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Theorem gcddvds 15225
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 K z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0z 11388 . . . . . 6 |- 0 e. ZZ
2 dvds0 14997 . . . . . 6 |- ( 0 e. ZZ -> 0 || 0 )
31, 2ax-mp 5 . . . . 5 |- 0 || 0
4 breq2 4657 . . . . . . 7 |- ( M = 0 -> ( 0 || M <-> 0 || 0 ) )
5 breq2 4657 . . . . . . 7 |- ( N = 0 -> ( 0 || N <-> 0 || 0 ) )
64, 5bi2anan9 917 . . . . . 6 |- ( ( M = 0 /\ N = 0 ) -> ( ( 0 || M /\ 0 || N ) <-> ( 0 || 0 /\ 0 || 0 ) ) )
7 anidm 676 . . . . . 6 |- ( ( 0 || 0 /\ 0 || 0 ) <-> 0 || 0 )
86, 7syl6bb 276 . . . . 5 |- ( ( M = 0 /\ N = 0 ) -> ( ( 0 || M /\ 0 || N ) <-> 0 || 0 ) )
93, 8mpbiri 248 . . . 4 |- ( ( M = 0 /\ N = 0 ) -> ( 0 || M /\ 0 || N ) )
10 oveq12 6659 . . . . . . 7 |- ( ( M = 0 /\ N = 0 ) -> ( M gcd N ) = ( 0 gcd 0 ) )
11 gcd0val 15219 . . . . . . 7 |- ( 0 gcd 0 ) = 0
1210, 11syl6eq 2672 . . . . . 6 |- ( ( M = 0 /\ N = 0 ) -> ( M gcd N ) = 0 )
1312breq1d 4663 . . . . 5 |- ( ( M = 0 /\ N = 0 ) -> ( ( M gcd N ) || M <-> 0 || M ) )
1412breq1d 4663 . . . . 5 |- ( ( M = 0 /\ N = 0 ) -> ( ( M gcd N ) || N <-> 0 || N ) )
1513, 14anbi12d 747 . . . 4 |- ( ( M = 0 /\ N = 0 ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || N ) <-> ( 0 || M /\ 0 || N ) ) )
169, 15mpbird 247 . . 3 |- ( ( M = 0 /\ N = 0 ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
1716adantl 482 . 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
18 eqid 2622 . . . . 5 |- { n e. ZZ | A. z e. { M , N } n || z } = { n e. ZZ | A. z e. { M , N } n || z }
19 eqid 2622 . . . . 5 |- { n e. ZZ | ( n || M /\ n || N ) } = { n e. ZZ | ( n || M /\ n || N ) }
2018, 19gcdcllem3 15223 . . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) e. NN /\ ( sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) || M /\ sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) || N ) /\ ( ( K e. ZZ /\ K || M /\ K || N ) -> K <_ sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) ) )
2120simp2d 1074 . . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) || M /\ sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) || N ) )
22 gcdn0val 15220 . . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) )
2322breq1d 4663 . . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( M gcd N ) || M <-> sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) || M ) )
2422breq1d 4663 . . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( M gcd N ) || N <-> sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) || N ) )
2523, 24anbi12d 747 . . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || N ) <-> ( sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) || M /\ sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) || N ) ) )
2621, 25mpbird 247 . 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
2717, 26pm2.61dan 832 1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   {cpr 4179   class class class wbr 4653  (class class class)co 6650   supcsup 8346   RRcr 9935   0cc0 9936   < clt 10074   <_ cle 10075   NNcn 11020   ZZcz 11377   || cdvds 14983   gcd cgcd 15216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217
This theorem is referenced by:  zeqzmulgcd  15232  divgcdz  15233  divgcdnn  15236  gcd0id  15240  gcdneg  15243  gcdaddmlem  15245  gcd1  15249  bezoutlem4  15259  dvdsgcdb  15262  dfgcd2  15263  mulgcd  15265  gcdzeq  15271  dvdsmulgcd  15274  sqgcd  15278  dvdssqlem  15279  bezoutr  15281  gcddvdslcm  15315  lcmgcdlem  15319  lcmgcdeq  15325  coprmgcdb  15362  mulgcddvds  15369  rpmulgcd2  15370  qredeu  15372  rpdvds  15374  divgcdcoprm0  15379  divgcdodd  15422  coprm  15423  rpexp  15432  divnumden  15456  phimullem  15484  hashgcdlem  15493  hashgcdeq  15494  phisum  15495  pythagtriplem4  15524  pythagtriplem19  15538  pcgcd1  15581  pc2dvds  15583  pockthlem  15609  odmulg  17973  odadd1  18251  odadd2  18252  znunit  19912  znrrg  19914  dvdsmulf1o  24920  2sqlem8  25151  2sqcoprm  29647  qqhval2lem  30025  goldbachthlem2  41458  divgcdoddALTV  41593
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