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Theorem gcddvds 11929
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 9235 . . . . . 6 |- 0 e. ZZ
2 dvds0 11779 . . . . . 6 |- ( 0 e. ZZ -> 0 || 0 )
31, 2ax-mp 5 . . . . 5 |- 0 || 0
4 breq2 4002 . . . . . . 7 |- ( M = 0 -> ( 0 || M <-> 0 || 0 ) )
5 breq2 4002 . . . . . . 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 5874 . . . . . . 7 |- ( ( M = 0 /\ N = 0 ) -> ( M gcd N ) = ( 0 gcd 0 ) )
11 gcd0val 11926 . . . . . . 7 |- ( 0 gcd 0 ) = 0
1210, 11eqtrdi 2224 . . . . . 6 |- ( ( M = 0 /\ N = 0 ) -> ( M gcd N ) = 0 )
1312breq1d 4008 . . . . 5 |- ( ( M = 0 /\ N = 0 ) -> ( ( M gcd N ) || M <-> 0 || M ) )
1412breq1d 4008 . . . . 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 11927 . . . 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 9231 . . . . . 6 |- ZZ C_ RR
20 gcdsupex 11923 . . . . . 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 3218 . . . . . 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 3238 . . . . . 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 9322 . . . 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 2252 . . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) e. { n e. ZZ | ( n || M /\ n || N ) } )
27 gcdn0cl 11928 . . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) e. NN )
2827nnzd 9345 . . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) e. ZZ )
29 breq1 4001 . . . . . 6 |- ( n = ( M gcd N ) -> ( n || M <-> ( M gcd N ) || M ) )
30 breq1 4001 . . . . . 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 2892 . . . 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 11910 . . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
36 exmiddc 836 . . 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 798 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 708  DECID wdc 834   = wceq 1353   e. wcel 2146   A.wral 2453   E.wrex 2454   {crab 2457   C_ wss 3127   class class class wbr 3998  (class class class)co 5865   supcsup 6971   RRcr 7785   0cc0 7786   < clt 7966   ZZcz 9224   || cdvds 11760   gcd cgcd 11908
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-sup 6973  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-fz 9978  df-fzo 10111  df-fl 10238  df-mod 10291  df-seqfrec 10414  df-exp 10488  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-dvds 11761  df-gcd 11909
This theorem is referenced by:  zeqzmulgcd  11936  divgcdz  11937  divgcdnn  11941  gcd0id  11945  gcdneg  11948  gcdaddm  11950  gcd1  11953  dvdsgcdb  11979  dfgcd2  11980  mulgcd  11982  gcdzeq  11988  dvdsmulgcd  11991  sqgcd  11995  dvdssqlem  11996  bezoutr  11998  gcddvdslcm  12038  lcmgcdlem  12042  lcmgcdeq  12048  coprmgcdb  12053  ncoprmgcdne1b  12054  mulgcddvds  12059  rpmulgcd2  12060  qredeu  12062  rpdvds  12064  divgcdcoprm0  12066  divgcdodd  12108  coprm  12109  rpexp  12118  divnumden  12161  phimullem  12190  hashgcdlem  12203  hashgcdeq  12204  phisum  12205  pythagtriplem4  12233  pythagtriplem19  12247  pcgcd1  12292  pc2dvds  12294  pockthlem  12319  2sqlem8  14010
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