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Theorem prmodvdslcmf 17080
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 17066 . . 3 (𝑁 ∈ ℕ0 → (#p𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
2 eqidd 2735 . . . . . 6 (𝑘 ∈ (1...𝑁) → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)))
3 simpr 484 . . . . . . . 8 ((𝑘 ∈ (1...𝑁) ∧ 𝑚 = 𝑘) → 𝑚 = 𝑘)
43eleq1d 2823 . . . . . . 7 ((𝑘 ∈ (1...𝑁) ∧ 𝑚 = 𝑘) → (𝑚 ∈ ℙ ↔ 𝑘 ∈ ℙ))
54, 3ifbieq1d 4554 . . . . . 6 ((𝑘 ∈ (1...𝑁) ∧ 𝑚 = 𝑘) → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
6 elfznn 13589 . . . . . 6 (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
7 1nn 12274 . . . . . . . 8 1 ∈ ℕ
87a1i 11 . . . . . . 7 (𝑘 ∈ (1...𝑁) → 1 ∈ ℕ)
96, 8ifcld 4576 . . . . . 6 (𝑘 ∈ (1...𝑁) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
102, 5, 6, 9fvmptd 7022 . . . . 5 (𝑘 ∈ (1...𝑁) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
1110eqcomd 2740 . . . 4 (𝑘 ∈ (1...𝑁) → if(𝑘 ∈ ℙ, 𝑘, 1) = ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))
1211prodeq2i 15950 . . 3 𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1) = ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)
131, 12eqtrdi 2790 . 2 (𝑁 ∈ ℕ0 → (#p𝑁) = ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))
14 fzfid 14010 . . . 4 (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin)
15 fz1ssnn 13591 . . . 4 (1...𝑁) ⊆ ℕ
1614, 15jctil 519 . . 3 (𝑁 ∈ ℕ0 → ((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin))
17 fzssz 13562 . . . . 5 (1...𝑁) ⊆ ℤ
1817a1i 11 . . . 4 (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ ℤ)
19 0nelfz1 13579 . . . . 5 0 ∉ (1...𝑁)
2019a1i 11 . . . 4 (𝑁 ∈ ℕ0 → 0 ∉ (1...𝑁))
21 lcmfn0cl 16659 . . . 4 (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin ∧ 0 ∉ (1...𝑁)) → (lcm‘(1...𝑁)) ∈ ℕ)
2218, 14, 20, 21syl3anc 1370 . . 3 (𝑁 ∈ ℕ0 → (lcm‘(1...𝑁)) ∈ ℕ)
23 id 22 . . . . . 6 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ)
247a1i 11 . . . . . 6 (𝑚 ∈ ℕ → 1 ∈ ℕ)
2523, 24ifcld 4576 . . . . 5 (𝑚 ∈ ℕ → if(𝑚 ∈ ℙ, 𝑚, 1) ∈ ℕ)
2625adantl 481 . . . 4 ((𝑁 ∈ ℕ0𝑚 ∈ ℕ) → if(𝑚 ∈ ℙ, 𝑚, 1) ∈ ℕ)
2726fmpttd 7134 . . 3 (𝑁 ∈ ℕ0 → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)):ℕ⟶ℕ)
28 simpr 484 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 𝑘 ∈ (1...𝑁))
2928adantr 480 . . . . . 6 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝑘})) → 𝑘 ∈ (1...𝑁))
30 eldifi 4140 . . . . . . 7 (𝑥 ∈ ((1...𝑁) ∖ {𝑘}) → 𝑥 ∈ (1...𝑁))
3130adantl 481 . . . . . 6 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝑘})) → 𝑥 ∈ (1...𝑁))
32 eldif 3972 . . . . . . . 8 (𝑥 ∈ ((1...𝑁) ∖ {𝑘}) ↔ (𝑥 ∈ (1...𝑁) ∧ ¬ 𝑥 ∈ {𝑘}))
33 velsn 4646 . . . . . . . . . . 11 (𝑥 ∈ {𝑘} ↔ 𝑥 = 𝑘)
3433biimpri 228 . . . . . . . . . 10 (𝑥 = 𝑘𝑥 ∈ {𝑘})
3534equcoms 2016 . . . . . . . . 9 (𝑘 = 𝑥𝑥 ∈ {𝑘})
3635necon3bi 2964 . . . . . . . 8 𝑥 ∈ {𝑘} → 𝑘𝑥)
3732, 36simplbiim 504 . . . . . . 7 (𝑥 ∈ ((1...𝑁) ∖ {𝑘}) → 𝑘𝑥)
3837adantl 481 . . . . . 6 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝑘})) → 𝑘𝑥)
39 eqid 2734 . . . . . . 7 (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))
4039fvprmselgcd1 17078 . . . . . 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 1370 . . . . 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 3143 . . . 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 3143 . . 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 2735 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)))
45 simpr 484 . . . . . . . 8 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑚 = 𝑘) → 𝑚 = 𝑘)
4645eleq1d 2823 . . . . . . 7 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑚 = 𝑘) → (𝑚 ∈ ℙ ↔ 𝑘 ∈ ℙ))
4746, 45ifbieq1d 4554 . . . . . 6 (((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) ∧ 𝑚 = 𝑘) → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
4815, 28sselid 3992 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ)
4917, 28sselid 3992 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℤ)
50 1zzd 12645 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 1 ∈ ℤ)
5149, 50ifcld 4576 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℤ)
5244, 47, 48, 51fvmptd 7022 . . . . 5 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
53 breq1 5150 . . . . . 6 (𝑥 = if(𝑘 ∈ ℙ, 𝑘, 1) → (𝑥 ∥ (lcm‘(1...𝑁)) ↔ if(𝑘 ∈ ℙ, 𝑘, 1) ∥ (lcm‘(1...𝑁))))
5416adantr 480 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → ((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin))
55172a1i 12 . . . . . . . 8 ((1...𝑁) ∈ Fin → ((1...𝑁) ⊆ ℕ → (1...𝑁) ⊆ ℤ))
5655imdistanri 569 . . . . . . 7 (((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin))
57 dvdslcmf 16664 . . . . . . 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 13565 . . . . . . . . 9 (𝑘 ∈ (1...𝑁) → 𝑁 ∈ (ℤ‘1))
6059adantl 481 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 𝑁 ∈ (ℤ‘1))
61 eluzfz1 13567 . . . . . . . 8 (𝑁 ∈ (ℤ‘1) → 1 ∈ (1...𝑁))
6260, 61syl 17 . . . . . . 7 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → 1 ∈ (1...𝑁))
6328, 62ifcld 4576 . . . . . 6 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ (1...𝑁))
6453, 58, 63rspcdva 3622 . . . . 5 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∥ (lcm‘(1...𝑁)))
6552, 64eqbrtrd 5169 . . . 4 ((𝑁 ∈ ℕ0𝑘 ∈ (1...𝑁)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
6665ralrimiva 3143 . . 3 (𝑁 ∈ ℕ0 → ∀𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
67 coprmproddvds 16696 . . 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 1378 . 2 (𝑁 ∈ ℕ0 → ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
6913, 68eqbrtrd 5169 1 (𝑁 ∈ ℕ0 → (#p𝑁) ∥ (lcm‘(1...𝑁)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1536  wcel 2105  wne 2937  wnel 3043  wral 3058  cdif 3959  wss 3962  ifcif 4530  {csn 4630   class class class wbr 5147  cmpt 5230  wf 6558  cfv 6562  (class class class)co 7430  Fincfn 8983  0cc0 11152  1c1 11153  cn 12263  0cn0 12523  cz 12610  cuz 12875  ...cfz 13543  cprod 15935  cdvds 16286   gcd cgcd 16527  lcmclcmf 16622  cprime 16704  #pcprmo 17064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-prod 15936  df-dvds 16287  df-gcd 16528  df-lcmf 16624  df-prm 16705  df-prmo 17065
This theorem is referenced by:  prmolelcmf  17081
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