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Theorem lcmval 11939
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 11829 and gcdval 11842. (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 11937 . . 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 2164 . . . . . 6  |-  ( x  =  M  ->  (
x  =  0  <->  M  =  0 ) )
43orbi1d 781 . . . . 5  |-  ( x  =  M  ->  (
( x  =  0  \/  y  =  0 )  <->  ( M  =  0  \/  y  =  0 ) ) )
5 breq1 3968 . . . . . . . 8  |-  ( x  =  M  ->  (
x  ||  n  <->  M  ||  n
) )
65anbi1d 461 . . . . . . 7  |-  ( x  =  M  ->  (
( x  ||  n  /\  y  ||  n )  <-> 
( M  ||  n  /\  y  ||  n ) ) )
76rabbidv 2701 . . . . . 6  |-  ( x  =  M  ->  { n  e.  NN  |  ( x 
||  n  /\  y  ||  n ) }  =  { n  e.  NN  |  ( M  ||  n  /\  y  ||  n
) } )
87infeq1d 6956 . . . . 5  |-  ( x  =  M  -> inf ( { n  e.  NN  | 
( x  ||  n  /\  y  ||  n ) } ,  RR ,  <  )  = inf ( { n  e.  NN  | 
( M  ||  n  /\  y  ||  n ) } ,  RR ,  <  ) )
94, 8ifbieq2d 3529 . . . 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 2164 . . . . . 6  |-  ( y  =  N  ->  (
y  =  0  <->  N  =  0 ) )
1110orbi2d 780 . . . . 5  |-  ( y  =  N  ->  (
( M  =  0  \/  y  =  0 )  <->  ( M  =  0  \/  N  =  0 ) ) )
12 breq1 3968 . . . . . . . 8  |-  ( y  =  N  ->  (
y  ||  n  <->  N  ||  n
) )
1312anbi2d 460 . . . . . . 7  |-  ( y  =  N  ->  (
( M  ||  n  /\  y  ||  n )  <-> 
( M  ||  n  /\  N  ||  n ) ) )
1413rabbidv 2701 . . . . . 6  |-  ( y  =  N  ->  { n  e.  NN  |  ( M 
||  n  /\  y  ||  n ) }  =  { n  e.  NN  |  ( M  ||  n  /\  N  ||  n
) } )
1514infeq1d 6956 . . . . 5  |-  ( y  =  N  -> inf ( { n  e.  NN  | 
( M  ||  n  /\  y  ||  n ) } ,  RR ,  <  )  = inf ( { n  e.  NN  | 
( M  ||  n  /\  N  ||  n ) } ,  RR ,  <  ) )
1611, 15ifbieq2d 3529 . . . 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 2210 . . 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 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 7872 . . . 4  |-  0  e.  _V
2221a1i 9 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  \/  N  =  0 ) )  -> 
0  e.  _V )
23 1zzd 9194 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  1  e.  ZZ )
24 nnuz 9474 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
25 rabeq 2704 . . . . . 6  |-  ( NN  =  ( ZZ>= `  1
)  ->  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) }  =  { n  e.  ( ZZ>=
`  1 )  |  ( M  ||  n  /\  N  ||  n ) } )
2624, 25ax-mp 5 . . . . 5  |-  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) }  =  { n  e.  ( ZZ>=
`  1 )  |  ( M  ||  n  /\  N  ||  n ) }
27 dvdsmul1 11708 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  ( M  x.  N ) )
2827adantr 274 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  ||  ( M  x.  N )
)
29 simpll 519 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  e.  ZZ )
30 simplr 520 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  e.  ZZ )
3129, 30zmulcld 9292 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M  x.  N )  e.  ZZ )
32 dvdsabsb 11705 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( M  ||  ( M  x.  N
)  <->  M  ||  ( abs `  ( M  x.  N
) ) ) )
3329, 31, 32syl2anc 409 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M  ||  ( M  x.  N
)  <->  M  ||  ( abs `  ( M  x.  N
) ) ) )
3428, 33mpbid 146 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  ||  ( abs `  ( M  x.  N ) ) )
35 dvdsmul2 11709 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  ||  ( M  x.  N ) )
3635adantr 274 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  ||  ( M  x.  N )
)
37 dvdsabsb 11705 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( N  ||  ( M  x.  N
)  <->  N  ||  ( abs `  ( M  x.  N
) ) ) )
3830, 31, 37syl2anc 409 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( N  ||  ( M  x.  N
)  <->  N  ||  ( abs `  ( M  x.  N
) ) ) )
3936, 38mpbid 146 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  ||  ( abs `  ( M  x.  N ) ) )
4029zcnd 9287 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  e.  CC )
4130zcnd 9287 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  e.  CC )
4240, 41absmuld 11094 . . . . . . . 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 742 . . . . . . . . . . . . 13  |-  ( -.  ( M  =  0  \/  N  =  0 )  <->  ( -.  M  =  0  /\  -.  N  =  0 ) )
4543, 44sylib 121 . . . . . . . . . . . 12  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( -.  M  =  0  /\  -.  N  =  0 ) )
4645simpld 111 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  -.  M  = 
0 )
4746neqned 2334 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  =/=  0
)
48 nnabscl 11000 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
4929, 47, 48syl2anc 409 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( abs `  M
)  e.  NN )
5045simprd 113 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  -.  N  = 
0 )
5150neqned 2334 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  =/=  0
)
52 nnabscl 11000 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
5330, 51, 52syl2anc 409 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( abs `  N
)  e.  NN )
5449, 53nnmulcld 8882 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( ( abs `  M )  x.  ( abs `  N ) )  e.  NN )
5542, 54eqeltrd 2234 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( abs `  ( M  x.  N )
)  e.  NN )
56 breq2 3969 . . . . . . . . 9  |-  ( n  =  ( abs `  ( M  x.  N )
)  ->  ( M  ||  n  <->  M  ||  ( abs `  ( M  x.  N
) ) ) )
57 breq2 3969 . . . . . . . . 9  |-  ( n  =  ( abs `  ( M  x.  N )
)  ->  ( N  ||  n  <->  N  ||  ( abs `  ( M  x.  N
) ) ) )
5856, 57anbi12d 465 . . . . . . . 8  |-  ( n  =  ( abs `  ( M  x.  N )
)  ->  ( ( M  ||  n  /\  N  ||  n )  <->  ( M  ||  ( abs `  ( M  x.  N )
)  /\  N  ||  ( abs `  ( M  x.  N ) ) ) ) )
5958elrab3 2869 . . . . . . 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 ) ) ) ) )
6055, 59syl 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 ) ) ) ) )
6134, 39, 60mpbir2and 929 . . . . 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 9928 . . . . . . 7  |-  ( n  e.  ( 1 ... ( abs `  ( M  x.  N )
) )  ->  n  e.  ZZ )
63 zdvdsdc 11707 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  n  e.  ZZ )  -> DECID  M 
||  n )
6429, 62, 63syl2an 287 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... ( abs `  ( M  x.  N ) ) ) )  -> DECID  M  ||  n )
65 zdvdsdc 11707 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  n  e.  ZZ )  -> DECID  N 
||  n )
6630, 62, 65syl2an 287 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... ( abs `  ( M  x.  N ) ) ) )  -> DECID  N  ||  n )
67 dcan 919 . . . . . 6  |-  (DECID  M  ||  n  ->  (DECID  N  ||  n  -> DECID  ( M  ||  n  /\  N  ||  n ) ) )
6864, 66, 67sylc 62 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... ( abs `  ( M  x.  N ) ) ) )  -> DECID  ( M  ||  n  /\  N  ||  n ) )
6923, 26, 61, 68infssuzcldc 11837 . . . 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
) } )
7069elexd 2725 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  -> inf ( { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) } ,  RR ,  <  )  e. 
_V )
71 lcmmndc 11938 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  \/  N  =  0 ) )
7222, 70, 71ifcldadc 3534 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  if ( ( M  =  0  \/  N  =  0 ) ,  0 , inf ( { n  e.  NN  | 
( M  ||  n  /\  N  ||  n ) } ,  RR ,  <  ) )  e.  _V )
732, 18, 19, 20, 72ovmpod 5948 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 103    <-> wb 104    \/ wo 698  DECID wdc 820    = wceq 1335    e. wcel 2128    =/= wne 2327   {crab 2439   _Vcvv 2712   ifcif 3505   class class class wbr 3965   ` cfv 5170  (class class class)co 5824    e. cmpo 5826  infcinf 6927   RRcr 7731   0cc0 7732   1c1 7733    x. cmul 7737    < clt 7912   NNcn 8833   ZZcz 9167   ZZ>=cuz 9439   ...cfz 9912   abscabs 10897    || cdvds 11683   lcm clcm 11936
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-isom 5179  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-sup 6928  df-inf 6929  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-q 9529  df-rp 9561  df-fz 9913  df-fzo 10042  df-fl 10169  df-mod 10222  df-seqfrec 10345  df-exp 10419  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-dvds 11684  df-lcm 11937
This theorem is referenced by:  lcmcom  11940  lcm0val  11941  lcmn0val  11942  lcmass  11961
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