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Theorem gcdass 13047
Description: Associative law for  gcd operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
gcdass  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N  gcd  M
)  gcd  P )  =  ( N  gcd  ( M  gcd  P ) ) )

Proof of Theorem gcdass
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 anass 632 . . 3  |-  ( ( ( N  =  0  /\  M  =  0 )  /\  P  =  0 )  <->  ( N  =  0  /\  ( M  =  0  /\  P  =  0 ) ) )
2 anass 632 . . . . . 6  |-  ( ( ( x  ||  N  /\  x  ||  M )  /\  x  ||  P
)  <->  ( x  ||  N  /\  ( x  ||  M  /\  x  ||  P
) ) )
32a1i 11 . . . . 5  |-  ( x  e.  ZZ  ->  (
( ( x  ||  N  /\  x  ||  M
)  /\  x  ||  P
)  <->  ( x  ||  N  /\  ( x  ||  M  /\  x  ||  P
) ) ) )
43rabbiia 2948 . . . 4  |-  { x  e.  ZZ  |  ( ( x  ||  N  /\  x  ||  M )  /\  x  ||  P ) }  =  { x  e.  ZZ  |  ( x 
||  N  /\  (
x  ||  M  /\  x  ||  P ) ) }
54supeq1i 7454 . . 3  |-  sup ( { x  e.  ZZ  |  ( ( x 
||  N  /\  x  ||  M )  /\  x  ||  P ) } ,  RR ,  <  )  =  sup ( { x  e.  ZZ  |  ( x 
||  N  /\  (
x  ||  M  /\  x  ||  P ) ) } ,  RR ,  <  )
61, 5ifbieq2i 3760 . 2  |-  if ( ( ( N  =  0  /\  M  =  0 )  /\  P  =  0 ) ,  0 ,  sup ( { x  e.  ZZ  |  ( ( x 
||  N  /\  x  ||  M )  /\  x  ||  P ) } ,  RR ,  <  ) )  =  if ( ( N  =  0  /\  ( M  =  0  /\  P  =  0 ) ) ,  0 ,  sup ( { x  e.  ZZ  | 
( x  ||  N  /\  ( x  ||  M  /\  x  ||  P ) ) } ,  RR ,  <  ) )
7 gcdcl 13019 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( N  gcd  M
)  e.  NN0 )
873adant3 978 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( N  gcd  M )  e. 
NN0 )
98nn0zd 10375 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( N  gcd  M )  e.  ZZ )
10 simp3 960 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  P  e.  ZZ )
11 gcdval 13010 . . . 4  |-  ( ( ( N  gcd  M
)  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( N  gcd  M )  gcd  P )  =  if ( ( ( N  gcd  M
)  =  0  /\  P  =  0 ) ,  0 ,  sup ( { x  e.  ZZ  |  ( x  ||  ( N  gcd  M )  /\  x  ||  P
) } ,  RR ,  <  ) ) )
129, 10, 11syl2anc 644 . . 3  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N  gcd  M
)  gcd  P )  =  if ( ( ( N  gcd  M )  =  0  /\  P  =  0 ) ,  0 ,  sup ( { x  e.  ZZ  |  ( x  ||  ( N  gcd  M )  /\  x  ||  P
) } ,  RR ,  <  ) ) )
13 gcdeq0 13023 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( N  gcd  M )  =  0  <->  ( N  =  0  /\  M  =  0 ) ) )
14133adant3 978 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N  gcd  M
)  =  0  <->  ( N  =  0  /\  M  =  0 ) ) )
1514anbi1d 687 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( ( N  gcd  M )  =  0  /\  P  =  0 )  <-> 
( ( N  =  0  /\  M  =  0 )  /\  P  =  0 ) ) )
1615bicomd 194 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( ( N  =  0  /\  M  =  0 )  /\  P  =  0 )  <->  ( ( N  gcd  M )  =  0  /\  P  =  0 ) ) )
17 simpr 449 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
18 simpl1 961 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  ZZ )  ->  N  e.  ZZ )
19 simpl2 962 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  ZZ )  ->  M  e.  ZZ )
20 dvdsgcdb 13046 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  ->  (
( x  ||  N  /\  x  ||  M )  <-> 
x  ||  ( N  gcd  M ) ) )
2117, 18, 19, 20syl3anc 1185 . . . . . . 7  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x 
||  N  /\  x  ||  M )  <->  x  ||  ( N  gcd  M ) ) )
2221anbi1d 687 . . . . . 6  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( ( x  ||  N  /\  x  ||  M )  /\  x  ||  P )  <->  ( x  ||  ( N  gcd  M
)  /\  x  ||  P
) ) )
2322rabbidva 2949 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  { x  e.  ZZ  |  ( ( x  ||  N  /\  x  ||  M )  /\  x  ||  P ) }  =  { x  e.  ZZ  |  ( x 
||  ( N  gcd  M )  /\  x  ||  P ) } )
2423supeq1d 7453 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  sup ( { x  e.  ZZ  |  ( ( x 
||  N  /\  x  ||  M )  /\  x  ||  P ) } ,  RR ,  <  )  =  sup ( { x  e.  ZZ  |  ( x 
||  ( N  gcd  M )  /\  x  ||  P ) } ,  RR ,  <  ) )
2516, 24ifbieq2d 3761 . . 3  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  if ( ( ( N  =  0  /\  M  =  0 )  /\  P  =  0 ) ,  0 ,  sup ( { x  e.  ZZ  |  ( ( x 
||  N  /\  x  ||  M )  /\  x  ||  P ) } ,  RR ,  <  ) )  =  if ( ( ( N  gcd  M
)  =  0  /\  P  =  0 ) ,  0 ,  sup ( { x  e.  ZZ  |  ( x  ||  ( N  gcd  M )  /\  x  ||  P
) } ,  RR ,  <  ) ) )
2612, 25eqtr4d 2473 . 2  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N  gcd  M
)  gcd  P )  =  if ( ( ( N  =  0  /\  M  =  0 )  /\  P  =  0 ) ,  0 ,  sup ( { x  e.  ZZ  |  ( ( x  ||  N  /\  x  ||  M )  /\  x  ||  P ) } ,  RR ,  <  ) ) )
27 simp1 958 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  N  e.  ZZ )
28 gcdcl 13019 . . . . . 6  |-  ( ( M  e.  ZZ  /\  P  e.  ZZ )  ->  ( M  gcd  P
)  e.  NN0 )
29283adant1 976 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( M  gcd  P )  e. 
NN0 )
3029nn0zd 10375 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( M  gcd  P )  e.  ZZ )
31 gcdval 13010 . . . 4  |-  ( ( N  e.  ZZ  /\  ( M  gcd  P )  e.  ZZ )  -> 
( N  gcd  ( M  gcd  P ) )  =  if ( ( N  =  0  /\  ( M  gcd  P
)  =  0 ) ,  0 ,  sup ( { x  e.  ZZ  |  ( x  ||  N  /\  x  ||  ( M  gcd  P ) ) } ,  RR ,  <  ) ) )
3227, 30, 31syl2anc 644 . . 3  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( N  gcd  ( M  gcd  P ) )  =  if ( ( N  =  0  /\  ( M  gcd  P )  =  0 ) ,  0 ,  sup ( { x  e.  ZZ  | 
( x  ||  N  /\  x  ||  ( M  gcd  P ) ) } ,  RR ,  <  ) ) )
33 gcdeq0 13023 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  P  e.  ZZ )  ->  ( ( M  gcd  P )  =  0  <->  ( M  =  0  /\  P  =  0 ) ) )
34333adant1 976 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( M  gcd  P
)  =  0  <->  ( M  =  0  /\  P  =  0 ) ) )
3534anbi2d 686 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N  =  0  /\  ( M  gcd  P )  =  0 )  <-> 
( N  =  0  /\  ( M  =  0  /\  P  =  0 ) ) ) )
3635bicomd 194 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N  =  0  /\  ( M  =  0  /\  P  =  0 ) )  <->  ( N  =  0  /\  ( M  gcd  P )  =  0 ) ) )
37 simpl3 963 . . . . . . . 8  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  ZZ )  ->  P  e.  ZZ )
38 dvdsgcdb 13046 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( x  ||  M  /\  x  ||  P )  <-> 
x  ||  ( M  gcd  P ) ) )
3917, 19, 37, 38syl3anc 1185 . . . . . . 7  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x 
||  M  /\  x  ||  P )  <->  x  ||  ( M  gcd  P ) ) )
4039anbi2d 686 . . . . . 6  |-  ( ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  /\  x  e.  ZZ )  ->  ( ( x 
||  N  /\  (
x  ||  M  /\  x  ||  P ) )  <-> 
( x  ||  N  /\  x  ||  ( M  gcd  P ) ) ) )
4140rabbidva 2949 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  { x  e.  ZZ  |  ( x 
||  N  /\  (
x  ||  M  /\  x  ||  P ) ) }  =  { x  e.  ZZ  |  ( x 
||  N  /\  x  ||  ( M  gcd  P
) ) } )
4241supeq1d 7453 . . . 4  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  sup ( { x  e.  ZZ  |  ( x  ||  N  /\  ( x  ||  M  /\  x  ||  P
) ) } ,  RR ,  <  )  =  sup ( { x  e.  ZZ  |  ( x 
||  N  /\  x  ||  ( M  gcd  P
) ) } ,  RR ,  <  ) )
4336, 42ifbieq2d 3761 . . 3  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  if ( ( N  =  0  /\  ( M  =  0  /\  P  =  0 ) ) ,  0 ,  sup ( { x  e.  ZZ  |  ( x  ||  N  /\  ( x  ||  M  /\  x  ||  P
) ) } ,  RR ,  <  ) )  =  if ( ( N  =  0  /\  ( M  gcd  P
)  =  0 ) ,  0 ,  sup ( { x  e.  ZZ  |  ( x  ||  N  /\  x  ||  ( M  gcd  P ) ) } ,  RR ,  <  ) ) )
4432, 43eqtr4d 2473 . 2  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  ( N  gcd  ( M  gcd  P ) )  =  if ( ( N  =  0  /\  ( M  =  0  /\  P  =  0 ) ) ,  0 ,  sup ( { x  e.  ZZ  |  ( x  ||  N  /\  ( x  ||  M  /\  x  ||  P
) ) } ,  RR ,  <  ) ) )
456, 26, 443eqtr4a 2496 1  |-  ( ( N  e.  ZZ  /\  M  e.  ZZ  /\  P  e.  ZZ )  ->  (
( N  gcd  M
)  gcd  P )  =  ( N  gcd  ( M  gcd  P ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {crab 2711   ifcif 3741   class class class wbr 4214  (class class class)co 6083   supcsup 7447   RRcr 8991   0cc0 8992    < clt 9122   NN0cn0 10223   ZZcz 10284    || cdivides 12854    gcd cgcd 13008
This theorem is referenced by:  rpmulgcd  13057  coprimeprodsq  13185  gcd32  25372  gcdabsorb  25373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-dvds 12855  df-gcd 13009
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