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Theorem lcmass 14658
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 533 . . 3  |-  ( ( ( N  =  0  \/  M  =  0 )  \/  P  =  0 )  <->  ( N  =  0  \/  ( M  =  0  \/  P  =  0 ) ) )
2 anass 661 . . . . . 6  |-  ( ( ( N  ||  x  /\  M  ||  x )  /\  P  ||  x
)  <->  ( N  ||  x  /\  ( M  ||  x  /\  P  ||  x
) ) )
32a1i 11 . . . . 5  |-  ( x  e.  NN  ->  (
( ( N  ||  x  /\  M  ||  x
)  /\  P  ||  x
)  <->  ( N  ||  x  /\  ( M  ||  x  /\  P  ||  x
) ) ) )
43rabbiia 3019 . . . 4  |-  { x  e.  NN  |  ( ( N  ||  x  /\  M  ||  x )  /\  P  ||  x ) }  =  { x  e.  NN  |  ( N 
||  x  /\  ( M  ||  x  /\  P  ||  x ) ) }
54infeq1i 8012 . . 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 3896 . 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 14645 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N lcm  M )  e.  NN0 )
873adant3 1050 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( N lcm  M )  e.  NN0 )
98nn0zd 11061 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( N lcm  M )  e.  ZZ )
10 simp3 1032 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  P  e.  ZZ )
11 lcmval 14634 . . . 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 673 . . 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 14644 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N lcm  M
)  =  0  <->  ( N  =  0  \/  M  =  0 ) ) )
14133adant3 1050 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N lcm  M )  =  0  <->  ( N  =  0  \/  M  =  0 ) ) )
1514orbi1d 717 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( ( N lcm  M
)  =  0  \/  P  =  0 )  <-> 
( ( N  =  0  \/  M  =  0 )  \/  P  =  0 ) ) )
1615bicomd 206 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( ( N  =  0  \/  M  =  0 )  \/  P  =  0 )  <->  ( ( N lcm  M )  =  0  \/  P  =  0 ) ) )
17 nnz 10983 . . . . . . . . 9  |-  ( x  e.  NN  ->  x  e.  ZZ )
1817adantl 473 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  x  e.  ZZ )
19 simp1 1030 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  N  e.  ZZ )
2019adantr 472 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  N  e.  ZZ )
21 simpl2 1034 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  M  e.  ZZ )
22 lcmdvdsb 14657 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  ->  (
( N  ||  x  /\  M  ||  x )  <-> 
( N lcm  M ) 
||  x ) )
2318, 20, 21, 22syl3anc 1292 . . . . . . 7  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  ( ( N 
||  x  /\  M  ||  x )  <->  ( N lcm  M )  ||  x ) )
2423anbi1d 719 . . . . . 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 3021 . . . . 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 8011 . . . 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 3897 . . 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 2508 . 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 14645 . . . . . 6  |-  ( ( M  e.  ZZ  /\  P  e.  ZZ )  ->  ( M lcm  P )  e.  NN0 )
30293adant1 1048 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( M lcm  P )  e.  NN0 )
3130nn0zd 11061 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( M lcm  P )  e.  ZZ )
32 lcmval 14634 . . . 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 673 . . 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 14644 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( M lcm  P
)  =  0  <->  ( M  =  0  \/  P  =  0 ) ) )
35343adant1 1048 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( M lcm  P )  =  0  <->  ( M  =  0  \/  P  =  0 ) ) )
3635orbi2d 716 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N  =  0  \/  ( M lcm  P
)  =  0 )  <-> 
( N  =  0  \/  ( M  =  0  \/  P  =  0 ) ) ) )
3736bicomd 206 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N  =  0  \/  ( M  =  0  \/  P  =  0 ) )  <->  ( N  =  0  \/  ( M lcm  P )  =  0 ) ) )
3810adantr 472 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  P  e.  ZZ )
39 lcmdvdsb 14657 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( M  ||  x  /\  P  ||  x )  <-> 
( M lcm  P ) 
||  x ) )
4018, 21, 38, 39syl3anc 1292 . . . . . . 7  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  NN )  ->  ( ( M 
||  x  /\  P  ||  x )  <->  ( M lcm  P )  ||  x ) )
4140anbi2d 718 . . . . . 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 3021 . . . . 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 8011 . . . 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 3897 . . 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 2508 . 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 2531 1  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N lcm  M ) lcm 
P )  =  ( N lcm  ( M lcm  P
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   {crab 2760   ifcif 3872   class class class wbr 4395  (class class class)co 6308  infcinf 7973   RRcr 9556   0cc0 9557    < clt 9693   NNcn 10631   NN0cn0 10893   ZZcz 10961    || cdvds 14382   lcm clcm 14626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-lcm 14630
This theorem is referenced by:  lcmfunsnlem2lem2  14691  lcmfun  14697
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