Theorem prmodvdslcmf 17083

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 17069	. . 3 ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
2		eqidd 2763	. . . . . 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 2847	. . . . . . 7 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑚 = 𝑘) → (𝑚 ∈ ℙ ↔ 𝑘 ∈ ℙ))
5	4, 3	ifbieq1d 4505	. . . . . 6 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑚 = 𝑘) → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
6		elfznn 13558	. . . . . 6 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
7		1nn 12221	. . . . . . . 8 ⊢ 1 ∈ ℕ
8	7	a1i 11	. . . . . . 7 ⊢ (𝑘 ∈ (1...𝑁) → 1 ∈ ℕ)
9	6, 8	ifcld 4527	. . . . . 6 ⊢ (𝑘 ∈ (1...𝑁) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
10	2, 5, 6, 9	fvmptd 6983	. . . . 5 ⊢ (𝑘 ∈ (1...𝑁) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
11	10	eqcomd 2768	. . . 4 ⊢ (𝑘 ∈ (1...𝑁) → if(𝑘 ∈ ℙ, 𝑘, 1) = ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))
12	11	prodeq2i 15948	. . 3 ⊢ ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1) = ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)
13	1, 12	eqtrdi 2813	. 2 ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) = ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))
14		fzfid 13986	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin)
15		fz1ssnn 13560	. . . 4 ⊢ (1...𝑁) ⊆ ℕ
16	14, 15	jctil 527	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin))
17		fzssz 13531	. . . . 5 ⊢ (1...𝑁) ⊆ ℤ
18	17	a1i 11	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ ℤ)
19		0nelfz1 13548	. . . . 5 ⊢ 0 ∉ (1...𝑁)
20	19	a1i 11	. . . 4 ⊢ (𝑁 ∈ ℕ0 → 0 ∉ (1...𝑁))
21		lcmfn0cl 16660	. . . 4 ⊢ (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin ∧ 0 ∉ (1...𝑁)) → (lcm‘(1...𝑁)) ∈ ℕ)
22	18, 14, 20, 21	syl3anc 1390	. . 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 4527	. . . . 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 7096	. . 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 4084	. . . . . . 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 3914	. . . . . . . 8 ⊢ (𝑥 ∈ ((1...𝑁) ∖ {𝑘}) ↔ (𝑥 ∈ (1...𝑁) ∧ ¬ 𝑥 ∈ {𝑘}))
33		velsn 4598	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑘} ↔ 𝑥 = 𝑘)
34	33	biimpri 230	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → 𝑥 ∈ {𝑘})
35	34	equcoms 2040	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → 𝑥 ∈ {𝑘})
36	35	necon3bi 2983	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ {𝑘} → 𝑘 ≠ 𝑥)
37	32, 36	simplbiim 512	. . . . . . 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 2762	. . . . . . 7 ⊢ (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))
40	39	fvprmselgcd1 17081	. . . . . 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 1390	. . . . 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 3154	. . . 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 3154	. . 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 2763	. . . . . 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 2847	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑚 = 𝑘) → (𝑚 ∈ ℙ ↔ 𝑘 ∈ ℙ))
47	46, 45	ifbieq1d 4505	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑚 = 𝑘) → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
48	15, 28	sselid 3934	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ)
49	17, 28	sselid 3934	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℤ)
50		1zzd 12602	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 1 ∈ ℤ)
51	49, 50	ifcld 4527	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℤ)
52	44, 47, 48, 51	fvmptd 6983	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
53		breq1 5103	. . . . . 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 577	. . . . . . 7 ⊢ (((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin))
57		dvdslcmf 16665	. . . . . . 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 13534	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → 𝑁 ∈ (ℤ≥‘1))
60	59	adantl 485	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑁 ∈ (ℤ≥‘1))
61		eluzfz1 13536	. . . . . . . 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 4527	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ (1...𝑁))
64	53, 58, 63	rspcdva 3582	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∥ (lcm‘(1...𝑁)))
65	52, 64	eqbrtrd 5122	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
66	65	ralrimiva 3154	. . 3 ⊢ (𝑁 ∈ ℕ0 → ∀𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
67		coprmproddvds 16697	. . 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 1398	. 2 ⊢ (𝑁 ∈ ℕ0 → ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
69	13, 68	eqbrtrd 5122	1 ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) ∥ (lcm‘(1...𝑁)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142   ≠ wne 2957   ∉ wnel 3061  ∀wral 3076   ∖ cdif 3901   ⊆ wss 3904  ifcif 4480  {csn 4582   class class class wbr 5100   ↦ cmpt 5181  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396  Fincfn 8927  0cc0 11073  1c1 11074  ℕcn 12210  ℕ₀cn0 12481  ℤcz 12568  ℤ_≥cuz 12839  ...cfz 13512  ∏cprod 15933   ∥ cdvds 16286   gcd cgcd 16528  lcmclcmf 16623  ℙcprime 16705  #_pcprmo 17067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-inf2 9596  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150  ax-pre-sup 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-sup 9388  df-inf 9389  df-oi 9458  df-card 9897  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-div 11845  df-nn 12211  df-2 12280  df-3 12281  df-n0 12482  df-z 12569  df-uz 12840  df-rp 12994  df-fz 13513  df-fzo 13660  df-fl 13802  df-mod 13880  df-seq 14015  df-exp 14075  df-hash 14344  df-cj 15126  df-re 15127  df-im 15128  df-sqrt 15262  df-abs 15263  df-clim 15515  df-prod 15934  df-dvds 16287  df-gcd 16529  df-lcmf 16625  df-prm 16706  df-prmo 17068

This theorem is referenced by:  prmolelcmf  17084