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Theorem gcddvds 11686
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 9088 . . . . . 6  |-  0  e.  ZZ
2 dvds0 11542 . . . . . 6  |-  ( 0  e.  ZZ  ->  0  ||  0 )
31, 2ax-mp 5 . . . . 5  |-  0  ||  0
4 breq2 3940 . . . . . . 7  |-  ( M  =  0  ->  (
0  ||  M  <->  0  ||  0 ) )
5 breq2 3940 . . . . . . 7  |-  ( N  =  0  ->  (
0  ||  N  <->  0  ||  0 ) )
64, 5bi2anan9 596 . . . . . 6  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( 0 
||  M  /\  0  ||  N )  <->  ( 0 
||  0  /\  0  ||  0 ) ) )
7 anidm 394 . . . . . 6  |-  ( ( 0  ||  0  /\  0  ||  0 )  <->  0  ||  0 )
86, 7syl6bb 195 . . . . 5  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( 0 
||  M  /\  0  ||  N )  <->  0  ||  0 ) )
93, 8mpbiri 167 . . . 4  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( 0  ||  M  /\  0  ||  N
) )
10 oveq12 5790 . . . . . . 7  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( M  gcd  N )  =  ( 0  gcd  0 ) )
11 gcd0val 11683 . . . . . . 7  |-  ( 0  gcd  0 )  =  0
1210, 11eqtrdi 2189 . . . . . 6  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( M  gcd  N )  =  0 )
1312breq1d 3946 . . . . 5  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( M  gcd  N )  ||  M 
<->  0  ||  M ) )
1412breq1d 3946 . . . . 5  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( M  gcd  N )  ||  N 
<->  0  ||  N ) )
1513, 14anbi12d 465 . . . 4  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( ( M  gcd  N ) 
||  M  /\  ( M  gcd  N )  ||  N )  <->  ( 0 
||  M  /\  0  ||  N ) ) )
169, 15mpbird 166 . . 3  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  N ) )
1716adantl 275 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  /\  N  =  0 ) )  -> 
( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N ) )
18 gcdn0val 11684 . . . 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 9084 . . . . . 6  |-  ZZ  C_  RR
20 gcdsupex 11680 . . . . . 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 3166 . . . . . 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 65 . . . . 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 3186 . . . . . 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 9172 . . . 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 2217 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  e.  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } )
27 gcdn0cl 11685 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  e.  NN )
2827nnzd 9195 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  e.  ZZ )
29 breq1 3939 . . . . . 6  |-  ( n  =  ( M  gcd  N )  ->  ( n  ||  M  <->  ( M  gcd  N )  ||  M ) )
30 breq1 3939 . . . . . 6  |-  ( n  =  ( M  gcd  N )  ->  ( n  ||  N  <->  ( M  gcd  N )  ||  N ) )
3129, 30anbi12d 465 . . . . 5  |-  ( n  =  ( M  gcd  N )  ->  ( (
n  ||  M  /\  n  ||  N )  <->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  N ) ) )
3231elrab3 2844 . . . 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 146 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  N ) )
35 gcdmndc 11671 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  /\  N  =  0 ) )
36 exmiddc 822 . . 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 788 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 103    <-> wb 104    \/ wo 698  DECID wdc 820    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418   {crab 2421    C_ wss 3075   class class class wbr 3936  (class class class)co 5781   supcsup 6876   RRcr 7642   0cc0 7643    < clt 7823   ZZcz 9077    || cdvds 11527    gcd cgcd 11669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761  ax-arch 7762  ax-caucvg 7763
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-frec 6295  df-sup 6878  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-inn 8744  df-2 8802  df-3 8803  df-4 8804  df-n0 9001  df-z 9078  df-uz 9350  df-q 9438  df-rp 9470  df-fz 9821  df-fzo 9950  df-fl 10073  df-mod 10126  df-seqfrec 10249  df-exp 10323  df-cj 10645  df-re 10646  df-im 10647  df-rsqrt 10801  df-abs 10802  df-dvds 11528  df-gcd 11670
This theorem is referenced by:  zeqzmulgcd  11693  divgcdz  11694  divgcdnn  11697  gcd0id  11701  gcdneg  11704  gcdaddm  11706  gcd1  11709  dvdsgcdb  11735  dfgcd2  11736  mulgcd  11738  gcdzeq  11744  dvdsmulgcd  11747  sqgcd  11751  dvdssqlem  11752  bezoutr  11754  gcddvdslcm  11788  lcmgcdlem  11792  lcmgcdeq  11798  coprmgcdb  11803  ncoprmgcdne1b  11804  mulgcddvds  11809  rpmulgcd2  11810  qredeu  11812  rpdvds  11814  divgcdcoprm0  11816  divgcdodd  11855  coprm  11856  rpexp  11865  divnumden  11908  phimullem  11935  hashgcdlem  11937  hashgcdeq  11938
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