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Theorem dvdsgcd 11443
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 11442 . . 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 964 . 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 11271 . . . . . . . . 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 1143 . . . . . . . 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 1153 . . . . . . 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 974 . . . . . . . . 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 977 . . . . . . . . 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 8853 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  x  e.  CC )
9 zcn 8853 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
10 mulcom 7568 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  M  e.  CC )  ->  ( x  x.  M
)  =  ( M  x.  x ) )
118, 9, 10syl2an 284 . . . . . . . . 9  |-  ( ( x  e.  ZZ  /\  M  e.  ZZ )  ->  ( x  x.  M
)  =  ( M  x.  x ) )
126, 7, 11syl2anc 404 . . . . . . . 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 975 . . . . . . . . 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 978 . . . . . . . . 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 8853 . . . . . . . . . 10  |-  ( y  e.  ZZ  ->  y  e.  CC )
16 zcn 8853 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  e.  CC )
17 mulcom 7568 . . . . . . . . . 10  |-  ( ( y  e.  CC  /\  N  e.  CC )  ->  ( y  x.  N
)  =  ( N  x.  y ) )
1815, 16, 17syl2an 284 . . . . . . . . 9  |-  ( ( y  e.  ZZ  /\  N  e.  ZZ )  ->  ( y  x.  N
)  =  ( N  x.  y ) )
1913, 14, 18syl2anc 404 . . . . . . . 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 5708 . . . . . . 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 3891 . . . . . 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 3871 . . . . . 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 1148 . . . 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 2509 . . 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 927    = wceq 1296    e. wcel 1445   E.wrex 2371   class class class wbr 3867  (class class class)co 5690   CCcc 7445    + caddc 7450    x. cmul 7452   ZZcz 8848    || cdvds 11238    gcd cgcd 11380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560  ax-arch 7561  ax-caucvg 7562
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-ilim 4220  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-frec 6194  df-sup 6759  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-2 8579  df-3 8580  df-4 8581  df-n0 8772  df-z 8849  df-uz 9119  df-q 9204  df-rp 9234  df-fz 9574  df-fzo 9703  df-fl 9826  df-mod 9879  df-iseq 10002  df-seq3 10003  df-exp 10086  df-cj 10407  df-re 10408  df-im 10409  df-rsqrt 10562  df-abs 10563  df-dvds 11239  df-gcd 11381
This theorem is referenced by:  dvdsgcdb  11444  dfgcd2  11445  mulgcd  11447  ncoprmgcdne1b  11513  mulgcddvds  11518  rpmulgcd2  11519  rpexp  11574
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