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Theorem gcddvds 11964
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 9264 . . . . . 6 |- 0 e. ZZ
2 dvds0 11813 . . . . . 6 |- ( 0 e. ZZ -> 0 || 0 )
31, 2ax-mp 5 . . . . 5 |- 0 || 0
4 breq2 4008 . . . . . . 7 |- ( M = 0 -> ( 0 || M <-> 0 || 0 ) )
5 breq2 4008 . . . . . . 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 5884 . . . . . . 7 |- ( ( M = 0 /\ N = 0 ) -> ( M gcd N ) = ( 0 gcd 0 ) )
11 gcd0val 11961 . . . . . . 7 |- ( 0 gcd 0 ) = 0
1210, 11eqtrdi 2226 . . . . . 6 |- ( ( M = 0 /\ N = 0 ) -> ( M gcd N ) = 0 )
1312breq1d 4014 . . . . 5 |- ( ( M = 0 /\ N = 0 ) -> ( ( M gcd N ) || M <-> 0 || M ) )
1412breq1d 4014 . . . . 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 11962 . . . 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 9260 . . . . . 6 |- ZZ C_ RR
20 gcdsupex 11958 . . . . . 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 3221 . . . . . 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 3241 . . . . . 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 9351 . . . 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 2254 . . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) e. { n e. ZZ | ( n || M /\ n || N ) } )
27 gcdn0cl 11963 . . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) e. NN )
2827nnzd 9374 . . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) e. ZZ )
29 breq1 4007 . . . . . 6 |- ( n = ( M gcd N ) -> ( n || M <-> ( M gcd N ) || M ) )
30 breq1 4007 . . . . . 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 2895 . . . 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 11945 . . 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 2148   A.wral 2455   E.wrex 2456   {crab 2459   C_ wss 3130   class class class wbr 4004  (class class class)co 5875   supcsup 6981   RRcr 7810   0cc0 7811   < clt 7992   ZZcz 9253   || cdvds 11794   gcd cgcd 11943
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931
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 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-sup 6983  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-fz 10009  df-fzo 10143  df-fl 10270  df-mod 10323  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-dvds 11795  df-gcd 11944
This theorem is referenced by:  zeqzmulgcd  11971  divgcdz  11972  divgcdnn  11976  gcd0id  11980  gcdneg  11983  gcdaddm  11985  gcd1  11988  dvdsgcdb  12014  dfgcd2  12015  mulgcd  12017  gcdzeq  12023  dvdsmulgcd  12026  sqgcd  12030  dvdssqlem  12031  bezoutr  12033  gcddvdslcm  12073  lcmgcdlem  12077  lcmgcdeq  12083  coprmgcdb  12088  ncoprmgcdne1b  12089  mulgcddvds  12094  rpmulgcd2  12095  qredeu  12097  rpdvds  12099  divgcdcoprm0  12101  divgcdodd  12143  coprm  12144  rpexp  12153  divnumden  12196  phimullem  12225  hashgcdlem  12238  hashgcdeq  12239  phisum  12240  pythagtriplem4  12268  pythagtriplem19  12282  pcgcd1  12327  pc2dvds  12329  pockthlem  12354  2sqlem8  14473
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