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Theorem dvdsflf1o 24958
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 dvdsflf1o.2 . . . . 5 (𝜑𝑁 ∈ ℕ)
3 elfznn 12408 . . . . 5 (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) → 𝑛 ∈ ℕ)
4 nnmulcl 11081 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑁 · 𝑛) ∈ ℕ)
52, 3, 4syl2an 493 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ∈ ℕ)
6 dvdsflf1o.1 . . . . . . . . 9 (𝜑𝐴 ∈ ℝ)
76, 2nndivred 11107 . . . . . . . 8 (𝜑 → (𝐴 / 𝑁) ∈ ℝ)
8 fznnfl 12701 . . . . . . . 8 ((𝐴 / 𝑁) ∈ ℝ → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝐴 / 𝑁))))
97, 8syl 17 . . . . . . 7 (𝜑 → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝐴 / 𝑁))))
109simplbda 653 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑛 ≤ (𝐴 / 𝑁))
113adantl 481 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑛 ∈ ℕ)
1211nnred 11073 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑛 ∈ ℝ)
136adantr 480 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝐴 ∈ ℝ)
142nnred 11073 . . . . . . . 8 (𝜑𝑁 ∈ ℝ)
1514adantr 480 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑁 ∈ ℝ)
162nngt0d 11102 . . . . . . . 8 (𝜑 → 0 < 𝑁)
1716adantr 480 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 0 < 𝑁)
18 lemuldiv2 10942 . . . . . . 7 ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑁 · 𝑛) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑁)))
1912, 13, 15, 17, 18syl112anc 1370 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → ((𝑁 · 𝑛) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑁)))
2010, 19mpbird 247 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ≤ 𝐴)
212nnzd 11519 . . . . . . 7 (𝜑𝑁 ∈ ℤ)
22 elfzelz 12380 . . . . . . 7 (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) → 𝑛 ∈ ℤ)
23 zmulcl 11464 . . . . . . 7 ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑁 · 𝑛) ∈ ℤ)
2421, 22, 23syl2an 493 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ∈ ℤ)
25 flge 12646 . . . . . 6 ((𝐴 ∈ ℝ ∧ (𝑁 · 𝑛) ∈ ℤ) → ((𝑁 · 𝑛) ≤ 𝐴 ↔ (𝑁 · 𝑛) ≤ (⌊‘𝐴)))
2613, 24, 25syl2anc 694 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → ((𝑁 · 𝑛) ≤ 𝐴 ↔ (𝑁 · 𝑛) ≤ (⌊‘𝐴)))
2720, 26mpbid 222 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ≤ (⌊‘𝐴))
286flcld 12639 . . . . . 6 (𝜑 → (⌊‘𝐴) ∈ ℤ)
2928adantr 480 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (⌊‘𝐴) ∈ ℤ)
30 fznn 12446 . . . . 5 ((⌊‘𝐴) ∈ ℤ → ((𝑁 · 𝑛) ∈ (1...(⌊‘𝐴)) ↔ ((𝑁 · 𝑛) ∈ ℕ ∧ (𝑁 · 𝑛) ≤ (⌊‘𝐴))))
3129, 30syl 17 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → ((𝑁 · 𝑛) ∈ (1...(⌊‘𝐴)) ↔ ((𝑁 · 𝑛) ∈ ℕ ∧ (𝑁 · 𝑛) ≤ (⌊‘𝐴))))
325, 27, 31mpbir2and 977 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ∈ (1...(⌊‘𝐴)))
33 dvdsmul1 15050 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑁 ∥ (𝑁 · 𝑛))
3421, 22, 33syl2an 493 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑁 ∥ (𝑁 · 𝑛))
35 breq2 4689 . . . 4 (𝑥 = (𝑁 · 𝑛) → (𝑁𝑥𝑁 ∥ (𝑁 · 𝑛)))
3635elrab 3396 . . 3 ((𝑁 · 𝑛) ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥} ↔ ((𝑁 · 𝑛) ∈ (1...(⌊‘𝐴)) ∧ 𝑁 ∥ (𝑁 · 𝑛)))
3732, 34, 36sylanbrc 699 . 2 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})
38 breq2 4689 . . . . . . 7 (𝑥 = 𝑚 → (𝑁𝑥𝑁𝑚))
3938elrab 3396 . . . . . 6 (𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥} ↔ (𝑚 ∈ (1...(⌊‘𝐴)) ∧ 𝑁𝑚))
4039simprbi 479 . . . . 5 (𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥} → 𝑁𝑚)
4140adantl 481 . . . 4 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑁𝑚)
42 elrabi 3391 . . . . . . 7 (𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥} → 𝑚 ∈ (1...(⌊‘𝐴)))
4342adantl 481 . . . . . 6 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑚 ∈ (1...(⌊‘𝐴)))
44 elfznn 12408 . . . . . 6 (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
4543, 44syl 17 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑚 ∈ ℕ)
462adantr 480 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑁 ∈ ℕ)
47 nndivdvds 15036 . . . . 5 ((𝑚 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁𝑚 ↔ (𝑚 / 𝑁) ∈ ℕ))
4845, 46, 47syl2anc 694 . . . 4 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝑁𝑚 ↔ (𝑚 / 𝑁) ∈ ℕ))
4941, 48mpbid 222 . . 3 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝑚 / 𝑁) ∈ ℕ)
50 fznnfl 12701 . . . . . . 7 (𝐴 ∈ ℝ → (𝑚 ∈ (1...(⌊‘𝐴)) ↔ (𝑚 ∈ ℕ ∧ 𝑚𝐴)))
516, 50syl 17 . . . . . 6 (𝜑 → (𝑚 ∈ (1...(⌊‘𝐴)) ↔ (𝑚 ∈ ℕ ∧ 𝑚𝐴)))
5251simplbda 653 . . . . 5 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚𝐴)
5342, 52sylan2 490 . . . 4 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑚𝐴)
5445nnred 11073 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑚 ∈ ℝ)
556adantr 480 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝐴 ∈ ℝ)
5614adantr 480 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑁 ∈ ℝ)
5716adantr 480 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 0 < 𝑁)
58 lediv1 10926 . . . . 5 ((𝑚 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (𝑚𝐴 ↔ (𝑚 / 𝑁) ≤ (𝐴 / 𝑁)))
5954, 55, 56, 57, 58syl112anc 1370 . . . 4 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝑚𝐴 ↔ (𝑚 / 𝑁) ≤ (𝐴 / 𝑁)))
6053, 59mpbid 222 . . 3 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝑚 / 𝑁) ≤ (𝐴 / 𝑁))
617adantr 480 . . . 4 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝐴 / 𝑁) ∈ ℝ)
62 fznnfl 12701 . . . 4 ((𝐴 / 𝑁) ∈ ℝ → ((𝑚 / 𝑁) ∈ (1...(⌊‘(𝐴 / 𝑁))) ↔ ((𝑚 / 𝑁) ∈ ℕ ∧ (𝑚 / 𝑁) ≤ (𝐴 / 𝑁))))
6361, 62syl 17 . . 3 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → ((𝑚 / 𝑁) ∈ (1...(⌊‘(𝐴 / 𝑁))) ↔ ((𝑚 / 𝑁) ∈ ℕ ∧ (𝑚 / 𝑁) ≤ (𝐴 / 𝑁))))
6449, 60, 63mpbir2and 977 . 2 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝑚 / 𝑁) ∈ (1...(⌊‘(𝐴 / 𝑁))))
6545nncnd 11074 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑚 ∈ ℂ)
6665adantrl 752 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → 𝑚 ∈ ℂ)
672nncnd 11074 . . . . 5 (𝜑𝑁 ∈ ℂ)
6867adantr 480 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → 𝑁 ∈ ℂ)
6911nncnd 11074 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑛 ∈ ℂ)
7069adantrr 753 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → 𝑛 ∈ ℂ)
712nnne0d 11103 . . . . 5 (𝜑𝑁 ≠ 0)
7271adantr 480 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → 𝑁 ≠ 0)
7366, 68, 70, 72divmuld 10861 . . 3 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → ((𝑚 / 𝑁) = 𝑛 ↔ (𝑁 · 𝑛) = 𝑚))
74 eqcom 2658 . . 3 (𝑛 = (𝑚 / 𝑁) ↔ (𝑚 / 𝑁) = 𝑛)
75 eqcom 2658 . . 3 (𝑚 = (𝑁 · 𝑛) ↔ (𝑁 · 𝑛) = 𝑚)
7673, 74, 753bitr4g 303 . 2 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → (𝑛 = (𝑚 / 𝑁) ↔ 𝑚 = (𝑁 · 𝑛)))
771, 37, 64, 76f1o2d 6929 1 (𝜑𝐹:(1...(⌊‘(𝐴 / 𝑁)))–1-1-onto→{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  {crab 2945   class class class wbr 4685  cmpt 4762  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  cc 9972  cr 9973  0cc0 9974  1c1 9975   · cmul 9979   < clt 10112  cle 10113   / cdiv 10722  cn 11058  cz 11415  ...cfz 12364  cfl 12631  cdvds 15027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fl 12633  df-dvds 15028
This theorem is referenced by:  dvdsflsumcom  24959  logfac2  24987
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