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Theorem lcmcllem 12098
Description: Lemma for lcmn0cl 12099 and dvdslcm 12100. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
Assertion
Ref Expression
lcmcllem  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M lcm  N
)  e.  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) } )
Distinct variable groups:    n, M    n, N

Proof of Theorem lcmcllem
StepHypRef Expression
1 lcmn0val 12097 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M lcm  N
)  = inf ( { n  e.  NN  | 
( M  ||  n  /\  N  ||  n ) } ,  RR ,  <  ) )
2 1zzd 9309 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  1  e.  ZZ )
3 nnuz 9592 . . . 4  |-  NN  =  ( ZZ>= `  1 )
43rabeqi 2745 . . 3  |-  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) }  =  { n  e.  ( ZZ>=
`  1 )  |  ( M  ||  n  /\  N  ||  n ) }
5 breq2 4022 . . . . 5  |-  ( n  =  ( abs `  ( M  x.  N )
)  ->  ( M  ||  n  <->  M  ||  ( abs `  ( M  x.  N
) ) ) )
6 breq2 4022 . . . . 5  |-  ( n  =  ( abs `  ( M  x.  N )
)  ->  ( N  ||  n  <->  N  ||  ( abs `  ( M  x.  N
) ) ) )
75, 6anbi12d 473 . . . 4  |-  ( n  =  ( abs `  ( M  x.  N )
)  ->  ( ( M  ||  n  /\  N  ||  n )  <->  ( M  ||  ( abs `  ( M  x.  N )
)  /\  N  ||  ( abs `  ( M  x.  N ) ) ) ) )
8 simpll 527 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  e.  ZZ )
9 simplr 528 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  e.  ZZ )
108, 9zmulcld 9410 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M  x.  N )  e.  ZZ )
118zcnd 9405 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  e.  CC )
129zcnd 9405 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  e.  CC )
13 ioran 753 . . . . . . . . . . . 12  |-  ( -.  ( M  =  0  \/  N  =  0 )  <->  ( -.  M  =  0  /\  -.  N  =  0 ) )
1413biimpi 120 . . . . . . . . . . 11  |-  ( -.  ( M  =  0  \/  N  =  0 )  ->  ( -.  M  =  0  /\  -.  N  =  0
) )
1514adantl 277 . . . . . . . . . 10  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( -.  M  =  0  /\  -.  N  =  0 ) )
1615simpld 112 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  -.  M  = 
0 )
1716neneqad 2439 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M  =/=  0
)
18 0zd 9294 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  0  e.  ZZ )
19 zapne 9356 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  0  e.  ZZ )  ->  ( M #  0  <->  M  =/=  0 ) )
208, 18, 19syl2anc 411 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M #  0  <-> 
M  =/=  0 ) )
2117, 20mpbird 167 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  M #  0 )
2215simprd 114 . . . . . . . . 9  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  -.  N  = 
0 )
2322neneqad 2439 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N  =/=  0
)
24 zapne 9356 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  ->  ( N #  0  <->  N  =/=  0 ) )
259, 18, 24syl2anc 411 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( N #  0  <-> 
N  =/=  0 ) )
2623, 25mpbird 167 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  N #  0 )
2711, 12, 21, 26mulap0d 8644 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M  x.  N ) #  0 )
28 zapne 9356 . . . . . . 7  |-  ( ( ( M  x.  N
)  e.  ZZ  /\  0  e.  ZZ )  ->  ( ( M  x.  N ) #  0  <->  ( M  x.  N )  =/=  0
) )
2910, 18, 28syl2anc 411 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( ( M  x.  N ) #  0  <-> 
( M  x.  N
)  =/=  0 ) )
3027, 29mpbid 147 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M  x.  N )  =/=  0
)
31 nnabscl 11140 . . . . 5  |-  ( ( ( M  x.  N
)  e.  ZZ  /\  ( M  x.  N
)  =/=  0 )  ->  ( abs `  ( M  x.  N )
)  e.  NN )
3210, 30, 31syl2anc 411 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( abs `  ( M  x.  N )
)  e.  NN )
33 dvdsmul1 11851 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  ( M  x.  N ) )
34 zmulcl 9335 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N
)  e.  ZZ )
35 dvdsabsb 11848 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( M  ||  ( M  x.  N
)  <->  M  ||  ( abs `  ( M  x.  N
) ) ) )
3634, 35syldan 282 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  ( M  x.  N )  <->  M 
||  ( abs `  ( M  x.  N )
) ) )
3733, 36mpbid 147 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  M  ||  ( abs `  ( M  x.  N
) ) )
38 dvdsmul2 11852 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  ||  ( M  x.  N ) )
39 dvdsabsb 11848 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  ( M  x.  N
)  e.  ZZ )  ->  ( N  ||  ( M  x.  N
)  <->  N  ||  ( abs `  ( M  x.  N
) ) ) )
4034, 39sylan2 286 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  ( M  e.  ZZ  /\  N  e.  ZZ ) )  ->  ( N  ||  ( M  x.  N
)  <->  N  ||  ( abs `  ( M  x.  N
) ) ) )
4140anabss7 583 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  ||  ( M  x.  N )  <->  N 
||  ( abs `  ( M  x.  N )
) ) )
4238, 41mpbid 147 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  N  ||  ( abs `  ( M  x.  N
) ) )
4337, 42jca 306 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  ( abs `  ( M  x.  N ) )  /\  N  ||  ( abs `  ( M  x.  N )
) ) )
4443adantr 276 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M  ||  ( abs `  ( M  x.  N ) )  /\  N  ||  ( abs `  ( M  x.  N ) ) ) )
457, 32, 44elrabd 2910 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( abs `  ( M  x.  N )
)  e.  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) } )
46 simplll 533 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... ( abs `  ( M  x.  N ) ) ) )  ->  M  e.  ZZ )
47 elfzelz 10054 . . . . . 6  |-  ( n  e.  ( 1 ... ( abs `  ( M  x.  N )
) )  ->  n  e.  ZZ )
4847adantl 277 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... ( abs `  ( M  x.  N ) ) ) )  ->  n  e.  ZZ )
49 zdvdsdc 11850 . . . . 5  |-  ( ( M  e.  ZZ  /\  n  e.  ZZ )  -> DECID  M 
||  n )
5046, 48, 49syl2anc 411 . . . 4  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... ( abs `  ( M  x.  N ) ) ) )  -> DECID  M  ||  n )
51 simpllr 534 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... ( abs `  ( M  x.  N ) ) ) )  ->  N  e.  ZZ )
52 zdvdsdc 11850 . . . . 5  |-  ( ( N  e.  ZZ  /\  n  e.  ZZ )  -> DECID  N 
||  n )
5351, 48, 52syl2anc 411 . . . 4  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... ( abs `  ( M  x.  N ) ) ) )  -> DECID  N  ||  n )
5450, 53dcand 934 . . 3  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0
) )  /\  n  e.  ( 1 ... ( abs `  ( M  x.  N ) ) ) )  -> DECID  ( M  ||  n  /\  N  ||  n ) )
552, 4, 45, 54infssuzcldc 11983 . 2  |-  ( ( ( 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
) } )
561, 55eqeltrd 2266 1  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M lcm  N
)  e.  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) } )
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   class class class wbr 4018   ` cfv 5235  (class class class)co 5895  infcinf 7011   RRcr 7839   0cc0 7840   1c1 7841    x. cmul 7845    < clt 8021   # cap 8567   NNcn 8948   ZZcz 9282   ZZ>=cuz 9557   ...cfz 10037   abscabs 11037    || cdvds 11825   lcm clcm 12091
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 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958  ax-arch 7959  ax-caucvg 7960
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 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-frec 6415  df-sup 7012  df-inf 7013  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568  df-div 8659  df-inn 8949  df-2 9007  df-3 9008  df-4 9009  df-n0 9206  df-z 9283  df-uz 9558  df-q 9649  df-rp 9683  df-fz 10038  df-fzo 10172  df-fl 10300  df-mod 10353  df-seqfrec 10476  df-exp 10550  df-cj 10882  df-re 10883  df-im 10884  df-rsqrt 11038  df-abs 11039  df-dvds 11826  df-lcm 12092
This theorem is referenced by:  lcmn0cl  12099  dvdslcm  12100
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