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Theorem prmodvdslcmf 16438
 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 16424 . . 3 (𝑁 ∈ ℕ0 → (#p𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
2 eqidd 2759 . . . . . 6 (𝑘 ∈ (1...𝑁) → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)))
3 simpr 488 . . . . . . . 8 ((𝑘 ∈ (1...𝑁) ∧ 𝑚 = 𝑘) → 𝑚 = 𝑘)
43eleq1d 2836 . . . . . . 7 ((𝑘 ∈ (1...𝑁) ∧ 𝑚 = 𝑘) → (𝑚 ∈ ℙ ↔ 𝑘 ∈ ℙ))
54, 3ifbieq1d 4444 . . . . . 6 ((𝑘 ∈ (1...𝑁) ∧ 𝑚 = 𝑘) → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
6 elfznn 12985 . . . . . 6 (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
7 1nn 11685 . . . . . . . 8 1 ∈ ℕ
87a1i 11 . . . . . . 7 (𝑘 ∈ (1...𝑁) → 1 ∈ ℕ)
96, 8ifcld 4466 . . . . . 6 (𝑘 ∈ (1...𝑁) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
102, 5, 6, 9fvmptd 6766 . . . . 5 (𝑘 ∈ (1...𝑁) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
1110eqcomd 2764 . . . 4 (𝑘 ∈ (1...𝑁) → if(𝑘 ∈ ℙ, 𝑘, 1) = ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))
1211prodeq2i 15321 . . 3 𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1) = ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)
131, 12eqtrdi 2809 . 2 (𝑁 ∈ ℕ0 → (#p𝑁) = ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))
14 fzfid 13390 . . . 4 (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin)
15 fz1ssnn 12987 . . . 4 (1...𝑁) ⊆ ℕ
1614, 15jctil 523 . . 3 (𝑁 ∈ ℕ0 → ((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin))
17 fzssz 12958 . . . . 5 (1...𝑁) ⊆ ℤ
1817a1i 11 . . . 4 (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ ℤ)
19 0nelfz1 12975 . . . . 5 0 ∉ (1...𝑁)
2019a1i 11 . . . 4 (𝑁 ∈ ℕ0 → 0 ∉ (1...𝑁))
21 lcmfn0cl 16022 . . . 4 (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin ∧ 0 ∉ (1...𝑁)) → (lcm‘(1...𝑁)) ∈ ℕ)
2218, 14, 20, 21syl3anc 1368 . . 3 (𝑁 ∈ ℕ0 → (lcm‘(1...𝑁)) ∈ ℕ)
23 id 22 . . . . . 6 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ)
247a1i 11 . . . . . 6 (𝑚 ∈ ℕ → 1 ∈ ℕ)
2523, 24ifcld 4466 . . . . 5 (𝑚 ∈ ℕ → if(𝑚 ∈ ℙ, 𝑚, 1) ∈ ℕ)
2625adantl 485 . . . 4 ((𝑁 ∈ ℕ0𝑚 ∈ ℕ) → if(𝑚 ∈ ℙ, 𝑚, 1) ∈ ℕ)
2726fmpttd 6870 . . 3 (𝑁 ∈ ℕ0 → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)):ℕ⟶ℕ)
28 simpr 488 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 𝑘 ∈ (1...𝑁))
2928adantr 484 . . . . . 6 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝑘})) → 𝑘 ∈ (1...𝑁))
30 eldifi 4032 . . . . . . 7 (𝑥 ∈ ((1...𝑁) ∖ {𝑘}) → 𝑥 ∈ (1...𝑁))
3130adantl 485 . . . . . 6 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝑘})) → 𝑥 ∈ (1...𝑁))
32 eldif 3868 . . . . . . . 8 (𝑥 ∈ ((1...𝑁) ∖ {𝑘}) ↔ (𝑥 ∈ (1...𝑁) ∧ ¬ 𝑥 ∈ {𝑘}))
33 velsn 4538 . . . . . . . . . . 11 (𝑥 ∈ {𝑘} ↔ 𝑥 = 𝑘)
3433biimpri 231 . . . . . . . . . 10 (𝑥 = 𝑘𝑥 ∈ {𝑘})
3534equcoms 2027 . . . . . . . . 9 (𝑘 = 𝑥𝑥 ∈ {𝑘})
3635necon3bi 2977 . . . . . . . 8 𝑥 ∈ {𝑘} → 𝑘𝑥)
3732, 36simplbiim 508 . . . . . . 7 (𝑥 ∈ ((1...𝑁) ∖ {𝑘}) → 𝑘𝑥)
3837adantl 485 . . . . . 6 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝑘})) → 𝑘𝑥)
39 eqid 2758 . . . . . . 7 (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))
4039fvprmselgcd1 16436 . . . . . 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)
4129, 31, 38, 40syl3anc 1368 . . . . 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)
4241ralrimiva 3113 . . . 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)
4342ralrimiva 3113 . . 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)
44 eqidd 2759 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)))
45 simpr 488 . . . . . . . 8 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑚 = 𝑘) → 𝑚 = 𝑘)
4645eleq1d 2836 . . . . . . 7 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑚 = 𝑘) → (𝑚 ∈ ℙ ↔ 𝑘 ∈ ℙ))
4746, 45ifbieq1d 4444 . . . . . 6 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑚 = 𝑘) → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
4815, 28sseldi 3890 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ)
4917, 28sseldi 3890 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℤ)
50 1zzd 12052 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 1 ∈ ℤ)
5149, 50ifcld 4466 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℤ)
5244, 47, 48, 51fvmptd 6766 . . . . 5 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
53 breq1 5035 . . . . . 6 (𝑥 = if(𝑘 ∈ ℙ, 𝑘, 1) → (𝑥 ∥ (lcm‘(1...𝑁)) ↔ if(𝑘 ∈ ℙ, 𝑘, 1) ∥ (lcm‘(1...𝑁))))
5416adantr 484 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → ((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin))
55172a1i 12 . . . . . . . 8 ((1...𝑁) ∈ Fin → ((1...𝑁) ⊆ ℕ → (1...𝑁) ⊆ ℤ))
5655imdistanri 573 . . . . . . 7 (((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin))
57 dvdslcmf 16027 . . . . . . 7 (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin) → ∀𝑥 ∈ (1...𝑁)𝑥 ∥ (lcm‘(1...𝑁)))
5854, 56, 573syl 18 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → ∀𝑥 ∈ (1...𝑁)𝑥 ∥ (lcm‘(1...𝑁)))
59 elfzuz2 12961 . . . . . . . . 9 (𝑘 ∈ (1...𝑁) → 𝑁 ∈ (ℤ‘1))
6059adantl 485 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 𝑁 ∈ (ℤ‘1))
61 eluzfz1 12963 . . . . . . . 8 (𝑁 ∈ (ℤ‘1) → 1 ∈ (1...𝑁))
6260, 61syl 17 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 1 ∈ (1...𝑁))
6328, 62ifcld 4466 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ (1...𝑁))
6453, 58, 63rspcdva 3543 . . . . 5 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∥ (lcm‘(1...𝑁)))
6552, 64eqbrtrd 5054 . . . 4 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
6665ralrimiva 3113 . . 3 (𝑁 ∈ ℕ0 → ∀𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
67 coprmproddvds 16059 . . 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)))
6816, 22, 27, 43, 66, 67syl122anc 1376 . 2 (𝑁 ∈ ℕ0 → ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
6913, 68eqbrtrd 5054 1 (𝑁 ∈ ℕ0 → (#p𝑁) ∥ (lcm‘(1...𝑁)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951   ∉ wnel 3055  ∀wral 3070   ∖ cdif 3855   ⊆ wss 3858  ifcif 4420  {csn 4522   class class class wbr 5032   ↦ cmpt 5112  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150  Fincfn 8527  0cc0 10575  1c1 10576  ℕcn 11674  ℕ0cn0 11934  ℤcz 12020  ℤ≥cuz 12282  ...cfz 12939  ∏cprod 15307   ∥ cdvds 15655   gcd cgcd 15893  lcmclcmf 15985  ℙcprime 16067  #pcprmo 16422 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-inf2 9137  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652  ax-pre-sup 10653 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-se 5484  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-isom 6344  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-2o 8113  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-sup 8939  df-inf 8940  df-oi 9007  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-div 11336  df-nn 11675  df-2 11737  df-3 11738  df-n0 11935  df-z 12021  df-uz 12283  df-rp 12431  df-fz 12940  df-fzo 13083  df-fl 13211  df-mod 13287  df-seq 13419  df-exp 13480  df-hash 13741  df-cj 14506  df-re 14507  df-im 14508  df-sqrt 14642  df-abs 14643  df-clim 14893  df-prod 15308  df-dvds 15656  df-gcd 15894  df-lcmf 15987  df-prm 16068  df-prmo 16423 This theorem is referenced by:  prmolelcmf  16439
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