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Theorem gcddvds 11996
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 9294 . . . . . 6 |- 0 e. ZZ
2 dvds0 11845 . . . . . 6 |- ( 0 e. ZZ -> 0 || 0 )
31, 2ax-mp 5 . . . . 5 |- 0 || 0
4 breq2 4022 . . . . . . 7 |- ( M = 0 -> ( 0 || M <-> 0 || 0 ) )
5 breq2 4022 . . . . . . 7 |- ( N = 0 -> ( 0 || N <-> 0 || 0 ) )
64, 5bi2anan9 606 . . . . . 6 |- ( ( M = 0 /\ N = 0 ) -> ( ( 0 || M /\ 0 || N ) <-> ( 0 || 0 /\ 0 || 0 ) ) )
7 anidm 396 . . . . . 6 |- ( ( 0 || 0 /\ 0 || 0 ) <-> 0 || 0 )
86, 7bitrdi 196 . . . . 5 |- ( ( M = 0 /\ N = 0 ) -> ( ( 0 || M /\ 0 || N ) <-> 0 || 0 ) )
93, 8mpbiri 168 . . . 4 |- ( ( M = 0 /\ N = 0 ) -> ( 0 || M /\ 0 || N ) )
10 oveq12 5905 . . . . . . 7 |- ( ( M = 0 /\ N = 0 ) -> ( M gcd N ) = ( 0 gcd 0 ) )
11 gcd0val 11993 . . . . . . 7 |- ( 0 gcd 0 ) = 0
1210, 11eqtrdi 2238 . . . . . 6 |- ( ( M = 0 /\ N = 0 ) -> ( M gcd N ) = 0 )
1312breq1d 4028 . . . . 5 |- ( ( M = 0 /\ N = 0 ) -> ( ( M gcd N ) || M <-> 0 || M ) )
1412breq1d 4028 . . . . 5 |- ( ( M = 0 /\ N = 0 ) -> ( ( M gcd N ) || N <-> 0 || N ) )
1513, 14anbi12d 473 . . . 4 |- ( ( M = 0 /\ N = 0 ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || N ) <-> ( 0 || M /\ 0 || N ) ) )
169, 15mpbird 167 . . 3 |- ( ( M = 0 /\ N = 0 ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
1716adantl 277 . 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
18 gcdn0val 11994 . . . 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 9290 . . . . . 6 |- ZZ C_ RR
20 gcdsupex 11990 . . . . . 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 3235 . . . . . 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 3255 . . . . . 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 9381 . . . 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 2266 . . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) e. { n e. ZZ | ( n || M /\ n || N ) } )
27 gcdn0cl 11995 . . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) e. NN )
2827nnzd 9404 . . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) e. ZZ )
29 breq1 4021 . . . . . 6 |- ( n = ( M gcd N ) -> ( n || M <-> ( M gcd N ) || M ) )
30 breq1 4021 . . . . . 6 |- ( n = ( M gcd N ) -> ( n || N <-> ( M gcd N ) || N ) )
3129, 30anbi12d 473 . . . . 5 |- ( n = ( M gcd N ) -> ( ( n || M /\ n || N ) <-> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) ) )
3231elrab3 2909 . . . 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 147 . 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
35 gcdmndc 11977 . . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
36 exmiddc 837 . . 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 799 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 104   <-> wb 105   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2160   A.wral 2468   E.wrex 2469   {crab 2472   C_ wss 3144   class class class wbr 4018  (class class class)co 5896   supcsup 7011   RRcr 7840   0cc0 7841   < clt 8022   ZZcz 9283   || cdvds 11826   gcd cgcd 11975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959  ax-arch 7960  ax-caucvg 7961
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-sup 7013  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-n0 9207  df-z 9284  df-uz 9559  df-q 9650  df-rp 9684  df-fz 10039  df-fzo 10173  df-fl 10301  df-mod 10354  df-seqfrec 10477  df-exp 10551  df-cj 10883  df-re 10884  df-im 10885  df-rsqrt 11039  df-abs 11040  df-dvds 11827  df-gcd 11976
This theorem is referenced by:  zeqzmulgcd  12003  divgcdz  12004  divgcdnn  12008  gcd0id  12012  gcdneg  12015  gcdaddm  12017  gcd1  12020  dvdsgcdb  12046  dfgcd2  12047  mulgcd  12049  gcdzeq  12055  dvdsmulgcd  12058  sqgcd  12062  dvdssqlem  12063  bezoutr  12065  gcddvdslcm  12105  lcmgcdlem  12109  lcmgcdeq  12115  coprmgcdb  12120  ncoprmgcdne1b  12121  mulgcddvds  12126  rpmulgcd2  12127  qredeu  12129  rpdvds  12131  divgcdcoprm0  12133  divgcdodd  12175  coprm  12176  rpexp  12185  divnumden  12228  phimullem  12257  hashgcdlem  12270  hashgcdeq  12271  phisum  12272  pythagtriplem4  12300  pythagtriplem19  12314  pcgcd1  12360  pc2dvds  12362  pockthlem  12388  2sqlem8  14931
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