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Theorem lcmass 12055
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 767 . . 3  |-  ( ( ( N  =  0  \/  M  =  0 )  \/  P  =  0 )  <->  ( N  =  0  \/  ( M  =  0  \/  P  =  0 ) ) )
2 anass 401 . . . . . 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 2722 . . . 4  |-  { x  e.  NN  |  ( ( N  ||  x  /\  M  ||  x )  /\  P  ||  x ) }  =  { x  e.  NN  |  ( N 
||  x  /\  ( M  ||  x  /\  P  ||  x ) ) }
54infeq1i 7005 . . 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 3557 . 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 12042 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N lcm  M )  e.  NN0 )
873adant3 1017 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( N lcm  M )  e.  NN0 )
98nn0zd 9349 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( N lcm  M )  e.  ZZ )
10 simp3 999 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  P  e.  ZZ )
11 lcmval 12033 . . . 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 411 . . 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 12041 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N lcm  M
)  =  0  <->  ( N  =  0  \/  M  =  0 ) ) )
14133adant3 1017 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N lcm  M )  =  0  <->  ( N  =  0  \/  M  =  0 ) ) )
1514orbi1d 791 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( ( N lcm  M
)  =  0  \/  P  =  0 )  <-> 
( ( N  =  0  \/  M  =  0 )  \/  P  =  0 ) ) )
1615bicomd 141 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( ( N  =  0  \/  M  =  0 )  \/  P  =  0 )  <->  ( ( N lcm  M )  =  0  \/  P  =  0 ) ) )
17 nnz 9248 . . . . . . . . 9  |-  ( x  e.  NN  ->  x  e.  ZZ )
1817adantl 277 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  x  e.  ZZ )
19 simp1 997 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  N  e.  ZZ )
2019adantr 276 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  N  e.  ZZ )
21 simpl2 1001 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  M  e.  ZZ )
22 lcmdvdsb 12054 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  ->  (
( N  ||  x  /\  M  ||  x )  <-> 
( N lcm  M ) 
||  x ) )
2318, 20, 21, 22syl3anc 1238 . . . . . . 7  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  ( ( N 
||  x  /\  M  ||  x )  <->  ( N lcm  M )  ||  x ) )
2423anbi1d 465 . . . . . 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 2725 . . . . 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 7004 . . . 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 3558 . . 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 2213 . 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 12042 . . . . . 6  |-  ( ( M  e.  ZZ  /\  P  e.  ZZ )  ->  ( M lcm  P )  e.  NN0 )
30293adant1 1015 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( M lcm  P )  e.  NN0 )
3130nn0zd 9349 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( M lcm  P )  e.  ZZ )
32 lcmval 12033 . . . 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 411 . . 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 12041 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( M lcm  P
)  =  0  <->  ( M  =  0  \/  P  =  0 ) ) )
35343adant1 1015 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( M lcm  P )  =  0  <->  ( M  =  0  \/  P  =  0 ) ) )
3635orbi2d 790 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N  =  0  \/  ( M lcm  P
)  =  0 )  <-> 
( N  =  0  \/  ( M  =  0  \/  P  =  0 ) ) ) )
3736bicomd 141 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N  =  0  \/  ( M  =  0  \/  P  =  0 ) )  <->  ( N  =  0  \/  ( M lcm  P )  =  0 ) ) )
3810adantr 276 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  P  e.  ZZ )
39 lcmdvdsb 12054 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( M  ||  x  /\  P  ||  x )  <-> 
( M lcm  P ) 
||  x ) )
4018, 21, 38, 39syl3anc 1238 . . . . . . 7  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  ( ( M 
||  x  /\  P  ||  x )  <->  ( M lcm  P )  ||  x ) )
4140anbi2d 464 . . . . . 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 2725 . . . . 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 7004 . . . 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 3558 . . 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 2213 . 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 2236 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 104    <-> wb 105    \/ wo 708    /\ w3a 978    = wceq 1353    e. wcel 2148   {crab 2459   ifcif 3534   class class class wbr 4000  (class class class)co 5868  infcinf 6975   RRcr 7788   0cc0 7789    < clt 7969   NNcn 8895   NN0cn0 9152   ZZcz 9229    || cdvds 11765   lcm clcm 12030
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906  ax-pre-mulext 7907  ax-arch 7908  ax-caucvg 7909
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-isom 5220  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-frec 6385  df-sup 6976  df-inf 6977  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516  df-div 8606  df-inn 8896  df-2 8954  df-3 8955  df-4 8956  df-n0 9153  df-z 9230  df-uz 9505  df-q 9596  df-rp 9628  df-fz 9983  df-fzo 10116  df-fl 10243  df-mod 10296  df-seqfrec 10419  df-exp 10493  df-cj 10822  df-re 10823  df-im 10824  df-rsqrt 10978  df-abs 10979  df-dvds 11766  df-gcd 11914  df-lcm 12031
This theorem is referenced by: (None)
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