Theorem lcmval 11733

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 11625 and gcdval 11637. (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 11731	. . 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 2144	. . . . . 6 |- ( x = M -> ( x = 0 <-> M = 0 ) )
4	3	orbi1d 780	. . . . 5 |- ( x = M -> ( ( x = 0 \/ y = 0 ) <-> ( M = 0 \/ y = 0 ) ) )
5		breq1 3927	. . . . . . . 8 |- ( x = M -> ( x || n <-> M || n ) )
6	5	anbi1d 460	. . . . . . 7 |- ( x = M -> ( ( x || n /\ y || n ) <-> ( M || n /\ y || n ) ) )
7	6	rabbidv 2670	. . . . . 6 |- ( x = M -> { n e. NN | ( x || n /\ y || n ) } = { n e. NN | ( M || n /\ y || n ) } )
8	7	infeq1d 6892	. . . . 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 3491	. . . 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 2144	. . . . . 6 |- ( y = N -> ( y = 0 <-> N = 0 ) )
11	10	orbi2d 779	. . . . 5 |- ( y = N -> ( ( M = 0 \/ y = 0 ) <-> ( M = 0 \/ N = 0 ) ) )
12		breq1 3927	. . . . . . . 8 |- ( y = N -> ( y || n <-> N || n ) )
13	12	anbi2d 459	. . . . . . 7 |- ( y = N -> ( ( M || n /\ y || n ) <-> ( M || n /\ N || n ) ) )
14	13	rabbidv 2670	. . . . . 6 |- ( y = N -> { n e. NN | ( M || n /\ y || n ) } = { n e. NN | ( M || n /\ N || n ) } )
15	14	infeq1d 6892	. . . . 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 3491	. . . 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 2190	. . 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 7753	. . . 4 |- 0 e. _V
22	21	a1i 9	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> 0 e. _V )
23		1zzd 9074	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> 1 e. ZZ )
24		nnuz 9354	. . . . . 6 |- NN = ( ZZ>= ` 1 )
25		rabeq 2673	. . . . . 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 11504	. . . . . . . 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 518	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M e. ZZ )
30		simplr 519	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N e. ZZ )
31	29, 30	zmulcld 9172	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M x. N ) e. ZZ )
32		dvdsabsb 11501	. . . . . . . 8 |- ( ( M e. ZZ /\ ( M x. N ) e. ZZ ) -> ( M || ( M x. N ) <-> M || ( abs ` ( M x. N ) ) ) )
33	29, 31, 32	syl2anc 408	. . . . . . 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 11505	. . . . . . . 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 11501	. . . . . . . 8 |- ( ( N e. ZZ /\ ( M x. N ) e. ZZ ) -> ( N || ( M x. N ) <-> N || ( abs ` ( M x. N ) ) ) )
38	30, 31, 37	syl2anc 408	. . . . . . 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 9167	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M e. CC )
41	30	zcnd 9167	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N e. CC )
42	40, 41	absmuld 10959	. . . . . . . 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 741	. . . . . . . . . . . . 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 2313	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> M =/= 0 )
48		nnabscl 10865	. . . . . . . . . 10 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
49	29, 47, 48	syl2anc 408	. . . . . . . . 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 2313	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> N =/= 0 )
52		nnabscl 10865	. . . . . . . . . 10 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
53	30, 51, 52	syl2anc 408	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` N ) e. NN )
54	49, 53	nnmulcld 8762	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( abs ` M ) x. ( abs ` N ) ) e. NN )
55	42, 54	eqeltrd 2214	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( abs ` ( M x. N ) ) e. NN )
56		breq2 3928	. . . . . . . . 9 |- ( n = ( abs ` ( M x. N ) ) -> ( M || n <-> M || ( abs ` ( M x. N ) ) ) )
57		breq2 3928	. . . . . . . . 9 |- ( n = ( abs ` ( M x. N ) ) -> ( N || n <-> N || ( abs ` ( M x. N ) ) ) )
58	56, 57	anbi12d 464	. . . . . . . 8 |- ( n = ( abs ` ( M x. N ) ) -> ( ( M || n /\ N || n ) <-> ( M || ( abs ` ( M x. N ) ) /\ N || ( abs ` ( M x. N ) ) ) ) )
59	58	elrab3 2836	. . . . . . 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 928	. . . . 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 9799	. . . . . . 7 |- ( n e. ( 1 ... ( abs ` ( M x. N ) ) ) -> n e. ZZ )
63		zdvdsdc 11503	. . . . . . 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 11503	. . . . . . 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		dcan 918	. . . . . 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 11633	. . . 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 2694	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> inf ( { n e. NN | ( M || n /\ N || n ) } , RR , < ) e. _V )
71		lcmmndc 11732	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
72	22, 70, 71	ifcldadc 3496	. 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 5891	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 697  DECID wdc 819   = wceq 1331   e. wcel 1480   =/= wne 2306   {crab 2418   _Vcvv 2681   ifcif 3469   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   e. cmpo 5769  infcinf 6863   RRcr 7612   0cc0 7613   1c1 7614   x. cmul 7618   < clt 7793   NNcn 8713   ZZcz 9047   ZZ>=cuz 9319   ...cfz 9783   abscabs 10762   || cdvds 11482   lcm clcm 11730

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-isom 5127  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-sup 6864  df-inf 6865  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-fz 9784  df-fzo 9913  df-fl 10036  df-mod 10089  df-seqfrec 10212  df-exp 10286  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-dvds 11483  df-lcm 11731

This theorem is referenced by:  lcmcom  11734  lcm0val  11735  lcmn0val  11736  lcmass  11755