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Theorem gcddvds 11579
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 9033 . . . . . 6  |-  0  e.  ZZ
2 dvds0 11435 . . . . . 6  |-  ( 0  e.  ZZ  ->  0  ||  0 )
31, 2ax-mp 5 . . . . 5  |-  0  ||  0
4 breq2 3903 . . . . . . 7  |-  ( M  =  0  ->  (
0  ||  M  <->  0  ||  0 ) )
5 breq2 3903 . . . . . . 7  |-  ( N  =  0  ->  (
0  ||  N  <->  0  ||  0 ) )
64, 5bi2anan9 580 . . . . . 6  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( 0 
||  M  /\  0  ||  N )  <->  ( 0 
||  0  /\  0  ||  0 ) ) )
7 anidm 393 . . . . . 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 5751 . . . . . . 7  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( M  gcd  N )  =  ( 0  gcd  0 ) )
11 gcd0val 11576 . . . . . . 7  |-  ( 0  gcd  0 )  =  0
1210, 11syl6eq 2166 . . . . . 6  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( M  gcd  N )  =  0 )
1312breq1d 3909 . . . . 5  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( M  gcd  N )  ||  M 
<->  0  ||  M ) )
1412breq1d 3909 . . . . 5  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( ( M  gcd  N )  ||  N 
<->  0  ||  N ) )
1513, 14anbi12d 464 . . . 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 11577 . . . 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 9029 . . . . . 6  |-  ZZ  C_  RR
20 gcdsupex 11573 . . . . . 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 3132 . . . . . 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 3152 . . . . . 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 9117 . . . 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 2194 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  e.  { n  e.  ZZ  |  ( n 
||  M  /\  n  ||  N ) } )
27 gcdn0cl 11578 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  e.  NN )
2827nnzd 9140 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  e.  ZZ )
29 breq1 3902 . . . . . 6  |-  ( n  =  ( M  gcd  N )  ->  ( n  ||  M  <->  ( M  gcd  N )  ||  M ) )
30 breq1 3902 . . . . . 6  |-  ( n  =  ( M  gcd  N )  ->  ( n  ||  N  <->  ( M  gcd  N )  ||  N ) )
3129, 30anbi12d 464 . . . . 5  |-  ( n  =  ( M  gcd  N )  ->  ( (
n  ||  M  /\  n  ||  N )  <->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  N ) ) )
3231elrab3 2814 . . . 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 11564 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  /\  N  =  0 ) )
36 exmiddc 806 . . 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 772 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 682  DECID wdc 804    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394   {crab 2397    C_ wss 3041   class class class wbr 3899  (class class class)co 5742   supcsup 6837   RRcr 7587   0cc0 7588    < clt 7768   ZZcz 9022    || cdvds 11420    gcd cgcd 11562
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-sup 6839  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-fz 9759  df-fzo 9888  df-fl 10011  df-mod 10064  df-seqfrec 10187  df-exp 10261  df-cj 10582  df-re 10583  df-im 10584  df-rsqrt 10738  df-abs 10739  df-dvds 11421  df-gcd 11563
This theorem is referenced by:  zeqzmulgcd  11586  divgcdz  11587  divgcdnn  11590  gcd0id  11594  gcdneg  11597  gcdaddm  11599  gcd1  11602  dvdsgcdb  11628  dfgcd2  11629  mulgcd  11631  gcdzeq  11637  dvdsmulgcd  11640  sqgcd  11644  dvdssqlem  11645  bezoutr  11647  gcddvdslcm  11681  lcmgcdlem  11685  lcmgcdeq  11691  coprmgcdb  11696  ncoprmgcdne1b  11697  mulgcddvds  11702  rpmulgcd2  11703  qredeu  11705  rpdvds  11707  divgcdcoprm0  11709  divgcdodd  11748  coprm  11749  rpexp  11758  divnumden  11801  phimullem  11828  hashgcdlem  11830  hashgcdeq  11831
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