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Theorem lcmval 12098
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 11979 and gcdval 11995. (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.
StepHypRef Expression
1 df-lcm 12096 . . 3  |- lcm  =  ( x  e.  ZZ , 
y  e.  ZZ  |->  if ( ( x  =  0  \/  y  =  0 ) ,  0 , inf ( { n  e.  NN  |  ( x 
||  n  /\  y  ||  n ) } ,  RR ,  <  ) ) )
21a1i 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 2196 . . . . . 6  |-  ( x  =  M  ->  (
x  =  0  <->  M  =  0 ) )
43orbi1d 792 . . . . 5  |-  ( x  =  M  ->  (
( x  =  0  \/  y  =  0 )  <->  ( M  =  0  \/  y  =  0 ) ) )
5 breq1 4021 . . . . . . . 8  |-  ( x  =  M  ->  (
x  ||  n  <->  M  ||  n
) )
65anbi1d 465 . . . . . . 7  |-  ( x  =  M  ->  (
( x  ||  n  /\  y  ||  n )  <-> 
( M  ||  n  /\  y  ||  n ) ) )
76rabbidv 2741 . . . . . 6  |-  ( x  =  M  ->  { n  e.  NN  |  ( x 
||  n  /\  y  ||  n ) }  =  { n  e.  NN  |  ( M  ||  n  /\  y  ||  n
) } )
87infeq1d 7042 . . . . 5  |-  ( x  =  M  -> inf ( { n  e.  NN  | 
( x  ||  n  /\  y  ||  n ) } ,  RR ,  <  )  = inf ( { n  e.  NN  | 
( M  ||  n  /\  y  ||  n ) } ,  RR ,  <  ) )
94, 8ifbieq2d 3573 . . . 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 2196 . . . . . 6  |-  ( y  =  N  ->  (
y  =  0  <->  N  =  0 ) )
1110orbi2d 791 . . . . 5  |-  ( y  =  N  ->  (
( M  =  0  \/  y  =  0 )  <->  ( M  =  0  \/  N  =  0 ) ) )
12 breq1 4021 . . . . . . . 8  |-  ( y  =  N  ->  (
y  ||  n  <->  N  ||  n
) )
1312anbi2d 464 . . . . . . 7  |-  ( y  =  N  ->  (
( M  ||  n  /\  y  ||  n )  <-> 
( M  ||  n  /\  N  ||  n ) ) )
1413rabbidv 2741 . . . . . 6  |-  ( y  =  N  ->  { n  e.  NN  |  ( M 
||  n  /\  y  ||  n ) }  =  { n  e.  NN  |  ( M  ||  n  /\  N  ||  n
) } )
1514infeq1d 7042 . . . . 5  |-  ( y  =  N  -> inf ( { n  e.  NN  | 
( M  ||  n  /\  y  ||  n ) } ,  RR ,  <  )  = inf ( { n  e.  NN  | 
( M  ||  n  /\  N  ||  n ) } ,  RR ,  <  ) )
1611, 15ifbieq2d 3573 . . . 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 ,  <  ) ) )
179, 16sylan9eq 2242 . . 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 ,  <  ) ) )
1817adantl 277 . 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 109 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
20 simpr 110 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
21 c0ex 7982 . . . 4  |-  0  e.  _V
2221a1i 9 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  \/  N  =  0 ) )  -> 
0  e.  _V )
23 1zzd 9311 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  1  e.  ZZ )
24 nnuz 9595 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
2524rabeqi 2745 . . . . 5  |-  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) }  =  { n  e.  ( ZZ>=
`  1 )  |  ( M  ||  n  /\  N  ||  n ) }
26 dvdsmul1 11855 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  ( M  x.  N ) )
2726adantr 276 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  ||  ( M  x.  N )
)
28 simpll 527 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  e.  ZZ )
29 simplr 528 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  e.  ZZ )
3028, 29zmulcld 9412 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M  x.  N )  e.  ZZ )
31 dvdsabsb 11852 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( M  ||  ( M  x.  N
)  <->  M  ||  ( abs `  ( M  x.  N
) ) ) )
3228, 30, 31syl2anc 411 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M  ||  ( M  x.  N
)  <->  M  ||  ( abs `  ( M  x.  N
) ) ) )
3327, 32mpbid 147 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  ||  ( abs `  ( M  x.  N ) ) )
34 dvdsmul2 11856 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  ||  ( M  x.  N ) )
3534adantr 276 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  ||  ( M  x.  N )
)
36 dvdsabsb 11852 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( N  ||  ( M  x.  N
)  <->  N  ||  ( abs `  ( M  x.  N
) ) ) )
3729, 30, 36syl2anc 411 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( N  ||  ( M  x.  N
)  <->  N  ||  ( abs `  ( M  x.  N
) ) ) )
3835, 37mpbid 147 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  ||  ( abs `  ( M  x.  N ) ) )
3928zcnd 9407 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  e.  CC )
4029zcnd 9407 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  e.  CC )
4139, 40absmuld 11238 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( abs `  ( M  x.  N )
)  =  ( ( abs `  M )  x.  ( abs `  N
) ) )
42 simpr 110 . . . . . . . . . . . . 13  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  -.  ( M  =  0  \/  N  =  0 ) )
43 ioran 753 . . . . . . . . . . . . 13  |-  ( -.  ( M  =  0  \/  N  =  0 )  <->  ( -.  M  =  0  /\  -.  N  =  0 ) )
4442, 43sylib 122 . . . . . . . . . . . 12  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( -.  M  =  0  /\  -.  N  =  0 ) )
4544simpld 112 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  -.  M  = 
0 )
4645neneqad 2439 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  =/=  0
)
47 nnabscl 11144 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
4828, 46, 47syl2anc 411 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( abs `  M
)  e.  NN )
4944simprd 114 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  -.  N  = 
0 )
5049neneqad 2439 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  =/=  0
)
51 nnabscl 11144 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
5229, 50, 51syl2anc 411 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( abs `  N
)  e.  NN )
5348, 52nnmulcld 8999 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( ( abs `  M )  x.  ( abs `  N ) )  e.  NN )
5441, 53eqeltrd 2266 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( abs `  ( M  x.  N )
)  e.  NN )
55 breq2 4022 . . . . . . . . 9  |-  ( n  =  ( abs `  ( M  x.  N )
)  ->  ( M  ||  n  <->  M  ||  ( abs `  ( M  x.  N
) ) ) )
56 breq2 4022 . . . . . . . . 9  |-  ( n  =  ( abs `  ( M  x.  N )
)  ->  ( N  ||  n  <->  N  ||  ( abs `  ( M  x.  N
) ) ) )
5755, 56anbi12d 473 . . . . . . . 8  |-  ( n  =  ( abs `  ( M  x.  N )
)  ->  ( ( M  ||  n  /\  N  ||  n )  <->  ( M  ||  ( abs `  ( M  x.  N )
)  /\  N  ||  ( abs `  ( M  x.  N ) ) ) ) )
5857elrab3 2909 . . . . . . 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 ) ) ) ) )
5954, 58syl 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 ) ) ) ) )
6033, 38, 59mpbir2and 946 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( abs `  ( M  x.  N )
)  e.  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) } )
61 elfzelz 10057 . . . . . . 7  |-  ( n  e.  ( 1 ... ( abs `  ( M  x.  N )
) )  ->  n  e.  ZZ )
62 zdvdsdc 11854 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  n  e.  ZZ )  -> DECID  M 
||  n )
6328, 61, 62syl2an 289 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... ( abs `  ( M  x.  N ) ) ) )  -> DECID  M  ||  n )
64 zdvdsdc 11854 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  n  e.  ZZ )  -> DECID  N 
||  n )
6529, 61, 64syl2an 289 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... ( abs `  ( M  x.  N ) ) ) )  -> DECID  N  ||  n )
6663, 65dcand 934 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... ( abs `  ( M  x.  N ) ) ) )  -> DECID  ( M  ||  n  /\  N  ||  n ) )
6723, 25, 60, 66infssuzcldc 11987 . . . 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
) } )
6867elexd 2765 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  -> inf ( { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) } ,  RR ,  <  )  e. 
_V )
69 lcmmndc 12097 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  \/  N  =  0 ) )
7022, 68, 69ifcldadc 3578 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  if ( ( M  =  0  \/  N  =  0 ) ,  0 , inf ( { n  e.  NN  | 
( M  ||  n  /\  N  ||  n ) } ,  RR ,  <  ) )  e.  _V )
712, 18, 19, 20, 70ovmpod 6025 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 ,  <  ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709  DECID wdc 835    = wceq 1364    e. wcel 2160    =/= wne 2360   {crab 2472   _Vcvv 2752   ifcif 3549   class class class wbr 4018   ` cfv 5235  (class class class)co 5897    e. cmpo 5899  infcinf 7013   RRcr 7841   0cc0 7842   1c1 7843    x. cmul 7847    < clt 8023   NNcn 8950   ZZcz 9284   ZZ>=cuz 9559   ...cfz 10040   abscabs 11041    || cdvds 11829   lcm clcm 12095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-frec 6417  df-sup 7014  df-inf 7015  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-z 9285  df-uz 9560  df-q 9652  df-rp 9686  df-fz 10041  df-fzo 10175  df-fl 10303  df-mod 10356  df-seqfrec 10479  df-exp 10554  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-dvds 11830  df-lcm 12096
This theorem is referenced by:  lcmcom  12099  lcm0val  12100  lcmn0val  12101  lcmass  12120
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