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Theorem gcddiv 11101
Description: Division law for GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
gcddiv  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  ( C  ||  A  /\  C  ||  B ) )  ->  ( ( A  gcd  B )  /  C )  =  ( ( A  /  C
)  gcd  ( B  /  C ) ) )

Proof of Theorem gcddiv
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnz 8739 . . . . . . 7  |-  ( C  e.  NN  ->  C  e.  ZZ )
213ad2ant3 966 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  C  e.  ZZ )
3 simp1 943 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  A  e.  ZZ )
4 divides 10891 . . . . . 6  |-  ( ( C  e.  ZZ  /\  A  e.  ZZ )  ->  ( C  ||  A  <->  E. a  e.  ZZ  (
a  x.  C )  =  A ) )
52, 3, 4syl2anc 403 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( C  ||  A  <->  E. a  e.  ZZ  ( a  x.  C )  =  A ) )
6 simp2 944 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  B  e.  ZZ )
7 divides 10891 . . . . . 6  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  ||  B  <->  E. b  e.  ZZ  (
b  x.  C )  =  B ) )
82, 6, 7syl2anc 403 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( C  ||  B  <->  E. b  e.  ZZ  ( b  x.  C )  =  B ) )
95, 8anbi12d 457 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( C  ||  A  /\  C  ||  B )  <-> 
( E. a  e.  ZZ  ( a  x.  C )  =  A  /\  E. b  e.  ZZ  ( b  x.  C )  =  B ) ) )
10 reeanv 2536 . . . 4  |-  ( E. a  e.  ZZ  E. b  e.  ZZ  (
( a  x.  C
)  =  A  /\  ( b  x.  C
)  =  B )  <-> 
( E. a  e.  ZZ  ( a  x.  C )  =  A  /\  E. b  e.  ZZ  ( b  x.  C )  =  B ) )
119, 10syl6bbr 196 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( C  ||  A  /\  C  ||  B )  <->  E. a  e.  ZZ  E. b  e.  ZZ  (
( a  x.  C
)  =  A  /\  ( b  x.  C
)  =  B ) ) )
12 gcdcl 11051 . . . . . . . . . . . 12  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  gcd  b
)  e.  NN0 )
1312nn0cnd 8698 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  gcd  b
)  e.  CC )
14133adant3 963 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
a  gcd  b )  e.  CC )
15 nncn 8402 . . . . . . . . . . 11  |-  ( C  e.  NN  ->  C  e.  CC )
16153ad2ant3 966 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  C  e.  CC )
17 simp3 945 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  C  e.  NN )
1817nnap0d 8439 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  C #  0 )
1914, 16, 18divcanap4d 8236 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( ( a  gcd  b )  x.  C
)  /  C )  =  ( a  gcd  b ) )
20 nnnn0 8650 . . . . . . . . . . 11  |-  ( C  e.  NN  ->  C  e.  NN0 )
21 mulgcdr 11100 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN0 )  ->  (
( a  x.  C
)  gcd  ( b  x.  C ) )  =  ( ( a  gcd  b )  x.  C
) )
2220, 21syl3an3 1209 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( a  x.  C
)  gcd  ( b  x.  C ) )  =  ( ( a  gcd  b )  x.  C
) )
2322oveq1d 5649 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( ( a  x.  C )  gcd  (
b  x.  C ) )  /  C )  =  ( ( ( a  gcd  b )  x.  C )  /  C ) )
24 zcn 8725 . . . . . . . . . . . 12  |-  ( a  e.  ZZ  ->  a  e.  CC )
25243ad2ant1 964 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  a  e.  CC )
2625, 16, 18divcanap4d 8236 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( a  x.  C
)  /  C )  =  a )
27 zcn 8725 . . . . . . . . . . . 12  |-  ( b  e.  ZZ  ->  b  e.  CC )
28273ad2ant2 965 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  b  e.  CC )
2928, 16, 18divcanap4d 8236 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( b  x.  C
)  /  C )  =  b )
3026, 29oveq12d 5652 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( ( a  x.  C )  /  C
)  gcd  ( (
b  x.  C )  /  C ) )  =  ( a  gcd  b ) )
3119, 23, 303eqtr4d 2130 . . . . . . . 8  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( ( a  x.  C )  gcd  (
b  x.  C ) )  /  C )  =  ( ( ( a  x.  C )  /  C )  gcd  ( ( b  x.  C )  /  C
) ) )
32 oveq12 5643 . . . . . . . . . 10  |-  ( ( ( a  x.  C
)  =  A  /\  ( b  x.  C
)  =  B )  ->  ( ( a  x.  C )  gcd  ( b  x.  C
) )  =  ( A  gcd  B ) )
3332oveq1d 5649 . . . . . . . . 9  |-  ( ( ( a  x.  C
)  =  A  /\  ( b  x.  C
)  =  B )  ->  ( ( ( a  x.  C )  gcd  ( b  x.  C ) )  /  C )  =  ( ( A  gcd  B
)  /  C ) )
34 oveq1 5641 . . . . . . . . . 10  |-  ( ( a  x.  C )  =  A  ->  (
( a  x.  C
)  /  C )  =  ( A  /  C ) )
35 oveq1 5641 . . . . . . . . . 10  |-  ( ( b  x.  C )  =  B  ->  (
( b  x.  C
)  /  C )  =  ( B  /  C ) )
3634, 35oveqan12d 5653 . . . . . . . . 9  |-  ( ( ( a  x.  C
)  =  A  /\  ( b  x.  C
)  =  B )  ->  ( ( ( a  x.  C )  /  C )  gcd  ( ( b  x.  C )  /  C
) )  =  ( ( A  /  C
)  gcd  ( B  /  C ) ) )
3733, 36eqeq12d 2102 . . . . . . . 8  |-  ( ( ( a  x.  C
)  =  A  /\  ( b  x.  C
)  =  B )  ->  ( ( ( ( a  x.  C
)  gcd  ( b  x.  C ) )  /  C )  =  ( ( ( a  x.  C )  /  C
)  gcd  ( (
b  x.  C )  /  C ) )  <-> 
( ( A  gcd  B )  /  C )  =  ( ( A  /  C )  gcd  ( B  /  C
) ) ) )
3831, 37syl5ibcom 153 . . . . . . 7  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( ( a  x.  C )  =  A  /\  ( b  x.  C )  =  B )  ->  ( ( A  gcd  B )  /  C )  =  ( ( A  /  C
)  gcd  ( B  /  C ) ) ) )
39383expa 1143 . . . . . 6  |-  ( ( ( a  e.  ZZ  /\  b  e.  ZZ )  /\  C  e.  NN )  ->  ( ( ( a  x.  C )  =  A  /\  (
b  x.  C )  =  B )  -> 
( ( A  gcd  B )  /  C )  =  ( ( A  /  C )  gcd  ( B  /  C
) ) ) )
4039expcom 114 . . . . 5  |-  ( C  e.  NN  ->  (
( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( ( ( a  x.  C )  =  A  /\  (
b  x.  C )  =  B )  -> 
( ( A  gcd  B )  /  C )  =  ( ( A  /  C )  gcd  ( B  /  C
) ) ) ) )
4140rexlimdvv 2495 . . . 4  |-  ( C  e.  NN  ->  ( E. a  e.  ZZ  E. b  e.  ZZ  (
( a  x.  C
)  =  A  /\  ( b  x.  C
)  =  B )  ->  ( ( A  gcd  B )  /  C )  =  ( ( A  /  C
)  gcd  ( B  /  C ) ) ) )
42413ad2ant3 966 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( E. a  e.  ZZ  E. b  e.  ZZ  (
( a  x.  C
)  =  A  /\  ( b  x.  C
)  =  B )  ->  ( ( A  gcd  B )  /  C )  =  ( ( A  /  C
)  gcd  ( B  /  C ) ) ) )
4311, 42sylbid 148 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( C  ||  A  /\  C  ||  B )  ->  ( ( A  gcd  B )  /  C )  =  ( ( A  /  C
)  gcd  ( B  /  C ) ) ) )
4443imp 122 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  /\  ( C  ||  A  /\  C  ||  B ) )  ->  ( ( A  gcd  B )  /  C )  =  ( ( A  /  C
)  gcd  ( B  /  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   E.wrex 2360   class class class wbr 3837  (class class class)co 5634   CCcc 7327    x. cmul 7334    / cdiv 8113   NNcn 8394   NN0cn0 8643   ZZcz 8720    || cdvds 10889    gcd cgcd 11031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-sup 6658  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-fl 9642  df-mod 9695  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-dvds 10890  df-gcd 11032
This theorem is referenced by:  sqgcd  11111  divgcdodd  11215  divnumden  11267  hashgcdlem  11296
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