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Theorem gcddiv 12744
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 10061 . . . . . . 7  |-  ( C  e.  NN  ->  C  e.  ZZ )
213ad2ant3 978 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  C  e.  ZZ )
3 simp1 955 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  A  e.  ZZ )
4 divides 12549 . . . . . 6  |-  ( ( C  e.  ZZ  /\  A  e.  ZZ )  ->  ( C  ||  A  <->  E. a  e.  ZZ  (
a  x.  C )  =  A ) )
52, 3, 4syl2anc 642 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( C  ||  A  <->  E. a  e.  ZZ  ( a  x.  C )  =  A ) )
6 simp2 956 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  B  e.  ZZ )
7 divides 12549 . . . . . 6  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  ||  B  <->  E. b  e.  ZZ  (
b  x.  C )  =  B ) )
82, 6, 7syl2anc 642 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( C  ||  B  <->  E. b  e.  ZZ  ( b  x.  C )  =  B ) )
95, 8anbi12d 691 . . . 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 2720 . . . 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 254 . . 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 12712 . . . . . . . . . . . 12  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  gcd  b
)  e.  NN0 )
1312nn0cnd 10036 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  gcd  b
)  e.  CC )
14133adant3 975 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
a  gcd  b )  e.  CC )
15 nncn 9770 . . . . . . . . . . 11  |-  ( C  e.  NN  ->  C  e.  CC )
16153ad2ant3 978 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  C  e.  CC )
17 nnne0 9794 . . . . . . . . . . 11  |-  ( C  e.  NN  ->  C  =/=  0 )
18173ad2ant3 978 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  C  =/=  0 )
1914, 16, 18divcan4d 9558 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( ( a  gcd  b )  x.  C
)  /  C )  =  ( a  gcd  b ) )
20 nnnn0 9988 . . . . . . . . . . 11  |-  ( C  e.  NN  ->  C  e.  NN0 )
21 mulgcdr 12743 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN0 )  ->  (
( a  x.  C
)  gcd  ( b  x.  C ) )  =  ( ( a  gcd  b )  x.  C
) )
2220, 21syl3an3 1217 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( a  x.  C
)  gcd  ( b  x.  C ) )  =  ( ( a  gcd  b )  x.  C
) )
2322oveq1d 5889 . . . . . . . . 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 10045 . . . . . . . . . . . 12  |-  ( a  e.  ZZ  ->  a  e.  CC )
25243ad2ant1 976 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  a  e.  CC )
2625, 16, 18divcan4d 9558 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( a  x.  C
)  /  C )  =  a )
27 zcn 10045 . . . . . . . . . . . 12  |-  ( b  e.  ZZ  ->  b  e.  CC )
28273ad2ant2 977 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  b  e.  CC )
2928, 16, 18divcan4d 9558 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( b  x.  C
)  /  C )  =  b )
3026, 29oveq12d 5892 . . . . . . . . 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 2338 . . . . . . . 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 5883 . . . . . . . . . 10  |-  ( ( ( a  x.  C
)  =  A  /\  ( b  x.  C
)  =  B )  ->  ( ( a  x.  C )  gcd  ( b  x.  C
) )  =  ( A  gcd  B ) )
3332oveq1d 5889 . . . . . . . . 9  |-  ( ( ( a  x.  C
)  =  A  /\  ( b  x.  C
)  =  B )  ->  ( ( ( a  x.  C )  gcd  ( b  x.  C ) )  /  C )  =  ( ( A  gcd  B
)  /  C ) )
34 oveq1 5881 . . . . . . . . . 10  |-  ( ( a  x.  C )  =  A  ->  (
( a  x.  C
)  /  C )  =  ( A  /  C ) )
35 oveq1 5881 . . . . . . . . . 10  |-  ( ( b  x.  C )  =  B  ->  (
( b  x.  C
)  /  C )  =  ( B  /  C ) )
3634, 35oveqan12d 5893 . . . . . . . . 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 2310 . . . . . . . 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 211 . . . . . . 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 1151 . . . . . 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 424 . . . . 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 2686 . . . 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 978 . . 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 206 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( C  ||  A  /\  C  ||  B )  ->  ( ( A  gcd  B )  /  C )  =  ( ( A  /  C
)  gcd  ( B  /  C ) ) ) )
4443imp 418 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039  (class class class)co 5874   CCcc 8751   0cc0 8753    x. cmul 8758    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040    || cdivides 12547    gcd cgcd 12701
This theorem is referenced by:  sqgcd  12753  divgcdodd  12814  divnumden  12835  pythagtriplem19  12902  hashgcdlem  27619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702
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