ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  lcmval Unicode version

Theorem lcmval 10652
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 10546 and gcdval 10558. (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 10650 . . 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 2089 . . . . . 6  |-  ( x  =  M  ->  (
x  =  0  <->  M  =  0 ) )
43orbi1d 738 . . . . 5  |-  ( x  =  M  ->  (
( x  =  0  \/  y  =  0 )  <->  ( M  =  0  \/  y  =  0 ) ) )
5 breq1 3808 . . . . . . . 8  |-  ( x  =  M  ->  (
x  ||  n  <->  M  ||  n
) )
65anbi1d 453 . . . . . . 7  |-  ( x  =  M  ->  (
( x  ||  n  /\  y  ||  n )  <-> 
( M  ||  n  /\  y  ||  n ) ) )
76rabbidv 2599 . . . . . 6  |-  ( x  =  M  ->  { n  e.  NN  |  ( x 
||  n  /\  y  ||  n ) }  =  { n  e.  NN  |  ( M  ||  n  /\  y  ||  n
) } )
87infeq1d 6519 . . . . 5  |-  ( x  =  M  -> inf ( { n  e.  NN  | 
( x  ||  n  /\  y  ||  n ) } ,  RR ,  <  )  = inf ( { n  e.  NN  | 
( M  ||  n  /\  y  ||  n ) } ,  RR ,  <  ) )
94, 8ifbieq2d 3390 . . . 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 2089 . . . . . 6  |-  ( y  =  N  ->  (
y  =  0  <->  N  =  0 ) )
1110orbi2d 737 . . . . 5  |-  ( y  =  N  ->  (
( M  =  0  \/  y  =  0 )  <->  ( M  =  0  \/  N  =  0 ) ) )
12 breq1 3808 . . . . . . . 8  |-  ( y  =  N  ->  (
y  ||  n  <->  N  ||  n
) )
1312anbi2d 452 . . . . . . 7  |-  ( y  =  N  ->  (
( M  ||  n  /\  y  ||  n )  <-> 
( M  ||  n  /\  N  ||  n ) ) )
1413rabbidv 2599 . . . . . 6  |-  ( y  =  N  ->  { n  e.  NN  |  ( M 
||  n  /\  y  ||  n ) }  =  { n  e.  NN  |  ( M  ||  n  /\  N  ||  n
) } )
1514infeq1d 6519 . . . . 5  |-  ( y  =  N  -> inf ( { n  e.  NN  | 
( M  ||  n  /\  y  ||  n ) } ,  RR ,  <  )  = inf ( { n  e.  NN  | 
( M  ||  n  /\  N  ||  n ) } ,  RR ,  <  ) )
1611, 15ifbieq2d 3390 . . . 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 2135 . . 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 271 . 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 107 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  M  e.  ZZ )
20 simpr 108 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  e.  ZZ )
21 c0ex 7227 . . . 4  |-  0  e.  _V
2221a1i 9 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  \/  N  =  0 ) )  -> 
0  e.  _V )
23 1zzd 8511 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  1  e.  ZZ )
24 nnuz 8787 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
25 rabeq 2602 . . . . . 6  |-  ( NN  =  ( ZZ>= `  1
)  ->  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) }  =  { n  e.  ( ZZ>=
`  1 )  |  ( M  ||  n  /\  N  ||  n ) } )
2624, 25ax-mp 7 . . . . 5  |-  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) }  =  { n  e.  ( ZZ>=
`  1 )  |  ( M  ||  n  /\  N  ||  n ) }
27 dvdsmul1 10425 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  ( M  x.  N ) )
2827adantr 270 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  ||  ( M  x.  N )
)
29 simpll 496 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  e.  ZZ )
30 simplr 497 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  e.  ZZ )
3129, 30zmulcld 8608 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M  x.  N )  e.  ZZ )
32 dvdsabsb 10422 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( M  ||  ( M  x.  N
)  <->  M  ||  ( abs `  ( M  x.  N
) ) ) )
3329, 31, 32syl2anc 403 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M  ||  ( M  x.  N
)  <->  M  ||  ( abs `  ( M  x.  N
) ) ) )
3428, 33mpbid 145 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  ||  ( abs `  ( M  x.  N ) ) )
35 dvdsmul2 10426 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  ||  ( M  x.  N ) )
3635adantr 270 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  ||  ( M  x.  N )
)
37 dvdsabsb 10422 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( N  ||  ( M  x.  N
)  <->  N  ||  ( abs `  ( M  x.  N
) ) ) )
3830, 31, 37syl2anc 403 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( N  ||  ( M  x.  N
)  <->  N  ||  ( abs `  ( M  x.  N
) ) ) )
3936, 38mpbid 145 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  ||  ( abs `  ( M  x.  N ) ) )
4029zcnd 8603 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  e.  CC )
4130zcnd 8603 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  e.  CC )
4240, 41absmuld 10281 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( abs `  ( M  x.  N )
)  =  ( ( abs `  M )  x.  ( abs `  N
) ) )
43 simpr 108 . . . . . . . . . . . . 13  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  -.  ( M  =  0  \/  N  =  0 ) )
44 ioran 702 . . . . . . . . . . . . 13  |-  ( -.  ( M  =  0  \/  N  =  0 )  <->  ( -.  M  =  0  /\  -.  N  =  0 ) )
4543, 44sylib 120 . . . . . . . . . . . 12  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( -.  M  =  0  /\  -.  N  =  0 ) )
4645simpld 110 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  -.  M  = 
0 )
4746neqned 2256 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  =/=  0
)
48 nnabscl 10187 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
4929, 47, 48syl2anc 403 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( abs `  M
)  e.  NN )
5045simprd 112 . . . . . . . . . . 11  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  -.  N  = 
0 )
5150neqned 2256 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  =/=  0
)
52 nnabscl 10187 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
5330, 51, 52syl2anc 403 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( abs `  N
)  e.  NN )
5449, 53nnmulcld 8206 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( ( abs `  M )  x.  ( abs `  N ) )  e.  NN )
5542, 54eqeltrd 2159 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( abs `  ( M  x.  N )
)  e.  NN )
56 breq2 3809 . . . . . . . . 9  |-  ( n  =  ( abs `  ( M  x.  N )
)  ->  ( M  ||  n  <->  M  ||  ( abs `  ( M  x.  N
) ) ) )
57 breq2 3809 . . . . . . . . 9  |-  ( n  =  ( abs `  ( M  x.  N )
)  ->  ( N  ||  n  <->  N  ||  ( abs `  ( M  x.  N
) ) ) )
5856, 57anbi12d 457 . . . . . . . 8  |-  ( n  =  ( abs `  ( M  x.  N )
)  ->  ( ( M  ||  n  /\  N  ||  n )  <->  ( M  ||  ( abs `  ( M  x.  N )
)  /\  N  ||  ( abs `  ( M  x.  N ) ) ) ) )
5958elrab3 2758 . . . . . . 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 886 . . . . 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 9173 . . . . . . 7  |-  ( n  e.  ( 1 ... ( abs `  ( M  x.  N )
) )  ->  n  e.  ZZ )
63 zdvdsdc 10424 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  n  e.  ZZ )  -> DECID  M 
||  n )
6429, 62, 63syl2an 283 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... ( abs `  ( M  x.  N ) ) ) )  -> DECID  M  ||  n )
65 zdvdsdc 10424 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  n  e.  ZZ )  -> DECID  N 
||  n )
6630, 62, 65syl2an 283 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... ( abs `  ( M  x.  N ) ) ) )  -> DECID  N  ||  n )
67 dcan 876 . . . . . 6  |-  (DECID  M  ||  n  ->  (DECID  N  ||  n  -> DECID  ( M  ||  n  /\  N  ||  n ) ) )
6864, 66, 67sylc 61 . . . . 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 10554 . . . 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 2621 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  -> inf ( { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) } ,  RR ,  <  )  e. 
_V )
71 lcmmndc 10651 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  \/  N  =  0 ) )
7222, 70, 71ifcldadc 3395 . 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, 72ovmpt2d 5679 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 102    <-> wb 103    \/ wo 662  DECID wdc 776    = wceq 1285    e. wcel 1434    =/= wne 2249   {crab 2357   _Vcvv 2610   ifcif 3368   class class class wbr 3805   ` cfv 4952  (class class class)co 5563    |-> cmpt2 5565  infcinf 6490   RRcr 7094   0cc0 7095   1c1 7096    x. cmul 7100    < clt 7267   NNcn 8158   ZZcz 8484   ZZ>=cuz 8752   ...cfz 9157   abscabs 10084    || cdvds 10403   lcm clcm 10649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-isom 4961  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-sup 6491  df-inf 6492  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fz 9158  df-fzo 9282  df-fl 9404  df-mod 9457  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086  df-dvds 10404  df-lcm 10650
This theorem is referenced by:  lcmcom  10653  lcm0val  10654  lcmn0val  10655  lcmass  10674
  Copyright terms: Public domain W3C validator