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Theorem lcmass 10699
Description: Associative law for lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
Assertion
Ref Expression
lcmass  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N lcm  M ) lcm 
P )  =  ( N lcm  ( M lcm  P
) ) )

Proof of Theorem lcmass
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 orass 717 . . 3  |-  ( ( ( N  =  0  \/  M  =  0 )  \/  P  =  0 )  <->  ( N  =  0  \/  ( M  =  0  \/  P  =  0 ) ) )
2 anass 393 . . . . . 6  |-  ( ( ( N  ||  x  /\  M  ||  x )  /\  P  ||  x
)  <->  ( N  ||  x  /\  ( M  ||  x  /\  P  ||  x
) ) )
32a1i 9 . . . . 5  |-  ( x  e.  NN  ->  (
( ( N  ||  x  /\  M  ||  x
)  /\  P  ||  x
)  <->  ( N  ||  x  /\  ( M  ||  x  /\  P  ||  x
) ) ) )
43rabbiia 2597 . . . 4  |-  { x  e.  NN  |  ( ( N  ||  x  /\  M  ||  x )  /\  P  ||  x ) }  =  { x  e.  NN  |  ( N 
||  x  /\  ( M  ||  x  /\  P  ||  x ) ) }
54infeq1i 6522 . . 3  |- inf ( { x  e.  NN  | 
( ( N  ||  x  /\  M  ||  x
)  /\  P  ||  x
) } ,  RR ,  <  )  = inf ( { x  e.  NN  |  ( N  ||  x  /\  ( M  ||  x  /\  P  ||  x
) ) } ,  RR ,  <  )
61, 5ifbieq2i 3390 . 2  |-  if ( ( ( N  =  0  \/  M  =  0 )  \/  P  =  0 ) ,  0 , inf ( { x  e.  NN  | 
( ( N  ||  x  /\  M  ||  x
)  /\  P  ||  x
) } ,  RR ,  <  ) )  =  if ( ( N  =  0  \/  ( M  =  0  \/  P  =  0 ) ) ,  0 , inf ( { x  e.  NN  |  ( N 
||  x  /\  ( M  ||  x  /\  P  ||  x ) ) } ,  RR ,  <  ) )
7 lcmcl 10686 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N lcm  M )  e.  NN0 )
873adant3 959 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( N lcm  M )  e.  NN0 )
98nn0zd 8625 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( N lcm  M )  e.  ZZ )
10 simp3 941 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  P  e.  ZZ )
11 lcmval 10677 . . . 4  |-  ( ( ( N lcm  M )  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N lcm  M ) lcm 
P )  =  if ( ( ( N lcm 
M )  =  0  \/  P  =  0 ) ,  0 , inf ( { x  e.  NN  |  ( ( N lcm  M )  ||  x  /\  P  ||  x
) } ,  RR ,  <  ) ) )
129, 10, 11syl2anc 403 . . 3  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N lcm  M ) lcm 
P )  =  if ( ( ( N lcm 
M )  =  0  \/  P  =  0 ) ,  0 , inf ( { x  e.  NN  |  ( ( N lcm  M )  ||  x  /\  P  ||  x
) } ,  RR ,  <  ) ) )
13 lcmeq0 10685 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N lcm  M
)  =  0  <->  ( N  =  0  \/  M  =  0 ) ) )
14133adant3 959 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N lcm  M )  =  0  <->  ( N  =  0  \/  M  =  0 ) ) )
1514orbi1d 738 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( ( N lcm  M
)  =  0  \/  P  =  0 )  <-> 
( ( N  =  0  \/  M  =  0 )  \/  P  =  0 ) ) )
1615bicomd 139 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( ( N  =  0  \/  M  =  0 )  \/  P  =  0 )  <->  ( ( N lcm  M )  =  0  \/  P  =  0 ) ) )
17 nnz 8528 . . . . . . . . 9  |-  ( x  e.  NN  ->  x  e.  ZZ )
1817adantl 271 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  x  e.  ZZ )
19 simp1 939 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  N  e.  ZZ )
2019adantr 270 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  N  e.  ZZ )
21 simpl2 943 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  M  e.  ZZ )
22 lcmdvdsb 10698 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  ->  (
( N  ||  x  /\  M  ||  x )  <-> 
( N lcm  M ) 
||  x ) )
2318, 20, 21, 22syl3anc 1170 . . . . . . 7  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  ( ( N 
||  x  /\  M  ||  x )  <->  ( N lcm  M )  ||  x ) )
2423anbi1d 453 . . . . . 6  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  ( ( ( N  ||  x  /\  M  ||  x )  /\  P  ||  x )  <->  ( ( N lcm  M )  ||  x  /\  P  ||  x ) ) )
2524rabbidva 2599 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  { x  e.  NN  |  ( ( N  ||  x  /\  M  ||  x )  /\  P  ||  x ) }  =  { x  e.  NN  |  ( ( N lcm  M )  ||  x  /\  P  ||  x
) } )
2625infeq1d 6521 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  -> inf ( { x  e.  NN  | 
( ( N  ||  x  /\  M  ||  x
)  /\  P  ||  x
) } ,  RR ,  <  )  = inf ( { x  e.  NN  |  ( ( N lcm 
M )  ||  x  /\  P  ||  x ) } ,  RR ,  <  ) )
2716, 26ifbieq2d 3391 . . 3  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  if ( ( ( N  =  0  \/  M  =  0 )  \/  P  =  0 ) ,  0 , inf ( { x  e.  NN  |  ( ( N 
||  x  /\  M  ||  x )  /\  P  ||  x ) } ,  RR ,  <  ) )  =  if ( ( ( N lcm  M )  =  0  \/  P  =  0 ) ,  0 , inf ( { x  e.  NN  | 
( ( N lcm  M
)  ||  x  /\  P  ||  x ) } ,  RR ,  <  ) ) )
2812, 27eqtr4d 2118 . 2  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N lcm  M ) lcm 
P )  =  if ( ( ( N  =  0  \/  M  =  0 )  \/  P  =  0 ) ,  0 , inf ( { x  e.  NN  |  ( ( N 
||  x  /\  M  ||  x )  /\  P  ||  x ) } ,  RR ,  <  ) ) )
29 lcmcl 10686 . . . . . 6  |-  ( ( M  e.  ZZ  /\  P  e.  ZZ )  ->  ( M lcm  P )  e.  NN0 )
30293adant1 957 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( M lcm  P )  e.  NN0 )
3130nn0zd 8625 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( M lcm  P )  e.  ZZ )
32 lcmval 10677 . . . 4  |-  ( ( N  e.  ZZ  /\  ( M lcm  P )  e.  ZZ )  ->  ( N lcm  ( M lcm  P ) )  =  if ( ( N  =  0  \/  ( M lcm  P
)  =  0 ) ,  0 , inf ( { x  e.  NN  |  ( N  ||  x  /\  ( M lcm  P
)  ||  x ) } ,  RR ,  <  ) ) )
3319, 31, 32syl2anc 403 . . 3  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( N lcm  ( M lcm  P ) )  =  if ( ( N  =  0  \/  ( M lcm  P
)  =  0 ) ,  0 , inf ( { x  e.  NN  |  ( N  ||  x  /\  ( M lcm  P
)  ||  x ) } ,  RR ,  <  ) ) )
34 lcmeq0 10685 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( M lcm  P
)  =  0  <->  ( M  =  0  \/  P  =  0 ) ) )
35343adant1 957 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( M lcm  P )  =  0  <->  ( M  =  0  \/  P  =  0 ) ) )
3635orbi2d 737 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N  =  0  \/  ( M lcm  P
)  =  0 )  <-> 
( N  =  0  \/  ( M  =  0  \/  P  =  0 ) ) ) )
3736bicomd 139 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N  =  0  \/  ( M  =  0  \/  P  =  0 ) )  <->  ( N  =  0  \/  ( M lcm  P )  =  0 ) ) )
3810adantr 270 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  P  e.  ZZ )
39 lcmdvdsb 10698 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( M  ||  x  /\  P  ||  x )  <-> 
( M lcm  P ) 
||  x ) )
4018, 21, 38, 39syl3anc 1170 . . . . . . 7  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  ( ( M 
||  x  /\  P  ||  x )  <->  ( M lcm  P )  ||  x ) )
4140anbi2d 452 . . . . . 6  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  ( ( N 
||  x  /\  ( M  ||  x  /\  P  ||  x ) )  <->  ( N  ||  x  /\  ( M lcm 
P )  ||  x
) ) )
4241rabbidva 2599 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  { x  e.  NN  |  ( N 
||  x  /\  ( M  ||  x  /\  P  ||  x ) ) }  =  { x  e.  NN  |  ( N 
||  x  /\  ( M lcm  P )  ||  x
) } )
4342infeq1d 6521 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  -> inf ( { x  e.  NN  | 
( N  ||  x  /\  ( M  ||  x  /\  P  ||  x ) ) } ,  RR ,  <  )  = inf ( { x  e.  NN  |  ( N  ||  x  /\  ( M lcm  P
)  ||  x ) } ,  RR ,  <  ) )
4437, 43ifbieq2d 3391 . . 3  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  if ( ( N  =  0  \/  ( M  =  0  \/  P  =  0 ) ) ,  0 , inf ( { x  e.  NN  |  ( N  ||  x  /\  ( M  ||  x  /\  P  ||  x
) ) } ,  RR ,  <  ) )  =  if ( ( N  =  0  \/  ( M lcm  P )  =  0 ) ,  0 , inf ( { x  e.  NN  | 
( N  ||  x  /\  ( M lcm  P ) 
||  x ) } ,  RR ,  <  ) ) )
4533, 44eqtr4d 2118 . 2  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( N lcm  ( M lcm  P ) )  =  if ( ( N  =  0  \/  ( M  =  0  \/  P  =  0 ) ) ,  0 , inf ( { x  e.  NN  | 
( N  ||  x  /\  ( M  ||  x  /\  P  ||  x ) ) } ,  RR ,  <  ) ) )
466, 28, 453eqtr4a 2141 1  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N lcm  M ) lcm 
P )  =  ( N lcm  ( M lcm  P
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    /\ w3a 920    = wceq 1285    e. wcel 1434   {crab 2357   ifcif 3369   class class class wbr 3806  (class class class)co 5565  infcinf 6492   RRcr 7119   0cc0 7120    < clt 7292   NNcn 8183   NN0cn0 8432   ZZcz 8509    || cdvds 10428   lcm clcm 10674
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 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358  ax-cnex 7206  ax-resscn 7207  ax-1cn 7208  ax-1re 7209  ax-icn 7210  ax-addcl 7211  ax-addrcl 7212  ax-mulcl 7213  ax-mulrcl 7214  ax-addcom 7215  ax-mulcom 7216  ax-addass 7217  ax-mulass 7218  ax-distr 7219  ax-i2m1 7220  ax-0lt1 7221  ax-1rid 7222  ax-0id 7223  ax-rnegex 7224  ax-precex 7225  ax-cnre 7226  ax-pre-ltirr 7227  ax-pre-ltwlin 7228  ax-pre-lttrn 7229  ax-pre-apti 7230  ax-pre-ltadd 7231  ax-pre-mulgt0 7232  ax-pre-mulext 7233  ax-arch 7234  ax-caucvg 7235
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 2613  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-if 3370  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-ilim 4153  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-isom 4962  df-riota 5521  df-ov 5568  df-oprab 5569  df-mpt2 5570  df-1st 5820  df-2nd 5821  df-recs 5976  df-frec 6062  df-sup 6493  df-inf 6494  df-pnf 7294  df-mnf 7295  df-xr 7296  df-ltxr 7297  df-le 7298  df-sub 7425  df-neg 7426  df-reap 7819  df-ap 7826  df-div 7905  df-inn 8184  df-2 8242  df-3 8243  df-4 8244  df-n0 8433  df-z 8510  df-uz 8778  df-q 8863  df-rp 8893  df-fz 9183  df-fzo 9307  df-fl 9429  df-mod 9482  df-iseq 9599  df-iexp 9650  df-cj 9955  df-re 9956  df-im 9957  df-rsqrt 10110  df-abs 10111  df-dvds 10429  df-gcd 10571  df-lcm 10675
This theorem is referenced by: (None)
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