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Theorem dvdsgcd 11707
Description: An integer which divides each of two others also divides their gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 30-May-2014.)
Assertion
Ref Expression
dvdsgcd  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  M  /\  K  ||  N )  ->  K  ||  ( M  gcd  N ) ) )

Proof of Theorem dvdsgcd
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bezout 11706 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( M  gcd  N )  =  ( ( M  x.  x )  +  ( N  x.  y ) ) )
213adant1 999 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( M  gcd  N )  =  ( ( M  x.  x )  +  ( N  x.  y ) ) )
3 dvds2ln 11533 . . . . . . . . 9  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( ( K  ||  M  /\  K  ||  N
)  ->  K  ||  (
( x  x.  M
)  +  ( y  x.  N ) ) ) )
433impia 1178 . . . . . . . 8  |-  ( ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  K  ||  ( ( x  x.  M )  +  ( y  x.  N
) ) )
543coml 1188 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  K  ||  ( ( x  x.  M )  +  ( y  x.  N
) ) )
6 simp3l 1009 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  x  e.  ZZ )
7 simp12 1012 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  M  e.  ZZ )
8 zcn 9066 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  x  e.  CC )
9 zcn 9066 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
10 mulcom 7756 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( x  x.  M
)  =  ( M  x.  x ) )
118, 9, 10syl2an 287 . . . . . . . . 9  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ )  ->  ( x  x.  M
)  =  ( M  x.  x ) )
126, 7, 11syl2anc 408 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( x  x.  M
)  =  ( M  x.  x ) )
13 simp3r 1010 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
y  e.  ZZ )
14 simp13 1013 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  N  e.  ZZ )
15 zcn 9066 . . . . . . . . . 10  |-  ( y  e.  ZZ  ->  y  e.  CC )
16 zcn 9066 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  e.  CC )
17 mulcom 7756 . . . . . . . . . 10  |-  ( ( y  e.  CC  /\  N  e.  CC )  ->  ( y  x.  N
)  =  ( N  x.  y ) )
1815, 16, 17syl2an 287 . . . . . . . . 9  |-  ( ( y  e.  ZZ  /\  N  e.  ZZ )  ->  ( y  x.  N
)  =  ( N  x.  y ) )
1913, 14, 18syl2anc 408 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( y  x.  N
)  =  ( N  x.  y ) )
2012, 19oveq12d 5792 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( ( x  x.  M )  +  ( y  x.  N ) )  =  ( ( M  x.  x )  +  ( N  x.  y ) ) )
215, 20breqtrd 3954 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  K  ||  ( ( M  x.  x )  +  ( N  x.  y
) ) )
22 breq2 3933 . . . . . 6  |-  ( ( M  gcd  N )  =  ( ( M  x.  x )  +  ( N  x.  y
) )  ->  ( K  ||  ( M  gcd  N )  <->  K  ||  ( ( M  x.  x )  +  ( N  x.  y ) ) ) )
2321, 22syl5ibrcom 156 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( ( M  gcd  N )  =  ( ( M  x.  x )  +  ( N  x.  y ) )  ->  K  ||  ( M  gcd  N ) ) )
24233expia 1183 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  ( (
x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( M  gcd  N )  =  ( ( M  x.  x )  +  ( N  x.  y ) )  ->  K  ||  ( M  gcd  N ) ) ) )
2524rexlimdvv 2556 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  ||  M  /\  K  ||  N ) )  ->  ( E. x  e.  ZZ  E. y  e.  ZZ  ( M  gcd  N )  =  ( ( M  x.  x )  +  ( N  x.  y ) )  ->  K  ||  ( M  gcd  N ) ) )
2625ex 114 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  M  /\  K  ||  N )  ->  ( E. x  e.  ZZ  E. y  e.  ZZ  ( M  gcd  N )  =  ( ( M  x.  x )  +  ( N  x.  y ) )  ->  K  ||  ( M  gcd  N ) ) ) )
272, 26mpid 42 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( K  ||  M  /\  K  ||  N )  ->  K  ||  ( M  gcd  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962    = wceq 1331    e. wcel 1480   E.wrex 2417   class class class wbr 3929  (class class class)co 5774   CCcc 7625    + caddc 7630    x. cmul 7632   ZZcz 9061    || cdvds 11500    gcd cgcd 11642
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745  ax-arch 7746  ax-caucvg 7747
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-n0 8985  df-z 9062  df-uz 9334  df-q 9419  df-rp 9449  df-fz 9798  df-fzo 9927  df-fl 10050  df-mod 10103  df-seqfrec 10226  df-exp 10300  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778  df-dvds 11501  df-gcd 11643
This theorem is referenced by:  dvdsgcdb  11708  dfgcd2  11709  mulgcd  11711  ncoprmgcdne1b  11777  mulgcddvds  11782  rpmulgcd2  11783  rpexp  11838
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