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Theorem gcddvds 10734
Description: The gcd of two integers divides each of them. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
gcddvds  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N ) )

Proof of Theorem gcddvds
Dummy variables  n  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0z 8656 . . . . . 6  |-  0  e.  ZZ
2 dvds0 10590 . . . . . 6  |-  ( 0  e.  ZZ  ->  0  ||  0 )
31, 2ax-mp 7 . . . . 5  |-  0  ||  0
4 breq2 3815 . . . . . . 7  |-  ( M  =  0  ->  (
0  ||  M  <->  0  ||  0 ) )
5 breq2 3815 . . . . . . 7  |-  ( N  =  0  ->  (
0  ||  N  <->  0  ||  0 ) )
64, 5bi2anan9 571 . . . . . 6  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( 0 
||  M  /\  0  ||  N )  <->  ( 0 
||  0  /\  0  ||  0 ) ) )
7 anidm 388 . . . . . 6  |-  ( ( 0  ||  0  /\  0  ||  0 )  <->  0  ||  0 )
86, 7syl6bb 194 . . . . 5  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( 0 
||  M  /\  0  ||  N )  <->  0  ||  0 ) )
93, 8mpbiri 166 . . . 4  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( 0  ||  M  /\  0  ||  N
) )
10 oveq12 5599 . . . . . . 7  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( M  gcd  N )  =  ( 0  gcd  0 ) )
11 gcd0val 10731 . . . . . . 7  |-  ( 0  gcd  0 )  =  0
1210, 11syl6eq 2131 . . . . . 6  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( M  gcd  N )  =  0 )
1312breq1d 3821 . . . . 5  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( M  gcd  N )  ||  M 
<->  0  ||  M ) )
1412breq1d 3821 . . . . 5  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( M  gcd  N )  ||  N 
<->  0  ||  N ) )
1513, 14anbi12d 457 . . . 4  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( ( M  gcd  N ) 
||  M  /\  ( M  gcd  N )  ||  N )  <->  ( 0 
||  M  /\  0  ||  N ) ) )
169, 15mpbird 165 . . 3  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  N ) )
1716adantl 271 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N ) )
18 gcdn0val 10732 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  sup ( { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } ,  RR ,  <  ) )
19 zssre 8652 . . . . . 6  |-  ZZ  C_  RR
20 gcdsupex 10728 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  E. x  e.  ZZ  ( A. y  e.  {
n  e.  ZZ  | 
( n  ||  M  /\  n  ||  N ) }  -.  x  < 
y  /\  A. y  e.  RR  ( y  < 
x  ->  E. z  e.  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } y  < 
z ) ) )
21 ssrexv 3070 . . . . . 6  |-  ( ZZ  C_  RR  ->  ( E. x  e.  ZZ  ( A. y  e.  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) }  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } y  <  z ) )  ->  E. x  e.  RR  ( A. y  e.  {
n  e.  ZZ  | 
( n  ||  M  /\  n  ||  N ) }  -.  x  < 
y  /\  A. y  e.  RR  ( y  < 
x  ->  E. z  e.  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } y  < 
z ) ) ) )
2219, 20, 21mpsyl 64 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  E. x  e.  RR  ( A. y  e.  {
n  e.  ZZ  | 
( n  ||  M  /\  n  ||  N ) }  -.  x  < 
y  /\  A. y  e.  RR  ( y  < 
x  ->  E. z  e.  { n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } y  < 
z ) ) )
23 ssrab2 3090 . . . . . 6  |-  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) }  C_  ZZ
2423a1i 9 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) }  C_  ZZ )
2522, 24suprzclex 8739 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  sup ( { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } ,  RR ,  <  )  e. 
{ n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) } )
2618, 25eqeltrd 2159 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  e.  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } )
27 gcdn0cl 10733 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  e.  NN )
2827nnzd 8762 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  e.  ZZ )
29 breq1 3814 . . . . . 6  |-  ( n  =  ( M  gcd  N )  ->  ( n  ||  M  <->  ( M  gcd  N )  ||  M ) )
30 breq1 3814 . . . . . 6  |-  ( n  =  ( M  gcd  N )  ->  ( n  ||  N  <->  ( M  gcd  N )  ||  N ) )
3129, 30anbi12d 457 . . . . 5  |-  ( n  =  ( M  gcd  N )  ->  ( (
n  ||  M  /\  n  ||  N )  <->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  N ) ) )
3231elrab3 2760 . . . 4  |-  ( ( M  gcd  N )  e.  ZZ  ->  (
( M  gcd  N
)  e.  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) }  <->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  N ) ) )
3328, 32syl 14 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( ( M  gcd  N )  e. 
{ n  e.  ZZ  |  ( n  ||  M  /\  n  ||  N
) }  <->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  N ) ) )
3426, 33mpbid 145 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  N ) )
35 gcdmndc 10719 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  /\  N  =  0 ) )
36 exmiddc 778 . . 3  |-  (DECID  ( M  =  0  /\  N  =  0 )  -> 
( ( M  =  0  /\  N  =  0 )  \/  -.  ( M  =  0  /\  N  =  0
) ) )
3735, 36syl 14 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  =  0  /\  N  =  0 )  \/  -.  ( M  =  0  /\  N  =  0
) ) )
3817, 34, 37mpjaodan 745 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662  DECID wdc 776    = wceq 1285    e. wcel 1434   A.wral 2353   E.wrex 2354   {crab 2357    C_ wss 2984   class class class wbr 3811  (class class class)co 5590   supcsup 6583   RRcr 7251   0cc0 7252    < clt 7424   ZZcz 8645    || cdvds 10575    gcd cgcd 10717
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 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365  ax-cnex 7338  ax-resscn 7339  ax-1cn 7340  ax-1re 7341  ax-icn 7342  ax-addcl 7343  ax-addrcl 7344  ax-mulcl 7345  ax-mulrcl 7346  ax-addcom 7347  ax-mulcom 7348  ax-addass 7349  ax-mulass 7350  ax-distr 7351  ax-i2m1 7352  ax-0lt1 7353  ax-1rid 7354  ax-0id 7355  ax-rnegex 7356  ax-precex 7357  ax-cnre 7358  ax-pre-ltirr 7359  ax-pre-ltwlin 7360  ax-pre-lttrn 7361  ax-pre-apti 7362  ax-pre-ltadd 7363  ax-pre-mulgt0 7364  ax-pre-mulext 7365  ax-arch 7366  ax-caucvg 7367
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 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-po 4086  df-iso 4087  df-iord 4156  df-on 4158  df-ilim 4159  df-suc 4161  df-iom 4368  df-xp 4406  df-rel 4407  df-cnv 4408  df-co 4409  df-dm 4410  df-rn 4411  df-res 4412  df-ima 4413  df-iota 4933  df-fun 4970  df-fn 4971  df-f 4972  df-f1 4973  df-fo 4974  df-f1o 4975  df-fv 4976  df-riota 5546  df-ov 5593  df-oprab 5594  df-mpt2 5595  df-1st 5845  df-2nd 5846  df-recs 6001  df-frec 6087  df-sup 6585  df-pnf 7426  df-mnf 7427  df-xr 7428  df-ltxr 7429  df-le 7430  df-sub 7557  df-neg 7558  df-reap 7951  df-ap 7958  df-div 8037  df-inn 8316  df-2 8374  df-3 8375  df-4 8376  df-n0 8565  df-z 8646  df-uz 8914  df-q 8999  df-rp 9029  df-fz 9319  df-fzo 9443  df-fl 9565  df-mod 9618  df-iseq 9740  df-iexp 9791  df-cj 10102  df-re 10103  df-im 10104  df-rsqrt 10257  df-abs 10258  df-dvds 10576  df-gcd 10718
This theorem is referenced by:  zeqzmulgcd  10741  divgcdz  10742  divgcdnn  10745  gcd0id  10749  gcdneg  10752  gcdaddm  10754  gcd1  10757  dvdsgcdb  10781  dfgcd2  10782  mulgcd  10784  gcdzeq  10790  dvdsmulgcd  10793  sqgcd  10797  dvdssqlem  10798  bezoutr  10800  gcddvdslcm  10834  lcmgcdlem  10838  lcmgcdeq  10844  coprmgcdb  10849  ncoprmgcdne1b  10850  mulgcddvds  10855  rpmulgcd2  10856  qredeu  10858  rpdvds  10860  divgcdcoprm0  10862  divgcdodd  10901  coprm  10902  rpexp  10911  divnumden  10953  phimullem  10980  hashgcdlem  10982  hashgcdeq  10983
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