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Theorem lcmval 11671
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 11563 and gcdval 11575. (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 11669 . . 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 2124 . . . . . 6  |-  ( x  =  M  ->  (
x  =  0  <->  M  =  0 ) )
43orbi1d 765 . . . . 5  |-  ( x  =  M  ->  (
( x  =  0  \/  y  =  0 )  <->  ( M  =  0  \/  y  =  0 ) ) )
5 breq1 3902 . . . . . . . 8  |-  ( x  =  M  ->  (
x  ||  n  <->  M  ||  n
) )
65anbi1d 460 . . . . . . 7  |-  ( x  =  M  ->  (
( x  ||  n  /\  y  ||  n )  <-> 
( M  ||  n  /\  y  ||  n ) ) )
76rabbidv 2649 . . . . . 6  |-  ( x  =  M  ->  { n  e.  NN  |  ( x 
||  n  /\  y  ||  n ) }  =  { n  e.  NN  |  ( M  ||  n  /\  y  ||  n
) } )
87infeq1d 6867 . . . . 5  |-  ( x  =  M  -> inf ( { n  e.  NN  | 
( x  ||  n  /\  y  ||  n ) } ,  RR ,  <  )  = inf ( { n  e.  NN  | 
( M  ||  n  /\  y  ||  n ) } ,  RR ,  <  ) )
94, 8ifbieq2d 3466 . . . 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 2124 . . . . . 6  |-  ( y  =  N  ->  (
y  =  0  <->  N  =  0 ) )
1110orbi2d 764 . . . . 5  |-  ( y  =  N  ->  (
( M  =  0  \/  y  =  0 )  <->  ( M  =  0  \/  N  =  0 ) ) )
12 breq1 3902 . . . . . . . 8  |-  ( y  =  N  ->  (
y  ||  n  <->  N  ||  n
) )
1312anbi2d 459 . . . . . . 7  |-  ( y  =  N  ->  (
( M  ||  n  /\  y  ||  n )  <-> 
( M  ||  n  /\  N  ||  n ) ) )
1413rabbidv 2649 . . . . . 6  |-  ( y  =  N  ->  { n  e.  NN  |  ( M 
||  n  /\  y  ||  n ) }  =  { n  e.  NN  |  ( M  ||  n  /\  N  ||  n
) } )
1514infeq1d 6867 . . . . 5  |-  ( y  =  N  -> inf ( { n  e.  NN  | 
( M  ||  n  /\  y  ||  n ) } ,  RR ,  <  )  = inf ( { n  e.  NN  | 
( M  ||  n  /\  N  ||  n ) } ,  RR ,  <  ) )
1611, 15ifbieq2d 3466 . . . 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 2170 . . 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 7728 . . . 4  |-  0  e.  _V
2221a1i 9 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  \/  N  =  0 ) )  -> 
0  e.  _V )
23 1zzd 9049 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  1  e.  ZZ )
24 nnuz 9329 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
25 rabeq 2652 . . . . . 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 11442 . . . . . . . 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 503 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  e.  ZZ )
30 simplr 504 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  e.  ZZ )
3129, 30zmulcld 9147 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M  x.  N )  e.  ZZ )
32 dvdsabsb 11439 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( M  ||  ( M  x.  N
)  <->  M  ||  ( abs `  ( M  x.  N
) ) ) )
3329, 31, 32syl2anc 408 . . . . . . 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 11443 . . . . . . . 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 11439 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( N  ||  ( M  x.  N
)  <->  N  ||  ( abs `  ( M  x.  N
) ) ) )
3830, 31, 37syl2anc 408 . . . . . . 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 9142 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  e.  CC )
4130zcnd 9142 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  e.  CC )
4240, 41absmuld 10934 . . . . . . . 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 726 . . . . . . . . . . . . 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 2292 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  =/=  0
)
48 nnabscl 10840 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
4929, 47, 48syl2anc 408 . . . . . . . . 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 2292 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  =/=  0
)
52 nnabscl 10840 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
5330, 51, 52syl2anc 408 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( abs `  N
)  e.  NN )
5449, 53nnmulcld 8737 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( ( abs `  M )  x.  ( abs `  N ) )  e.  NN )
5542, 54eqeltrd 2194 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( abs `  ( M  x.  N )
)  e.  NN )
56 breq2 3903 . . . . . . . . 9  |-  ( n  =  ( abs `  ( M  x.  N )
)  ->  ( M  ||  n  <->  M  ||  ( abs `  ( M  x.  N
) ) ) )
57 breq2 3903 . . . . . . . . 9  |-  ( n  =  ( abs `  ( M  x.  N )
)  ->  ( N  ||  n  <->  N  ||  ( abs `  ( M  x.  N
) ) ) )
5856, 57anbi12d 464 . . . . . . . 8  |-  ( n  =  ( abs `  ( M  x.  N )
)  ->  ( ( M  ||  n  /\  N  ||  n )  <->  ( M  ||  ( abs `  ( M  x.  N )
)  /\  N  ||  ( abs `  ( M  x.  N ) ) ) ) )
5958elrab3 2814 . . . . . . 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 913 . . . . 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 9774 . . . . . . 7  |-  ( n  e.  ( 1 ... ( abs `  ( M  x.  N )
) )  ->  n  e.  ZZ )
63 zdvdsdc 11441 . . . . . . 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 11441 . . . . . . 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 903 . . . . . 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 11571 . . . 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 2673 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  -> inf ( { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) } ,  RR ,  <  )  e. 
_V )
71 lcmmndc 11670 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  \/  N  =  0 ) )
7222, 70, 71ifcldadc 3471 . 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 5866 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 682  DECID wdc 804    = wceq 1316    e. wcel 1465    =/= wne 2285   {crab 2397   _Vcvv 2660   ifcif 3444   class class class wbr 3899   ` cfv 5093  (class class class)co 5742    e. cmpo 5744  infcinf 6838   RRcr 7587   0cc0 7588   1c1 7589    x. cmul 7593    < clt 7768   NNcn 8688   ZZcz 9022   ZZ>=cuz 9294   ...cfz 9758   abscabs 10737    || cdvds 11420   lcm clcm 11668
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-isom 5102  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-sup 6839  df-inf 6840  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-fz 9759  df-fzo 9888  df-fl 10011  df-mod 10064  df-seqfrec 10187  df-exp 10261  df-cj 10582  df-re 10583  df-im 10584  df-rsqrt 10738  df-abs 10739  df-dvds 11421  df-lcm 11669
This theorem is referenced by:  lcmcom  11672  lcm0val  11673  lcmn0val  11674  lcmass  11693
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