Theorem prmodvdslcmf 17102

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 17088	. . 3 ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
2		eqidd 2764	. . . . . 6 ⊢ (𝑘 ∈ (1...𝑁) → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)))
3		simpr 489	. . . . . . . 8 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑚 = 𝑘) → 𝑚 = 𝑘)
4	3	eleq1d 2848	. . . . . . 7 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑚 = 𝑘) → (𝑚 ∈ ℙ ↔ 𝑘 ∈ ℙ))
5	4, 3	ifbieq1d 4512	. . . . . 6 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑚 = 𝑘) → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
6		elfznn 13577	. . . . . 6 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
7		1nn 12239	. . . . . . . 8 ⊢ 1 ∈ ℕ
8	7	a1i 11	. . . . . . 7 ⊢ (𝑘 ∈ (1...𝑁) → 1 ∈ ℕ)
9	6, 8	ifcld 4534	. . . . . 6 ⊢ (𝑘 ∈ (1...𝑁) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℕ)
10	2, 5, 6, 9	fvmptd 6997	. . . . 5 ⊢ (𝑘 ∈ (1...𝑁) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
11	10	eqcomd 2769	. . . 4 ⊢ (𝑘 ∈ (1...𝑁) → if(𝑘 ∈ ℙ, 𝑘, 1) = ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))
12	11	prodeq2i 15968	. . 3 ⊢ ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1) = ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘)
13	1, 12	eqtrdi 2814	. 2 ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) = ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘))
14		fzfid 14005	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin)
15		fz1ssnn 13579	. . . 4 ⊢ (1...𝑁) ⊆ ℕ
16	14, 15	jctil 528	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin))
17		fzssz 13549	. . . . 5 ⊢ (1...𝑁) ⊆ ℤ
18	17	a1i 11	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ ℤ)
19		0nelfz1 13566	. . . . 5 ⊢ 0 ∉ (1...𝑁)
20	19	a1i 11	. . . 4 ⊢ (𝑁 ∈ ℕ0 → 0 ∉ (1...𝑁))
21		lcmfn0cl 16679	. . . 4 ⊢ (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin ∧ 0 ∉ (1...𝑁)) → (lcm‘(1...𝑁)) ∈ ℕ)
22	18, 14, 20, 21	syl3anc 1398	. . 3 ⊢ (𝑁 ∈ ℕ0 → (lcm‘(1...𝑁)) ∈ ℕ)
23		id 23	. . . . . 6 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ)
24	7	a1i 11	. . . . . 6 ⊢ (𝑚 ∈ ℕ → 1 ∈ ℕ)
25	23, 24	ifcld 4534	. . . . 5 ⊢ (𝑚 ∈ ℕ → if(𝑚 ∈ ℙ, 𝑚, 1) ∈ ℕ)
26	25	adantl 486	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑚 ∈ ℕ) → if(𝑚 ∈ ℙ, 𝑚, 1) ∈ ℕ)
27	26	fmpttd 7110	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)):ℕ⟶ℕ)
28		simpr 489	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ (1...𝑁))
29	28	adantr 485	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝑘})) → 𝑘 ∈ (1...𝑁))
30		eldifi 4085	. . . . . . 7 ⊢ (𝑥 ∈ ((1...𝑁) ∖ {𝑘}) → 𝑥 ∈ (1...𝑁))
31	30	adantl 486	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝑘})) → 𝑥 ∈ (1...𝑁))
32		eldif 3915	. . . . . . . 8 ⊢ (𝑥 ∈ ((1...𝑁) ∖ {𝑘}) ↔ (𝑥 ∈ (1...𝑁) ∧ ¬ 𝑥 ∈ {𝑘}))
33		velsn 4605	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑘} ↔ 𝑥 = 𝑘)
34	33	biimpri 231	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → 𝑥 ∈ {𝑘})
35	34	equcoms 2050	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → 𝑥 ∈ {𝑘})
36	35	necon3bi 2984	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ {𝑘} → 𝑘 ≠ 𝑥)
37	32, 36	simplbiim 513	. . . . . . 7 ⊢ (𝑥 ∈ ((1...𝑁) ∖ {𝑘}) → 𝑘 ≠ 𝑥)
38	37	adantl 486	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑥 ∈ ((1...𝑁) ∖ {𝑘})) → 𝑘 ≠ 𝑥)
39		eqid 2763	. . . . . . 7 ⊢ (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))
40	39	fvprmselgcd1 17100	. . . . . 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 1398	. . . . 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 3157	. . . 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 3157	. . 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 2764	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)))
45		simpr 489	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑚 = 𝑘) → 𝑚 = 𝑘)
46	45	eleq1d 2848	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑚 = 𝑘) → (𝑚 ∈ ℙ ↔ 𝑘 ∈ ℙ))
47	46, 45	ifbieq1d 4512	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑚 = 𝑘) → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑘 ∈ ℙ, 𝑘, 1))
48	15, 28	sselid 3935	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ)
49	17, 28	sselid 3935	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℤ)
50		1zzd 12620	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 1 ∈ ℤ)
51	49, 50	ifcld 4534	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 ∈ ℙ, 𝑘, 1) ∈ ℤ)
52	44, 47, 48, 51	fvmptd 6997	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) = if(𝑘 ∈ ℙ, 𝑘, 1))
53		breq1 5112	. . . . . 6 ⊢ (𝑥 = if(𝑘 ∈ ℙ, 𝑘, 1) → (𝑥 ∥ (lcm‘(1...𝑁)) ↔ if(𝑘 ∈ ℙ, 𝑘, 1) ∥ (lcm‘(1...𝑁))))
54	16	adantr 485	. . . . . . 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 579	. . . . . . 7 ⊢ (((1...𝑁) ⊆ ℕ ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin))
57		dvdslcmf 16684	. . . . . . 7 ⊢ (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin) → ∀𝑥 ∈ (1...𝑁)𝑥 ∥ (lcm‘(1...𝑁)))
58	54, 56, 57	3syl 19	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → ∀𝑥 ∈ (1...𝑁)𝑥 ∥ (lcm‘(1...𝑁)))
59		elfzuz2 13552	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → 𝑁 ∈ (ℤ≥‘1))
60	59	adantl 486	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑁 ∈ (ℤ≥‘1))
61		eluzfz1 13554	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁))
62	60, 61	syl 18	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 1 ∈ (1...𝑁))
63	28, 62	ifcld 4534	. . . . . 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 5133	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → ((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
66	65	ralrimiva 3157	. . 3 ⊢ (𝑁 ∈ ℕ0 → ∀𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
67		coprmproddvds 16716	. . 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 1406	. 2 ⊢ (𝑁 ∈ ℕ0 → ∏𝑘 ∈ (1...𝑁)((𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))‘𝑘) ∥ (lcm‘(1...𝑁)))
69	13, 68	eqbrtrd 5133	1 ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) ∥ (lcm‘(1...𝑁)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958   ∉ wnel 3064  ∀wral 3079   ∖ cdif 3902   ⊆ wss 3905  ifcif 4487  {csn 4589   class class class wbr 5109   ↦ cmpt 5192  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  Fincfn 8939  0cc0 11095  1c1 11096  ℕcn 12228  ℕ₀cn0 12499  ℤcz 12586  ℤ_≥cuz 12857  ...cfz 13530  ∏cprod 15953   ∥ cdvds 16305   gcd cgcd 16547  lcmclcmf 16642  ℙcprime 16724  #_pcprmo 17086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-oi 9468  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-n0 12500  df-z 12587  df-uz 12858  df-rp 13012  df-fz 13531  df-fzo 13679  df-fl 13821  df-mod 13899  df-seq 14034  df-exp 14094  df-hash 14363  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-clim 15535  df-prod 15954  df-dvds 16306  df-gcd 16548  df-lcmf 16644  df-prm 16725  df-prmo 17087

This theorem is referenced by:  prmolelcmf  17103