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Theorem prmodvdslcmf 15694
Description: The primorial of a nonnegative integer divides the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) (Revised by AV, 29-Aug-2020.)
Assertion
Ref Expression
prmodvdslcmf (𝑁 ∈ ℕ0 → (#p𝑁) ∥ (lcm‘(1...𝑁)))

Proof of Theorem prmodvdslcmf
Dummy variables 𝑘 𝑚 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prmoval 15680 . . 3 (𝑁 ∈ ℕ0 → (#p𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
2 eqidd 2622 . . . . . 6 (𝑘 ∈ (1...𝑁) → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)))
3 simpr 477 . . . . . . . 8 ((𝑘 ∈ (1...𝑁) ∧ 𝑚 = 𝑘) → 𝑚 = 𝑘)
43eleq1d 2683 . . . . . . 7 ((𝑘 ∈ (1...𝑁) ∧ 𝑚 = 𝑘) → (𝑚 ∈ ℙ ↔ 𝑘 ∈ ℙ))
54, 3ifbieq1d 4087 . . . . . 6 ((𝑘 ∈ (1...𝑁) ∧ 𝑚 = 𝑘) → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
6 elfznn 12328 . . . . . 6 (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
7 1nn 10991 . . . . . . . 8 1 ∈ ℕ
87a1i 11 . . . . . . 7 (𝑘 ∈ (1...𝑁) → 1 ∈ ℕ)
96, 8ifcld 4109 . . . . . 6 (𝑘 ∈ (1...𝑁) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
102, 5, 6, 9fvmptd 6255 . . . . 5 (𝑘 ∈ (1...𝑁) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
1110eqcomd 2627 . . . 4 (𝑘 ∈ (1...𝑁) → if(𝑘 ∈ ℙ, 𝑘, 1) = ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))
1211prodeq2i 14593 . . 3 𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1) = ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)
131, 12syl6eq 2671 . 2 (𝑁 ∈ ℕ0 → (#p𝑁) = ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))
14 fzfid 12728 . . . 4 (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin)
15 fz1ssnn 12330 . . . 4 (1...𝑁) ⊆ ℕ
1614, 15jctil 559 . . 3 (𝑁 ∈ ℕ0 → ((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin))
17 fzssz 12301 . . . . 5 (1...𝑁) ⊆ ℤ
1817a1i 11 . . . 4 (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ ℤ)
19 0nelfz1 12318 . . . . 5 0 ∉ (1...𝑁)
2019a1i 11 . . . 4 (𝑁 ∈ ℕ0 → 0 ∉ (1...𝑁))
21 lcmfn0cl 15282 . . . 4 (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin ∧ 0 ∉ (1...𝑁)) → (lcm‘(1...𝑁)) ∈ ℕ)
2218, 14, 20, 21syl3anc 1323 . . 3 (𝑁 ∈ ℕ0 → (lcm‘(1...𝑁)) ∈ ℕ)
23 id 22 . . . . . 6 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ)
247a1i 11 . . . . . 6 (𝑚 ∈ ℕ → 1 ∈ ℕ)
2523, 24ifcld 4109 . . . . 5 (𝑚 ∈ ℕ → if(𝑚 ∈ ℙ, 𝑚, 1) ∈ ℕ)
2625adantl 482 . . . 4 ((𝑁 ∈ ℕ0𝑚 ∈ ℕ) → if(𝑚 ∈ ℙ, 𝑚, 1) ∈ ℕ)
27 eqid 2621 . . . 4 (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))
2826, 27fmptd 6351 . . 3 (𝑁 ∈ ℕ0 → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)):ℕ⟶ℕ)
29 simpr 477 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 𝑘 ∈ (1...𝑁))
3029adantr 481 . . . . . 6 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝑘})) → 𝑘 ∈ (1...𝑁))
31 eldifi 3716 . . . . . . 7 (𝑥 ∈ ((1...𝑁) ∖ {𝑘}) → 𝑥 ∈ (1...𝑁))
3231adantl 482 . . . . . 6 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝑘})) → 𝑥 ∈ (1...𝑁))
33 eldif 3570 . . . . . . . 8 (𝑥 ∈ ((1...𝑁) ∖ {𝑘}) ↔ (𝑥 ∈ (1...𝑁) ∧ ¬ 𝑥 ∈ {𝑘}))
34 velsn 4171 . . . . . . . . . . . 12 (𝑥 ∈ {𝑘} ↔ 𝑥 = 𝑘)
3534biimpri 218 . . . . . . . . . . 11 (𝑥 = 𝑘𝑥 ∈ {𝑘})
3635equcoms 1944 . . . . . . . . . 10 (𝑘 = 𝑥𝑥 ∈ {𝑘})
3736necon3bi 2816 . . . . . . . . 9 𝑥 ∈ {𝑘} → 𝑘𝑥)
3837adantl 482 . . . . . . . 8 ((𝑥 ∈ (1...𝑁) ∧ ¬ 𝑥 ∈ {𝑘}) → 𝑘𝑥)
3933, 38sylbi 207 . . . . . . 7 (𝑥 ∈ ((1...𝑁) ∖ {𝑘}) → 𝑘𝑥)
4039adantl 482 . . . . . 6 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝑘})) → 𝑘𝑥)
4127fvprmselgcd1 15692 . . . . . 6 ((𝑘 ∈ (1...𝑁) ∧ 𝑥 ∈ (1...𝑁) ∧ 𝑘𝑥) → (((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) gcd ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑥)) = 1)
4230, 32, 40, 41syl3anc 1323 . . . . 5 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝑘})) → (((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) gcd ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑥)) = 1)
4342ralrimiva 2962 . . . 4 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → ∀𝑥 ∈ ((1...𝑁) ∖ {𝑘})(((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) gcd ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑥)) = 1)
4443ralrimiva 2962 . . 3 (𝑁 ∈ ℕ0 → ∀𝑘 ∈ (1...𝑁)∀𝑥 ∈ ((1...𝑁) ∖ {𝑘})(((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) gcd ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑥)) = 1)
45 eqidd 2622 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)))
46 simpr 477 . . . . . . . 8 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑚 = 𝑘) → 𝑚 = 𝑘)
4746eleq1d 2683 . . . . . . 7 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑚 = 𝑘) → (𝑚 ∈ ℙ ↔ 𝑘 ∈ ℙ))
4847, 46ifbieq1d 4087 . . . . . 6 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑚 = 𝑘) → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
4915, 29sseldi 3586 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ)
5017, 29sseldi 3586 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℤ)
51 1zzd 11368 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 1 ∈ ℤ)
5250, 51ifcld 4109 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℤ)
5345, 48, 49, 52fvmptd 6255 . . . . 5 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
54 elfzuz2 12304 . . . . . . . . 9 (𝑘 ∈ (1...𝑁) → 𝑁 ∈ (ℤ‘1))
5554adantl 482 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 𝑁 ∈ (ℤ‘1))
56 eluzfz1 12306 . . . . . . . 8 (𝑁 ∈ (ℤ‘1) → 1 ∈ (1...𝑁))
5755, 56syl 17 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 1 ∈ (1...𝑁))
5829, 57ifcld 4109 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ (1...𝑁))
5916adantr 481 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → ((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin))
60172a1i 12 . . . . . . . 8 ((1...𝑁) ∈ Fin → ((1...𝑁) ⊆ ℕ → (1...𝑁) ⊆ ℤ))
6160imdistanri 726 . . . . . . 7 (((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin))
62 dvdslcmf 15287 . . . . . . 7 (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin) → ∀𝑥 ∈ (1...𝑁)𝑥 ∥ (lcm‘(1...𝑁)))
6359, 61, 623syl 18 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → ∀𝑥 ∈ (1...𝑁)𝑥 ∥ (lcm‘(1...𝑁)))
64 breq1 4626 . . . . . . 7 (𝑥 = if(𝑘 ∈ ℙ, 𝑘, 1) → (𝑥 ∥ (lcm‘(1...𝑁)) ↔ if(𝑘 ∈ ℙ, 𝑘, 1) ∥ (lcm‘(1...𝑁))))
6564rspcv 3295 . . . . . 6 (if(𝑘 ∈ ℙ, 𝑘, 1) ∈ (1...𝑁) → (∀𝑥 ∈ (1...𝑁)𝑥 ∥ (lcm‘(1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∥ (lcm‘(1...𝑁))))
6658, 63, 65sylc 65 . . . . 5 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∥ (lcm‘(1...𝑁)))
6753, 66eqbrtrd 4645 . . . 4 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
6867ralrimiva 2962 . . 3 (𝑁 ∈ ℕ0 → ∀𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
69 coprmproddvds 15320 . . 3 ((((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin) ∧ ((lcm‘(1...𝑁)) ∈ ℕ ∧ (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)):ℕ⟶ℕ) ∧ (∀𝑘 ∈ (1...𝑁)∀𝑥 ∈ ((1...𝑁) ∖ {𝑘})(((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) gcd ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑥)) = 1 ∧ ∀𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))) → ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
7016, 22, 28, 44, 68, 69syl122anc 1332 . 2 (𝑁 ∈ ℕ0 → ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
7113, 70eqbrtrd 4645 1 (𝑁 ∈ ℕ0 → (#p𝑁) ∥ (lcm‘(1...𝑁)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  wnel 2893  wral 2908  cdif 3557  wss 3560  ifcif 4064  {csn 4155   class class class wbr 4623  cmpt 4683  wf 5853  cfv 5857  (class class class)co 6615  Fincfn 7915  0cc0 9896  1c1 9897  cn 10980  0cn0 11252  cz 11337  cuz 11647  ...cfz 12284  cprod 14579  cdvds 14926   gcd cgcd 15159  lcmclcmf 15245  cprime 15328  #pcprmo 15678
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 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
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 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-sup 8308  df-inf 8309  df-oi 8375  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-n0 11253  df-z 11338  df-uz 11648  df-rp 11793  df-fz 12285  df-fzo 12423  df-fl 12549  df-mod 12625  df-seq 12758  df-exp 12817  df-hash 13074  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-clim 14169  df-prod 14580  df-dvds 14927  df-gcd 15160  df-lcmf 15247  df-prm 15329  df-prmo 15679
This theorem is referenced by:  prmolelcmf  15695
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