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Theorem dvdsflf1o 26536
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 5109 . . 3 (𝑥 = (𝑁 · 𝑛) → (𝑁𝑥𝑁 ∥ (𝑁 · 𝑛)))
3 dvdsflf1o.2 . . . . 5 (𝜑𝑁 ∈ ℕ)
4 elfznn 13470 . . . . 5 (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) → 𝑛 ∈ ℕ)
5 nnmulcl 12177 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑁 · 𝑛) ∈ ℕ)
63, 4, 5syl2an 596 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ∈ ℕ)
7 dvdsflf1o.1 . . . . . . . . 9 (𝜑𝐴 ∈ ℝ)
87, 3nndivred 12207 . . . . . . . 8 (𝜑 → (𝐴 / 𝑁) ∈ ℝ)
9 fznnfl 13767 . . . . . . . 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 500 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑛 ≤ (𝐴 / 𝑁))
124adantl 482 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑛 ∈ ℕ)
1312nnred 12168 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑛 ∈ ℝ)
147adantr 481 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝐴 ∈ ℝ)
153nnred 12168 . . . . . . . 8 (𝜑𝑁 ∈ ℝ)
1615adantr 481 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑁 ∈ ℝ)
173nngt0d 12202 . . . . . . . 8 (𝜑 → 0 < 𝑁)
1817adantr 481 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 0 < 𝑁)
19 lemuldiv2 12036 . . . . . . 7 ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → ((𝑁 · 𝑛) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑁)))
2013, 14, 16, 18, 19syl112anc 1374 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → ((𝑁 · 𝑛) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑁)))
2111, 20mpbird 256 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ≤ 𝐴)
223nnzd 12526 . . . . . . 7 (𝜑𝑁 ∈ ℤ)
23 elfzelz 13441 . . . . . . 7 (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) → 𝑛 ∈ ℤ)
24 zmulcl 12552 . . . . . . 7 ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑁 · 𝑛) ∈ ℤ)
2522, 23, 24syl2an 596 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ∈ ℤ)
26 flge 13710 . . . . . 6 ((𝐴 ∈ ℝ ∧ (𝑁 · 𝑛) ∈ ℤ) → ((𝑁 · 𝑛) ≤ 𝐴 ↔ (𝑁 · 𝑛) ≤ (⌊‘𝐴)))
2714, 25, 26syl2anc 584 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → ((𝑁 · 𝑛) ≤ 𝐴 ↔ (𝑁 · 𝑛) ≤ (⌊‘𝐴)))
2821, 27mpbid 231 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ≤ (⌊‘𝐴))
297flcld 13703 . . . . . 6 (𝜑 → (⌊‘𝐴) ∈ ℤ)
3029adantr 481 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (⌊‘𝐴) ∈ ℤ)
31 fznn 13509 . . . . 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 711 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ∈ (1...(⌊‘𝐴)))
34 dvdsmul1 16160 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑁 ∥ (𝑁 · 𝑛))
3522, 23, 34syl2an 596 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑁 ∥ (𝑁 · 𝑛))
362, 33, 35elrabd 3647 . 2 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → (𝑁 · 𝑛) ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})
37 breq2 5109 . . . . . . 7 (𝑥 = 𝑚 → (𝑁𝑥𝑁𝑚))
3837elrab 3645 . . . . . 6 (𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥} ↔ (𝑚 ∈ (1...(⌊‘𝐴)) ∧ 𝑁𝑚))
3938simprbi 497 . . . . 5 (𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥} → 𝑁𝑚)
4039adantl 482 . . . 4 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑁𝑚)
41 elrabi 3639 . . . . . . 7 (𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥} → 𝑚 ∈ (1...(⌊‘𝐴)))
4241adantl 482 . . . . . 6 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑚 ∈ (1...(⌊‘𝐴)))
43 elfznn 13470 . . . . . 6 (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
4442, 43syl 17 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑚 ∈ ℕ)
453adantr 481 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑁 ∈ ℕ)
46 nndivdvds 16145 . . . . 5 ((𝑚 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁𝑚 ↔ (𝑚 / 𝑁) ∈ ℕ))
4744, 45, 46syl2anc 584 . . . 4 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝑁𝑚 ↔ (𝑚 / 𝑁) ∈ ℕ))
4840, 47mpbid 231 . . 3 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝑚 / 𝑁) ∈ ℕ)
49 fznnfl 13767 . . . . . . 7 (𝐴 ∈ ℝ → (𝑚 ∈ (1...(⌊‘𝐴)) ↔ (𝑚 ∈ ℕ ∧ 𝑚𝐴)))
507, 49syl 17 . . . . . 6 (𝜑 → (𝑚 ∈ (1...(⌊‘𝐴)) ↔ (𝑚 ∈ ℕ ∧ 𝑚𝐴)))
5150simplbda 500 . . . . 5 ((𝜑𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚𝐴)
5241, 51sylan2 593 . . . 4 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑚𝐴)
5344nnred 12168 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑚 ∈ ℝ)
547adantr 481 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝐴 ∈ ℝ)
5515adantr 481 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑁 ∈ ℝ)
5617adantr 481 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 0 < 𝑁)
57 lediv1 12020 . . . . 5 ((𝑚 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (𝑚𝐴 ↔ (𝑚 / 𝑁) ≤ (𝐴 / 𝑁)))
5853, 54, 55, 56, 57syl112anc 1374 . . . 4 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝑚𝐴 ↔ (𝑚 / 𝑁) ≤ (𝐴 / 𝑁)))
5952, 58mpbid 231 . . 3 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝑚 / 𝑁) ≤ (𝐴 / 𝑁))
608adantr 481 . . . 4 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝐴 / 𝑁) ∈ ℝ)
61 fznnfl 13767 . . . 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 711 . 2 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → (𝑚 / 𝑁) ∈ (1...(⌊‘(𝐴 / 𝑁))))
6444nncnd 12169 . . . . 5 ((𝜑𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥}) → 𝑚 ∈ ℂ)
6564adantrl 714 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → 𝑚 ∈ ℂ)
663nncnd 12169 . . . . 5 (𝜑𝑁 ∈ ℂ)
6766adantr 481 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → 𝑁 ∈ ℂ)
6812nncnd 12169 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁)))) → 𝑛 ∈ ℂ)
6968adantrr 715 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → 𝑛 ∈ ℂ)
703nnne0d 12203 . . . . 5 (𝜑𝑁 ≠ 0)
7170adantr 481 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → 𝑁 ≠ 0)
7265, 67, 69, 71divmuld 11953 . . 3 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → ((𝑚 / 𝑁) = 𝑛 ↔ (𝑁 · 𝑛) = 𝑚))
73 eqcom 2743 . . 3 (𝑛 = (𝑚 / 𝑁) ↔ (𝑚 / 𝑁) = 𝑛)
74 eqcom 2743 . . 3 (𝑚 = (𝑁 · 𝑛) ↔ (𝑁 · 𝑛) = 𝑚)
7572, 73, 743bitr4g 313 . 2 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑁))) ∧ 𝑚 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})) → (𝑛 = (𝑚 / 𝑁) ↔ 𝑚 = (𝑁 · 𝑛)))
761, 36, 63, 75f1o2d 7607 1 (𝜑𝐹:(1...(⌊‘(𝐴 / 𝑁)))–1-1-onto→{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑁𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  {crab 3407   class class class wbr 5105  cmpt 5188  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  cc 11049  cr 11050  0cc0 11051  1c1 11052   · cmul 11056   < clt 11189  cle 11190   / cdiv 11812  cn 12153  cz 12499  ...cfz 13424  cfl 13695  cdvds 16136
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fl 13697  df-dvds 16137
This theorem is referenced by:  dvdsflsumcom  26537  logfac2  26565
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