Theorem lcmval 12017

Description: Value of the lcm operator. ( M lcm N ) is the least common multiple of M and N. If either M or N is 0, the result is defined conventionally as 0. Contrast with df-gcd 11898 and gcdval 11914. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)

Assertion

Ref	Expression
lcmval	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) )

Distinct variable groups:   n, M   n, N

Proof of Theorem lcmval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lcm 12015	. . 3 |- lcm = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) )
2	1	a1i 9	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> lcm = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) ) )
3		eqeq1 2177	. . . . . 6 |- ( x = M -> ( x = 0 <-> M = 0 ) )
4	3	orbi1d 786	. . . . 5 |- ( x = M -> ( ( x = 0 \/ y = 0 ) <-> ( M = 0 \/ y = 0 ) ) )
5		breq1 3992	. . . . . . . 8 |- ( x = M -> ( x || n <-> M || n ) )
6	5	anbi1d 462	. . . . . . 7 |- ( x = M -> ( ( x || n /\ y || n ) <-> ( M || n /\ y || n ) ) )
7	6	rabbidv 2719	. . . . . 6 |- ( x = M -> { n e. NN | ( x || n /\ y || n ) } = { n e. NN | ( M || n /\ y || n ) } )
8	7	infeq1d 6989	. . . . 5 |- ( x = M -> inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) = inf ( { n e. NN | ( M || n /\ y || n ) } , RR , < ) )
9	4, 8	ifbieq2d 3550	. . . 4 |- ( x = M -> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) = if ( ( M = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ y || n ) } , RR , < ) ) )
10		eqeq1 2177	. . . . . 6 |- ( y = N -> ( y = 0 <-> N = 0 ) )
11	10	orbi2d 785	. . . . 5 |- ( y = N -> ( ( M = 0 \/ y = 0 ) <-> ( M = 0 \/ N = 0 ) ) )
12		breq1 3992	. . . . . . . 8 |- ( y = N -> ( y || n <-> N || n ) )
13	12	anbi2d 461	. . . . . . 7 |- ( y = N -> ( ( M || n /\ y || n ) <-> ( M || n /\ N || n ) ) )
14	13	rabbidv 2719	. . . . . 6 |- ( y = N -> { n e. NN | ( M || n /\ y || n ) } = { n e. NN | ( M || n /\ N || n ) } )
15	14	infeq1d 6989	. . . . 5 |- ( y = N -> inf ( { n e. NN | ( M || n /\ y || n ) } , RR , < ) = inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) )
16	11, 15	ifbieq2d 3550	. . . 4 |- ( y = N -> if ( ( M = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ y || n ) } , RR , < ) ) = if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) )
17	9, 16	sylan9eq 2223	. . 3 |- ( ( x = M /\ y = N ) -> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) = if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) )
18	17	adantl 275	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( x = M /\ y = N ) ) -> if ( ( x = 0 \/ y = 0 ) , 0 , inf ( { n e. NN | ( x || n /\ y || n ) } , RR , < ) ) = if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) )
19		simpl 108	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
20		simpr 109	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
21		c0ex 7914	. . . 4 |- 0 e. _V
22	21	a1i 9	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> 0 e. _V )
23		1zzd 9239	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> 1 e. ZZ )
24		nnuz 9522	. . . . . 6 |- NN = ( ZZ>= ` 1 )
25		rabeq 2722	. . . . . 6 |- ( NN = ( ZZ>= ` 1 ) -> { n e. NN | ( M || n /\ N || n ) } = { n e. ( ZZ>= ` 1 ) | ( M || n /\ N || n ) } )
26	24, 25	ax-mp 5	. . . . 5 |- { n e. NN | ( M || n /\ N || n ) } = { n e. ( ZZ>= ` 1 ) | ( M || n /\ N || n ) }
27		dvdsmul1 11775	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> M || ( M x. N ) )
28	27	adantr 274	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M || ( M x. N ) )
29		simpll 524	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M e. ZZ )
30		simplr 525	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N e. ZZ )
31	29, 30	zmulcld 9340	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M x. N ) e. ZZ )
32		dvdsabsb 11772	. . . . . . . 8 |- ( ( M e. ZZ /\ ( M x. N ) e. ZZ ) -> ( M || ( M x. N ) <-> M || ( abs ` ( M x. N ) ) ) )
33	29, 31, 32	syl2anc 409	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M || ( M x. N ) <-> M || ( abs ` ( M x. N ) ) ) )
34	28, 33	mpbid 146	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M || ( abs ` ( M x. N ) ) )
35		dvdsmul2 11776	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> N || ( M x. N ) )
36	35	adantr 274	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N || ( M x. N ) )
37		dvdsabsb 11772	. . . . . . . 8 |- ( ( N e. ZZ /\ ( M x. N ) e. ZZ ) -> ( N || ( M x. N ) <-> N || ( abs ` ( M x. N ) ) ) )
38	30, 31, 37	syl2anc 409	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( N || ( M x. N ) <-> N || ( abs ` ( M x. N ) ) ) )
39	36, 38	mpbid 146	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N || ( abs ` ( M x. N ) ) )
40	29	zcnd 9335	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M e. CC )
41	30	zcnd 9335	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N e. CC )
42	40, 41	absmuld 11158	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` ( M x. N ) ) = ( ( abs ` M ) x. ( abs ` N ) ) )
43		simpr 109	. . . . . . . . . . . . 13 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. ( M = 0 \/ N = 0 ) )
44		ioran 747	. . . . . . . . . . . . 13 |- ( -. ( M = 0 \/ N = 0 ) <-> ( -. M = 0 /\ -. N = 0 ) )
45	43, 44	sylib 121	. . . . . . . . . . . 12 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( -. M = 0 /\ -. N = 0 ) )
46	45	simpld 111	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. M = 0 )
47	46	neqned 2347	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M =/= 0 )
48		nnabscl 11064	. . . . . . . . . 10 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
49	29, 47, 48	syl2anc 409	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` M ) e. NN )
50	45	simprd 113	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> -. N = 0 )
51	50	neqned 2347	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N =/= 0 )
52		nnabscl 11064	. . . . . . . . . 10 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
53	30, 51, 52	syl2anc 409	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` N ) e. NN )
54	49, 53	nnmulcld 8927	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( abs ` M ) x. ( abs ` N ) ) e. NN )
55	42, 54	eqeltrd 2247	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` ( M x. N ) ) e. NN )
56		breq2 3993	. . . . . . . . 9 |- ( n = ( abs ` ( M x. N ) ) -> ( M || n <-> M || ( abs ` ( M x. N ) ) ) )
57		breq2 3993	. . . . . . . . 9 |- ( n = ( abs ` ( M x. N ) ) -> ( N || n <-> N || ( abs ` ( M x. N ) ) ) )
58	56, 57	anbi12d 470	. . . . . . . 8 |- ( n = ( abs ` ( M x. N ) ) -> ( ( M || n /\ N || n ) <-> ( M || ( abs ` ( M x. N ) ) /\ N || ( abs ` ( M x. N ) ) ) ) )
59	58	elrab3 2887	. . . . . . 7 |- ( ( abs ` ( M x. N ) ) e. NN -> ( ( abs ` ( M x. N ) ) e. { n e. NN | ( M || n /\ N || n ) } <-> ( M || ( abs ` ( M x. N ) ) /\ N || ( abs ` ( M x. N ) ) ) ) )
60	55, 59	syl 14	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( abs ` ( M x. N ) ) e. { n e. NN | ( M || n /\ N || n ) } <-> ( M || ( abs ` ( M x. N ) ) /\ N || ( abs ` ( M x. N ) ) ) ) )
61	34, 39, 60	mpbir2and 939	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` ( M x. N ) ) e. { n e. NN | ( M || n /\ N || n ) } )
62		elfzelz 9981	. . . . . . 7 |- ( n e. ( 1 ... ( abs ` ( M x. N ) ) ) -> n e. ZZ )
63		zdvdsdc 11774	. . . . . . 7 |- ( ( M e. ZZ /\ n e. ZZ ) -> DECID M || n )
64	29, 62, 63	syl2an 287	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> DECID M || n )
65		zdvdsdc 11774	. . . . . . 7 |- ( ( N e. ZZ /\ n e. ZZ ) -> DECID N || n )
66	30, 62, 65	syl2an 287	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> DECID N || n )
67		dcan2 929	. . . . . 6 |- (DECID M || n -> (DECID N || n -> DECID ( M || n /\ N || n ) ) )
68	64, 66, 67	sylc 62	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) /\ n e. ( 1 ... ( abs ` ( M x. N ) ) ) ) -> DECID ( M || n /\ N || n ) )
69	23, 26, 61, 68	infssuzcldc 11906	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) e. { n e. NN | ( M || n /\ N || n ) } )
70	69	elexd 2743	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) e. _V )
71		lcmmndc 12016	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
72	22, 70, 71	ifcldadc 3555	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) e. _V )
73	2, 18, 19, 20, 72	ovmpod 5980	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = if ( ( M = 0 \/ N = 0 ) , 0 , inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703  DECID wdc 829   = wceq 1348   e. wcel 2141   =/= wne 2340   {crab 2452   _Vcvv 2730   ifcif 3526   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   e. cmpo 5855  infcinf 6960   RRcr 7773   0cc0 7774   1c1 7775   x. cmul 7779   < clt 7954   NNcn 8878   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965   abscabs 10961   || cdvds 11749   lcm clcm 12014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-lcm 12015

This theorem is referenced by:  lcmcom  12018  lcm0val  12019  lcmn0val  12020  lcmass  12039