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Theorem lcmval 31122
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 14021 and gcdval 14022. (Contributed by Steve Rodriguez, 20-Jan-2020.)
Assertion
Ref Expression
lcmval  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  =  if ( ( M  =  0  \/  N  =  0 ) ,  0 ,  sup ( { 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 eqeq1 2471 . . . 4  |-  ( x  =  M  ->  (
x  =  0  <->  M  =  0 ) )
21orbi1d 702 . . 3  |-  ( x  =  M  ->  (
( x  =  0  \/  y  =  0 )  <->  ( M  =  0  \/  y  =  0 ) ) )
3 breq1 4456 . . . . . 6  |-  ( x  =  M  ->  (
x  ||  n  <->  M  ||  n
) )
43anbi1d 704 . . . . 5  |-  ( x  =  M  ->  (
( x  ||  n  /\  y  ||  n )  <-> 
( M  ||  n  /\  y  ||  n ) ) )
54rabbidv 3110 . . . 4  |-  ( x  =  M  ->  { n  e.  NN  |  ( x 
||  n  /\  y  ||  n ) }  =  { n  e.  NN  |  ( M  ||  n  /\  y  ||  n
) } )
65supeq1d 7918 . . 3  |-  ( x  =  M  ->  sup ( { n  e.  NN  |  ( x  ||  n  /\  y  ||  n
) } ,  RR ,  `'  <  )  =  sup ( { n  e.  NN  |  ( M 
||  n  /\  y  ||  n ) } ,  RR ,  `'  <  ) )
72, 6ifbieq2d 3970 . 2  |-  ( x  =  M  ->  if ( ( x  =  0  \/  y  =  0 ) ,  0 ,  sup ( { n  e.  NN  | 
( x  ||  n  /\  y  ||  n ) } ,  RR ,  `'  <  ) )  =  if ( ( M  =  0  \/  y  =  0 ) ,  0 ,  sup ( { n  e.  NN  |  ( M  ||  n  /\  y  ||  n
) } ,  RR ,  `'  <  ) ) )
8 eqeq1 2471 . . . 4  |-  ( y  =  N  ->  (
y  =  0  <->  N  =  0 ) )
98orbi2d 701 . . 3  |-  ( y  =  N  ->  (
( M  =  0  \/  y  =  0 )  <->  ( M  =  0  \/  N  =  0 ) ) )
10 breq1 4456 . . . . . 6  |-  ( y  =  N  ->  (
y  ||  n  <->  N  ||  n
) )
1110anbi2d 703 . . . . 5  |-  ( y  =  N  ->  (
( M  ||  n  /\  y  ||  n )  <-> 
( M  ||  n  /\  N  ||  n ) ) )
1211rabbidv 3110 . . . 4  |-  ( y  =  N  ->  { n  e.  NN  |  ( M 
||  n  /\  y  ||  n ) }  =  { n  e.  NN  |  ( M  ||  n  /\  N  ||  n
) } )
1312supeq1d 7918 . . 3  |-  ( y  =  N  ->  sup ( { n  e.  NN  |  ( M  ||  n  /\  y  ||  n
) } ,  RR ,  `'  <  )  =  sup ( { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) } ,  RR ,  `'  <  ) )
149, 13ifbieq2d 3970 . 2  |-  ( y  =  N  ->  if ( ( M  =  0  \/  y  =  0 ) ,  0 ,  sup ( { n  e.  NN  | 
( M  ||  n  /\  y  ||  n ) } ,  RR ,  `'  <  ) )  =  if ( ( M  =  0  \/  N  =  0 ) ,  0 ,  sup ( { n  e.  NN  |  ( M  ||  n  /\  N  ||  n
) } ,  RR ,  `'  <  ) ) )
15 df-lcm 31121 . 2  |- lcm  =  ( x  e.  ZZ , 
y  e.  ZZ  |->  if ( ( x  =  0  \/  y  =  0 ) ,  0 ,  sup ( { n  e.  NN  | 
( x  ||  n  /\  y  ||  n ) } ,  RR ,  `'  <  ) ) )
16 c0ex 9602 . . 3  |-  0  e.  _V
17 gtso 9678 . . . 4  |-  `'  <  Or  RR
1817supex 7935 . . 3  |-  sup ( { n  e.  NN  |  ( M  ||  n  /\  N  ||  n
) } ,  RR ,  `'  <  )  e. 
_V
1916, 18ifex 4014 . 2  |-  if ( ( M  =  0  \/  N  =  0 ) ,  0 ,  sup ( { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) } ,  RR ,  `'  <  ) )  e.  _V
207, 14, 15, 19ovmpt2 6433 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  =  if ( ( M  =  0  \/  N  =  0 ) ,  0 ,  sup ( { n  e.  NN  |  ( M  ||  n  /\  N  ||  n
) } ,  RR ,  `'  <  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2821   ifcif 3945   class class class wbr 4453   `'ccnv 5004  (class class class)co 6295   supcsup 7912   RRcr 9503   0cc0 9504    < clt 9640   NNcn 10548   ZZcz 10876    || cdivides 13864   lcm clcm 31120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-mulcl 9566  ax-i2m1 9572  ax-pre-lttri 9578  ax-pre-lttrn 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-ltxr 9645  df-lcm 31121
This theorem is referenced by:  lcmcom  31123  lcm0val  31124  lcmn0val  31125  lcmass  31142
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