Theorem prmodvdslcmf 15751

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 e. NN0 -> (#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 15737	. . 3 |- ( N e. NN0 -> (#p ` N ) = prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) )
2		eqidd 2623	. . . . . 6 |- ( k e. ( 1 ... N ) -> ( m e. NN |-> if ( m e. Prime , m , 1 ) ) = ( m e. NN |-> if ( m e. Prime , m , 1 ) ) )
3		simpr 477	. . . . . . . 8 |- ( ( k e. ( 1 ... N ) /\ m = k ) -> m = k )
4	3	eleq1d 2686	. . . . . . 7 |- ( ( k e. ( 1 ... N ) /\ m = k ) -> ( m e. Prime <-> k e. Prime ) )
5	4, 3	ifbieq1d 4109	. . . . . 6 |- ( ( k e. ( 1 ... N ) /\ m = k ) -> if ( m e. Prime , m , 1 ) = if ( k e. Prime , k , 1 ) )
6		elfznn 12370	. . . . . 6 |- ( k e. ( 1 ... N ) -> k e. NN )
7		1nn 11031	. . . . . . . 8 |- 1 e. NN
8	7	a1i 11	. . . . . . 7 |- ( k e. ( 1 ... N ) -> 1 e. NN )
9	6, 8	ifcld 4131	. . . . . 6 |- ( k e. ( 1 ... N ) -> if ( k e. Prime , k , 1 ) e. NN )
10	2, 5, 6, 9	fvmptd 6288	. . . . 5 |- ( k e. ( 1 ... N ) -> ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` k ) = if ( k e. Prime , k , 1 ) )
11	10	eqcomd 2628	. . . 4 |- ( k e. ( 1 ... N ) -> if ( k e. Prime , k , 1 ) = ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` k ) )
12	11	prodeq2i 14649	. . 3 |- prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) = prod_ k e. ( 1 ... N ) ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` k )
13	1, 12	syl6eq 2672	. 2 |- ( N e. NN0 -> (#p ` N ) = prod_ k e. ( 1 ... N ) ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` k ) )
14		fzfid 12772	. . . 4 |- ( N e. NN0 -> ( 1 ... N ) e. Fin )
15		fz1ssnn 12372	. . . 4 |- ( 1 ... N ) C_ NN
16	14, 15	jctil 560	. . 3 |- ( N e. NN0 -> ( ( 1 ... N ) C_ NN /\ ( 1 ... N ) e. Fin ) )
17		fzssz 12343	. . . . 5 |- ( 1 ... N ) C_ ZZ
18	17	a1i 11	. . . 4 |- ( N e. NN0 -> ( 1 ... N ) C_ ZZ )
19		0nelfz1 12360	. . . . 5 |- 0 e/ ( 1 ... N )
20	19	a1i 11	. . . 4 |- ( N e. NN0 -> 0 e/ ( 1 ... N ) )
21		lcmfn0cl 15339	. . . 4 |- ( ( ( 1 ... N ) C_ ZZ /\ ( 1 ... N ) e. Fin /\ 0 e/ ( 1 ... N ) ) -> (lcm ` ( 1 ... N ) ) e. NN )
22	18, 14, 20, 21	syl3anc 1326	. . 3 |- ( N e. NN0 -> (lcm ` ( 1 ... N ) ) e. NN )
23		id 22	. . . . . 6 |- ( m e. NN -> m e. NN )
24	7	a1i 11	. . . . . 6 |- ( m e. NN -> 1 e. NN )
25	23, 24	ifcld 4131	. . . . 5 |- ( m e. NN -> if ( m e. Prime , m , 1 ) e. NN )
26	25	adantl 482	. . . 4 |- ( ( N e. NN0 /\ m e. NN ) -> if ( m e. Prime , m , 1 ) e. NN )
27		eqid 2622	. . . 4 |- ( m e. NN |-> if ( m e. Prime , m , 1 ) ) = ( m e. NN |-> if ( m e. Prime , m , 1 ) )
28	26, 27	fmptd 6385	. . 3 |- ( N e. NN0 -> ( m e. NN |-> if ( m e. Prime , m , 1 ) ) : NN --> NN )
29		simpr 477	. . . . . . 7 |- ( ( N e. NN0 /\ k e. ( 1 ... N ) ) -> k e. ( 1 ... N ) )
30	29	adantr 481	. . . . . 6 |- ( ( ( N e. NN0 /\ k e. ( 1 ... N ) ) /\ x e. ( ( 1 ... N ) \ { k } ) ) -> k e. ( 1 ... N ) )
31		eldifi 3732	. . . . . . 7 |- ( x e. ( ( 1 ... N ) \ { k } ) -> x e. ( 1 ... N ) )
32	31	adantl 482	. . . . . 6 |- ( ( ( N e. NN0 /\ k e. ( 1 ... N ) ) /\ x e. ( ( 1 ... N ) \ { k } ) ) -> x e. ( 1 ... N ) )
33		eldif 3584	. . . . . . . 8 |- ( x e. ( ( 1 ... N ) \ { k } ) <-> ( x e. ( 1 ... N ) /\ -. x e. { k } ) )
34		velsn 4193	. . . . . . . . . . . 12 |- ( x e. { k } <-> x = k )
35	34	biimpri 218	. . . . . . . . . . 11 |- ( x = k -> x e. { k } )
36	35	equcoms 1947	. . . . . . . . . 10 |- ( k = x -> x e. { k } )
37	36	necon3bi 2820	. . . . . . . . 9 |- ( -. x e. { k } -> k =/= x )
38	37	adantl 482	. . . . . . . 8 |- ( ( x e. ( 1 ... N ) /\ -. x e. { k } ) -> k =/= x )
39	33, 38	sylbi 207	. . . . . . 7 |- ( x e. ( ( 1 ... N ) \ { k } ) -> k =/= x )
40	39	adantl 482	. . . . . 6 |- ( ( ( N e. NN0 /\ k e. ( 1 ... N ) ) /\ x e. ( ( 1 ... N ) \ { k } ) ) -> k =/= x )
41	27	fvprmselgcd1 15749	. . . . . 6 |- ( ( k e. ( 1 ... N ) /\ x e. ( 1 ... N ) /\ k =/= x ) -> ( ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` k ) gcd ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` x ) ) = 1 )
42	30, 32, 40, 41	syl3anc 1326	. . . . 5 |- ( ( ( N e. NN0 /\ k e. ( 1 ... N ) ) /\ x e. ( ( 1 ... N ) \ { k } ) ) -> ( ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` k ) gcd ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` x ) ) = 1 )
43	42	ralrimiva 2966	. . . 4 |- ( ( N e. NN0 /\ k e. ( 1 ... N ) ) -> A. x e. ( ( 1 ... N ) \ { k } ) ( ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` k ) gcd ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` x ) ) = 1 )
44	43	ralrimiva 2966	. . 3 |- ( N e. NN0 -> A. k e. ( 1 ... N ) A. x e. ( ( 1 ... N ) \ { k } ) ( ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` k ) gcd ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` x ) ) = 1 )
45		eqidd 2623	. . . . . 6 |- ( ( N e. NN0 /\ k e. ( 1 ... N ) ) -> ( m e. NN |-> if ( m e. Prime , m , 1 ) ) = ( m e. NN |-> if ( m e. Prime , m , 1 ) ) )
46		simpr 477	. . . . . . . 8 |- ( ( ( N e. NN0 /\ k e. ( 1 ... N ) ) /\ m = k ) -> m = k )
47	46	eleq1d 2686	. . . . . . 7 |- ( ( ( N e. NN0 /\ k e. ( 1 ... N ) ) /\ m = k ) -> ( m e. Prime <-> k e. Prime ) )
48	47, 46	ifbieq1d 4109	. . . . . 6 |- ( ( ( N e. NN0 /\ k e. ( 1 ... N ) ) /\ m = k ) -> if ( m e. Prime , m , 1 ) = if ( k e. Prime , k , 1 ) )
49	15, 29	sseldi 3601	. . . . . 6 |- ( ( N e. NN0 /\ k e. ( 1 ... N ) ) -> k e. NN )
50	17, 29	sseldi 3601	. . . . . . 7 |- ( ( N e. NN0 /\ k e. ( 1 ... N ) ) -> k e. ZZ )
51		1zzd 11408	. . . . . . 7 |- ( ( N e. NN0 /\ k e. ( 1 ... N ) ) -> 1 e. ZZ )
52	50, 51	ifcld 4131	. . . . . 6 |- ( ( N e. NN0 /\ k e. ( 1 ... N ) ) -> if ( k e. Prime , k , 1 ) e. ZZ )
53	45, 48, 49, 52	fvmptd 6288	. . . . 5 |- ( ( N e. NN0 /\ k e. ( 1 ... N ) ) -> ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` k ) = if ( k e. Prime , k , 1 ) )
54		elfzuz2 12346	. . . . . . . . 9 |- ( k e. ( 1 ... N ) -> N e. ( ZZ>= ` 1 ) )
55	54	adantl 482	. . . . . . . 8 |- ( ( N e. NN0 /\ k e. ( 1 ... N ) ) -> N e. ( ZZ>= ` 1 ) )
56		eluzfz1 12348	. . . . . . . 8 |- ( N e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... N ) )
57	55, 56	syl 17	. . . . . . 7 |- ( ( N e. NN0 /\ k e. ( 1 ... N ) ) -> 1 e. ( 1 ... N ) )
58	29, 57	ifcld 4131	. . . . . 6 |- ( ( N e. NN0 /\ k e. ( 1 ... N ) ) -> if ( k e. Prime , k , 1 ) e. ( 1 ... N ) )
59	16	adantr 481	. . . . . . 7 |- ( ( N e. NN0 /\ k e. ( 1 ... N ) ) -> ( ( 1 ... N ) C_ NN /\ ( 1 ... N ) e. Fin ) )
60	17	2a1i 12	. . . . . . . 8 |- ( ( 1 ... N ) e. Fin -> ( ( 1 ... N ) C_ NN -> ( 1 ... N ) C_ ZZ ) )
61	60	imdistanri 727	. . . . . . 7 |- ( ( ( 1 ... N ) C_ NN /\ ( 1 ... N ) e. Fin ) -> ( ( 1 ... N ) C_ ZZ /\ ( 1 ... N ) e. Fin ) )
62		dvdslcmf 15344	. . . . . . 7 |- ( ( ( 1 ... N ) C_ ZZ /\ ( 1 ... N ) e. Fin ) -> A. x e. ( 1 ... N ) x || (lcm ` ( 1 ... N ) ) )
63	59, 61, 62	3syl 18	. . . . . 6 |- ( ( N e. NN0 /\ k e. ( 1 ... N ) ) -> A. x e. ( 1 ... N ) x || (lcm ` ( 1 ... N ) ) )
64		breq1 4656	. . . . . . 7 |- ( x = if ( k e. Prime , k , 1 ) -> ( x || (lcm ` ( 1 ... N ) ) <-> if ( k e. Prime , k , 1 ) || (lcm ` ( 1 ... N ) ) ) )
65	64	rspcv 3305	. . . . . 6 |- ( if ( k e. Prime , k , 1 ) e. ( 1 ... N ) -> ( A. x e. ( 1 ... N ) x || (lcm ` ( 1 ... N ) ) -> if ( k e. Prime , k , 1 ) || (lcm ` ( 1 ... N ) ) ) )
66	58, 63, 65	sylc 65	. . . . 5 |- ( ( N e. NN0 /\ k e. ( 1 ... N ) ) -> if ( k e. Prime , k , 1 ) || (lcm ` ( 1 ... N ) ) )
67	53, 66	eqbrtrd 4675	. . . 4 |- ( ( N e. NN0 /\ k e. ( 1 ... N ) ) -> ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` k ) || (lcm ` ( 1 ... N ) ) )
68	67	ralrimiva 2966	. . 3 |- ( N e. NN0 -> A. k e. ( 1 ... N ) ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` k ) || (lcm ` ( 1 ... N ) ) )
69		coprmproddvds 15377	. . 3 |- ( ( ( ( 1 ... N ) C_ NN /\ ( 1 ... N ) e. Fin ) /\ ( (lcm ` ( 1 ... N ) ) e. NN /\ ( m e. NN |-> if ( m e. Prime , m , 1 ) ) : NN --> NN ) /\ ( A. k e. ( 1 ... N ) A. x e. ( ( 1 ... N ) \ { k } ) ( ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` k ) gcd ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` x ) ) = 1 /\ A. k e. ( 1 ... N ) ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` k ) || (lcm ` ( 1 ... N ) ) ) ) -> prod_ k e. ( 1 ... N ) ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` k ) || (lcm ` ( 1 ... N ) ) )
70	16, 22, 28, 44, 68, 69	syl122anc 1335	. 2 |- ( N e. NN0 -> prod_ k e. ( 1 ... N ) ( ( m e. NN |-> if ( m e. Prime , m , 1 ) ) ` k ) || (lcm ` ( 1 ... N ) ) )
71	13, 70	eqbrtrd 4675	1 |- ( N e. NN0 -> (#p ` N ) || (lcm ` ( 1 ... N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   A.wral 2912   \ cdif 3571   C_ wss 3574   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   1c1 9937   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   prod_cprod 14635   || cdvds 14983   gcd cgcd 15216  lcmclcmf 15302   Primecprime 15385  #_pcprmo 15735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-prod 14636  df-dvds 14984  df-gcd 15217  df-lcmf 15304  df-prm 15386  df-prmo 15736

This theorem is referenced by:  prmolelcmf  15752