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Theorem gcddiv 10599
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 8487 . . . . . . 7  |-  ( C  e.  NN  ->  C  e.  ZZ )
213ad2ant3 962 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  C  e.  ZZ )
3 simp1 939 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  A  e.  ZZ )
4 divides 10389 . . . . . 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 940 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  B  e.  ZZ )
7 divides 10389 . . . . . 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 2528 . . . 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 10549 . . . . . . . . . . . 12  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  gcd  b
)  e.  NN0 )
1312nn0cnd 8446 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  gcd  b
)  e.  CC )
14133adant3 959 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
a  gcd  b )  e.  CC )
15 nncn 8150 . . . . . . . . . . 11  |-  ( C  e.  NN  ->  C  e.  CC )
16153ad2ant3 962 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  C  e.  CC )
17 simp3 941 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  C  e.  NN )
1817nnap0d 8187 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  C #  0 )
1914, 16, 18divcanap4d 7986 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( ( a  gcd  b )  x.  C
)  /  C )  =  ( a  gcd  b ) )
20 nnnn0 8398 . . . . . . . . . . 11  |-  ( C  e.  NN  ->  C  e.  NN0 )
21 mulgcdr 10598 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN0 )  ->  (
( a  x.  C
)  gcd  ( b  x.  C ) )  =  ( ( a  gcd  b )  x.  C
) )
2220, 21syl3an3 1205 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( a  x.  C
)  gcd  ( b  x.  C ) )  =  ( ( a  gcd  b )  x.  C
) )
2322oveq1d 5579 . . . . . . . . 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 8473 . . . . . . . . . . . 12  |-  ( a  e.  ZZ  ->  a  e.  CC )
25243ad2ant1 960 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  a  e.  CC )
2625, 16, 18divcanap4d 7986 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( a  x.  C
)  /  C )  =  a )
27 zcn 8473 . . . . . . . . . . . 12  |-  ( b  e.  ZZ  ->  b  e.  CC )
28273ad2ant2 961 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  b  e.  CC )
2928, 16, 18divcanap4d 7986 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( b  x.  C
)  /  C )  =  b )
3026, 29oveq12d 5582 . . . . . . . . 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 2125 . . . . . . . 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 5573 . . . . . . . . . 10  |-  ( ( ( a  x.  C
)  =  A  /\  ( b  x.  C
)  =  B )  ->  ( ( a  x.  C )  gcd  ( b  x.  C
) )  =  ( A  gcd  B ) )
3332oveq1d 5579 . . . . . . . . 9  |-  ( ( ( a  x.  C
)  =  A  /\  ( b  x.  C
)  =  B )  ->  ( ( ( a  x.  C )  gcd  ( b  x.  C ) )  /  C )  =  ( ( A  gcd  B
)  /  C ) )
34 oveq1 5571 . . . . . . . . . 10  |-  ( ( a  x.  C )  =  A  ->  (
( a  x.  C
)  /  C )  =  ( A  /  C ) )
35 oveq1 5571 . . . . . . . . . 10  |-  ( ( b  x.  C )  =  B  ->  (
( b  x.  C
)  /  C )  =  ( B  /  C ) )
3634, 35oveqan12d 5583 . . . . . . . . 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 2097 . . . . . . . 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 1139 . . . . . 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 2488 . . . 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 962 . . 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 920    = wceq 1285    e. wcel 1434   E.wrex 2354   class class class wbr 3806  (class class class)co 5564   CCcc 7077    x. cmul 7084    / cdiv 7863   NNcn 8142   NN0cn0 8391   ZZcz 8468    || cdvds 10387    gcd cgcd 10529
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358  ax-cnex 7165  ax-resscn 7166  ax-1cn 7167  ax-1re 7168  ax-icn 7169  ax-addcl 7170  ax-addrcl 7171  ax-mulcl 7172  ax-mulrcl 7173  ax-addcom 7174  ax-mulcom 7175  ax-addass 7176  ax-mulass 7177  ax-distr 7178  ax-i2m1 7179  ax-0lt1 7180  ax-1rid 7181  ax-0id 7182  ax-rnegex 7183  ax-precex 7184  ax-cnre 7185  ax-pre-ltirr 7186  ax-pre-ltwlin 7187  ax-pre-lttrn 7188  ax-pre-apti 7189  ax-pre-ltadd 7190  ax-pre-mulgt0 7191  ax-pre-mulext 7192  ax-arch 7193  ax-caucvg 7194
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-if 3370  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-ilim 4153  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-sup 6492  df-pnf 7253  df-mnf 7254  df-xr 7255  df-ltxr 7256  df-le 7257  df-sub 7384  df-neg 7385  df-reap 7778  df-ap 7785  df-div 7864  df-inn 8143  df-2 8201  df-3 8202  df-4 8203  df-n0 8392  df-z 8469  df-uz 8737  df-q 8822  df-rp 8852  df-fz 9142  df-fzo 9266  df-fl 9388  df-mod 9441  df-iseq 9558  df-iexp 9609  df-cj 9914  df-re 9915  df-im 9916  df-rsqrt 10069  df-abs 10070  df-dvds 10388  df-gcd 10530
This theorem is referenced by:  sqgcd  10609  divgcdodd  10713  divnumden  10765  hashgcdlem  10794
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