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 → (#p‘𝑁) ∥ (lcm‘(1...𝑁)))

Proof of Theorem prmodvdslcmf

Dummy variables 𝑘 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmoval 16424	. . 3 ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) = ∏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)
4	3	eleq1d 2836	. . . . . . 7 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑚 = 𝑘) → (𝑚 ∈ ℙ ↔ 𝑘 ∈ ℙ))
5	4, 3	ifbieq1d 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
8	7	a1i 11	. . . . . . 7 ⊢ (𝑘 ∈ (1...𝑁) → 1 ∈ ℕ)
9	6, 8	ifcld 4466	. . . . . 6 ⊢ (𝑘 ∈ (1...𝑁) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
10	2, 5, 6, 9	fvmptd 6766	. . . . 5 ⊢ (𝑘 ∈ (1...𝑁) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
11	10	eqcomd 2764	. . . 4 ⊢ (𝑘 ∈ (1...𝑁) → if(𝑘 ∈ ℙ, 𝑘, 1) = ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))
12	11	prodeq2i 15321	. . 3 ⊢ ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1) = ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)
13	1, 12	eqtrdi 2809	. 2 ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) = ∏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
16	14, 15	jctil 523	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin))
17		fzssz 12958	. . . . 5 ⊢ (1...𝑁) ⊆ ℤ
18	17	a1i 11	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ ℤ)
19		0nelfz1 12975	. . . . 5 ⊢ 0 ∉ (1...𝑁)
20	19	a1i 11	. . . 4 ⊢ (𝑁 ∈ ℕ0 → 0 ∉ (1...𝑁))
21		lcmfn0cl 16022	. . . 4 ⊢ (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin ∧ 0 ∉ (1...𝑁)) → (lcm‘(1...𝑁)) ∈ ℕ)
22	18, 14, 20, 21	syl3anc 1368	. . 3 ⊢ (𝑁 ∈ ℕ0 → (lcm‘(1...𝑁)) ∈ ℕ)
23		id 22	. . . . . 6 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ)
24	7	a1i 11	. . . . . 6 ⊢ (𝑚 ∈ ℕ → 1 ∈ ℕ)
25	23, 24	ifcld 4466	. . . . 5 ⊢ (𝑚 ∈ ℕ → if(𝑚 ∈ ℙ, 𝑚, 1) ∈ ℕ)
26	25	adantl 485	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑚 ∈ ℕ) → if(𝑚 ∈ ℙ, 𝑚, 1) ∈ ℕ)
27	26	fmpttd 6870	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)):ℕ⟶ℕ)
28		simpr 488	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ (1...𝑁))
29	28	adantr 484	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝑘})) → 𝑘 ∈ (1...𝑁))
30		eldifi 4032	. . . . . . 7 ⊢ (𝑥 ∈ ((1...𝑁) ∖ {𝑘}) → 𝑥 ∈ (1...𝑁))
31	30	adantl 485	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (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)
34	33	biimpri 231	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → 𝑥 ∈ {𝑘})
35	34	equcoms 2027	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → 𝑥 ∈ {𝑘})
36	35	necon3bi 2977	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ {𝑘} → 𝑘 ≠ 𝑥)
37	32, 36	simplbiim 508	. . . . . . 7 ⊢ (𝑥 ∈ ((1...𝑁) ∖ {𝑘}) → 𝑘 ≠ 𝑥)
38	37	adantl 485	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝑘})) → 𝑘 ≠ 𝑥)
39		eqid 2758	. . . . . . 7 ⊢ (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))
40	39	fvprmselgcd1 16436	. . . . . 6 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑥 ∈ (1...𝑁) ∧ 𝑘 ≠ 𝑥) → (((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) gcd ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑥)) = 1)
41	29, 31, 38, 40	syl3anc 1368	. . . . 5 ⊢ (((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)
42	41	ralrimiva 3113	. . . 4 ⊢ ((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)
43	42	ralrimiva 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 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)))
45		simpr 488	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑚 = 𝑘) → 𝑚 = 𝑘)
46	45	eleq1d 2836	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑚 = 𝑘) → (𝑚 ∈ ℙ ↔ 𝑘 ∈ ℙ))
47	46, 45	ifbieq1d 4444	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑚 = 𝑘) → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
48	15, 28	sseldi 3890	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ)
49	17, 28	sseldi 3890	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℤ)
50		1zzd 12052	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 1 ∈ ℤ)
51	49, 50	ifcld 4466	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℤ)
52	44, 47, 48, 51	fvmptd 6766	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (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))))
54	16	adantr 484	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → ((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin))
55	17	2a1i 12	. . . . . . . 8 ⊢ ((1...𝑁) ∈ Fin → ((1...𝑁) ⊆ ℕ → (1...𝑁) ⊆ ℤ))
56	55	imdistanri 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)))
58	54, 56, 57	3syl 18	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → ∀𝑥 ∈ (1...𝑁)𝑥 ∥ (lcm‘(1...𝑁)))
59		elfzuz2 12961	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → 𝑁 ∈ (ℤ≥‘1))
60	59	adantl 485	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑁 ∈ (ℤ≥‘1))
61		eluzfz1 12963	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁))
62	60, 61	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 1 ∈ (1...𝑁))
63	28, 62	ifcld 4466	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ (1...𝑁))
64	53, 58, 63	rspcdva 3543	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∥ (lcm‘(1...𝑁)))
65	52, 64	eqbrtrd 5054	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
66	65	ralrimiva 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)))
68	16, 22, 27, 43, 66, 67	syl122anc 1376	. 2 ⊢ (𝑁 ∈ ℕ0 → ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
69	13, 68	eqbrtrd 5054	1 ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) ∥ (lcm‘(1...𝑁)))

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