹:(1...(⌊‘(𝐴 / 𝑁)))–1-1-onto→{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})
Distinct variable groups:   𝑥,𝑛,𝐴   𝑛,𝑁,𝑥   𝜑,𝑛
Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥,𝑛)

Proof of Theorem dvdsflf1o
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 dvdsflf1o.f . 2 𝐹 = (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ↦ (𝑁 · 𝑛))
2 breq2 5057 . . 3 (𝑥 = (𝑁 · 𝑛) → (𝑁𝑥𝑁 ∥ (𝑁 · 𝑛)))
3 dvdsflf1o.2 . . . . 5 (𝜑𝑁 ∈ ℕ)
4 elfznn 12942 . . . . 5 (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) → 𝑛 ∈ ℕ)
5 nnmulcl 11660 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑁 · 𝑛) ∈ ℕ)
63, 4, 5syl2an 598 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ∈ ℕ)
7 dvdsflf1o.1 . . . . . . . . 9 (𝜑𝐴 ∈ ℝ)
87, 3nndivred 11690 . . . . . . . 8 (𝜑 → (𝐴 / 𝑁) ∈ ℝ)
9 fznnfl 13236 . . . . . . . 8 ((𝐴 / 𝑁) ∈ ℝ → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝐴 / 𝑁))))
108, 9syl 17 . . . . . . 7 (𝜑 → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝐴 / 𝑁))))
1110simplbda 503 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑛 ≤ (𝐴 / 𝑁))
124adantl 485 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑛 ∈ ℕ)
1312nnred 11651 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑛 ∈ ℝ)
147adantr 484 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝐴 ∈ ℝ)
153nnred 11651 . . . . . . . 8 (𝜑𝑁 ∈ ℝ)
1615adantr 484 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑁 ∈ ℝ)
173nngt0d 11685 . . . . . . . 8 (𝜑 → 0 < 𝑁)
1817adantr 484 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 0 < 𝑁)
19 lemuldiv2 11521 . . . . . . 7 ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑁 · 𝑛) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑁)))
2013, 14, 16, 18, 19syl112anc 1371 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → ((𝑁 · 𝑛) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑁)))
2111, 20mpbird 260 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ≤ 𝐴)
223nnzd 12085 . . . . . . 7 (𝜑𝑁 ∈ ℤ)
23 elfzelz 12913 . . . . . . 7 (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) → 𝑛 ∈ ℤ)
24 zmulcl 12030 . . . . . . 7 ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑁 · 𝑛) ∈ ℤ)
2522, 23, 24syl2an 598 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ∈ ℤ)
26 flge 13181 . . . . . 6 ((𝐴 ∈ ℝ ∧ (𝑁 · 𝑛) ∈ ℤ) → ((𝑁 · 𝑛) ≤ 𝐴 ↔ (𝑁 · 𝑛) ≤ (⌊‘𝐴)))
2714, 25, 26syl2anc 587 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → ((𝑁 · 𝑛) ≤ 𝐴 ↔ (𝑁 · 𝑛) ≤ (⌊‘𝐴)))
2821, 27mpbid 235 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ≤ (⌊‘𝐴))
297flcld 13174 . . . . . 6 (𝜑 → (⌊‘𝐴) ∈ ℤ)
3029adantr 484 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (⌊‘𝐴) ∈ ℤ)
31 fznn 12981 . . . . 5 ((⌊‘𝐴) ∈ ℤ → ((𝑁 · 𝑛) ∈ (1...(⌊‘𝐴)) ↔ ((𝑁 · 𝑛) ∈ ℕ ∧ (𝑁 · 𝑛) ≤ (⌊‘𝐴))))
3230, 31syl 17 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → ((𝑁 · 𝑛) ∈ (1...(⌊‘𝐴)) ↔ ((𝑁 · 𝑛) ∈ ℕ ∧ (𝑁 · 𝑛) ≤ (⌊‘𝐴))))
336, 28, 32mpbir2and 712 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ∈ (1...(⌊‘𝐴)))
34 dvdsmul1 15633 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑁 ∥ (𝑁 · 𝑛))
3522, 23, 34syl2an 598 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑁 ∥ (𝑁 · 𝑛))
362, 33, 35elrabd 3668 . 2 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})
37 breq2 5057 . . . . . . 7 (𝑥 = 𝑚 → (𝑁𝑥𝑁𝑚))
3837elrab 3666 . . . . . 6 (𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥} ↔ (𝑚 ∈ (1...(⌊‘𝐴)) ∧ 𝑁𝑚))
3938simprbi 500 . . . . 5 (𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥} → 𝑁𝑚)
4039adantl 485 . . . 4 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑁𝑚)
41 elrabi 3661 . . . . . . 7 (𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥} → 𝑚 ∈ (1...(⌊‘𝐴)))
4241adantl 485 . . . . . 6 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑚 ∈ (1...(⌊‘𝐴)))
43 elfznn 12942 . . . . . 6 (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
4442, 43syl 17 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑚 ∈ ℕ)
453adantr 484 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑁 ∈ ℕ)
46 nndivdvds 15618 . . . . 5 ((𝑚 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁𝑚 ↔ (𝑚 / 𝑁) ∈ ℕ))
4744, 45, 46syl2anc 587 . . . 4 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝑁𝑚 ↔ (𝑚 / 𝑁) ∈ ℕ))
4840, 47mpbid 235 . . 3 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝑚 / 𝑁) ∈ ℕ)
49 fznnfl 13236 . . . . . . 7 (𝐴 ∈ ℝ → (𝑚 ∈ (1...(⌊‘𝐴)) ↔ (𝑚 ∈ ℕ ∧ 𝑚𝐴)))
507, 49syl 17 . . . . . 6 (𝜑 → (𝑚 ∈ (1...(⌊‘𝐴)) ↔ (𝑚 ∈ ℕ ∧ 𝑚𝐴)))
5150simplbda 503 . . . . 5 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚𝐴)
5241, 51sylan2 595 . . . 4 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑚𝐴)
5344nnred 11651 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑚 ∈ ℝ)
547adantr 484 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝐴 ∈ ℝ)
5515adantr 484 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑁 ∈ ℝ)
5617adantr 484 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 0 < 𝑁)
57 lediv1 11505 . . . . 5 ((𝑚 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (𝑚𝐴 ↔ (𝑚 / 𝑁) ≤ (𝐴 / 𝑁)))
5853, 54, 55, 56, 57syl112anc 1371 . . . 4 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝑚𝐴 ↔ (𝑚 / 𝑁) ≤ (𝐴 / 𝑁)))
5952, 58mpbid 235 . . 3 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝑚 / 𝑁) ≤ (𝐴 / 𝑁))
608adantr 484 . . . 4 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝐴 / 𝑁) ∈ ℝ)
61 fznnfl 13236 . . . 4 ((𝐴 / 𝑁) ∈ ℝ → ((𝑚 / 𝑁) ∈ (1...(⌊‘(𝐴 / 𝑁))) ↔ ((𝑚 / 𝑁) ∈ ℕ ∧ (𝑚 / 𝑁) ≤ (𝐴 / 𝑁))))
6260, 61syl 17 . . 3 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → ((𝑚 / 𝑁) ∈ (1...(⌊‘(𝐴 / 𝑁))) ↔ ((𝑚 / 𝑁) ∈ ℕ ∧ (𝑚 / 𝑁) ≤ (𝐴 / 𝑁))))
6348, 59, 62mpbir2and 712 . 2 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝑚 / 𝑁) ∈ (1...(⌊‘(𝐴 / 𝑁))))
6444nncnd 11652 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑚 ∈ ℂ)
6564adantrl 715 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → 𝑚 ∈ ℂ)
663nncnd 11652 . . . . 5 (𝜑𝑁 ∈ ℂ)
6766adantr 484 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → 𝑁 ∈ ℂ)
6812nncnd 11652 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑛 ∈ ℂ)
6968adantrr 716 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → 𝑛 ∈ ℂ)
703nnne0d 11686 . . . . 5 (𝜑𝑁 ≠ 0)
7170adantr 484 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → 𝑁 ≠ 0)
7265, 67, 69, 71divmuld 11438 . . 3 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → ((𝑚 / 𝑁) = 𝑛 ↔ (𝑁 · 𝑛) = 𝑚))
73 eqcom 2831 . . 3 (𝑛 = (𝑚 / 𝑁) ↔ (𝑚 / 𝑁) = 𝑛)
74 eqcom 2831 . . 3 (𝑚 = (𝑁 · 𝑛) ↔ (𝑁 · 𝑛) = 𝑚)
7572, 73, 743bitr4g 317 . 2 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → (𝑛 = (𝑚 / 𝑁) ↔ 𝑚 = (𝑁 · 𝑛)))
761, 36, 63, 75f1o2d 7395 1 (𝜑𝐹:(1...(⌊‘(𝐴 / 𝑁)))–1-1-onto→{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3014  {crab 3137   class class class wbr 5053  cmpt 5133  1-1-ontowf1o 6344  cfv 6345  (class class class)co 7151  cc 10535  cr 10536  0cc0 10537  1c1 10538   · cmul 10542   < clt 10675  cle 10676   / cdiv 11297  cn 11636  cz 11980  ...cfz 12896  cfl 13166  cdvds 15609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-1st 7686  df-2nd 7687  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-er 8287  df-en 8508  df-dom 8509  df-sdom 8510  df-sup 8905  df-inf 8906  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11637  df-n0 11897  df-z 11981  df-uz 12243  df-fz 12897  df-fl 13168  df-dvds 15610
This theorem is referenced by:  dvdsflsumcom  25782  logfac2  25810
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