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Theorem gcddiv 12969
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 10228 . . . . . . 7  |-  ( C  e.  NN  ->  C  e.  ZZ )
213ad2ant3 980 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  C  e.  ZZ )
3 simp1 957 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  A  e.  ZZ )
4 divides 12774 . . . . . 6  |-  ( ( C  e.  ZZ  /\  A  e.  ZZ )  ->  ( C  ||  A  <->  E. a  e.  ZZ  (
a  x.  C )  =  A ) )
52, 3, 4syl2anc 643 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( C  ||  A  <->  E. a  e.  ZZ  ( a  x.  C )  =  A ) )
6 simp2 958 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  B  e.  ZZ )
7 divides 12774 . . . . . 6  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  ||  B  <->  E. b  e.  ZZ  (
b  x.  C )  =  B ) )
82, 6, 7syl2anc 643 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( C  ||  B  <->  E. b  e.  ZZ  ( b  x.  C )  =  B ) )
95, 8anbi12d 692 . . . 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 2811 . . . 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 255 . . 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 12937 . . . . . . . . . . . 12  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  gcd  b
)  e.  NN0 )
1312nn0cnd 10201 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  gcd  b
)  e.  CC )
14133adant3 977 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
a  gcd  b )  e.  CC )
15 nncn 9933 . . . . . . . . . . 11  |-  ( C  e.  NN  ->  C  e.  CC )
16153ad2ant3 980 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  C  e.  CC )
17 nnne0 9957 . . . . . . . . . . 11  |-  ( C  e.  NN  ->  C  =/=  0 )
18173ad2ant3 980 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  C  =/=  0 )
1914, 16, 18divcan4d 9721 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( ( a  gcd  b )  x.  C
)  /  C )  =  ( a  gcd  b ) )
20 nnnn0 10153 . . . . . . . . . . 11  |-  ( C  e.  NN  ->  C  e.  NN0 )
21 mulgcdr 12968 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN0 )  ->  (
( a  x.  C
)  gcd  ( b  x.  C ) )  =  ( ( a  gcd  b )  x.  C
) )
2220, 21syl3an3 1219 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( a  x.  C
)  gcd  ( b  x.  C ) )  =  ( ( a  gcd  b )  x.  C
) )
2322oveq1d 6028 . . . . . . . . 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 10212 . . . . . . . . . . . 12  |-  ( a  e.  ZZ  ->  a  e.  CC )
25243ad2ant1 978 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  a  e.  CC )
2625, 16, 18divcan4d 9721 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( a  x.  C
)  /  C )  =  a )
27 zcn 10212 . . . . . . . . . . . 12  |-  ( b  e.  ZZ  ->  b  e.  CC )
28273ad2ant2 979 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  b  e.  CC )
2928, 16, 18divcan4d 9721 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( b  x.  C
)  /  C )  =  b )
3026, 29oveq12d 6031 . . . . . . . . 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 2422 . . . . . . . 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 6022 . . . . . . . . . 10  |-  ( ( ( a  x.  C
)  =  A  /\  ( b  x.  C
)  =  B )  ->  ( ( a  x.  C )  gcd  ( b  x.  C
) )  =  ( A  gcd  B ) )
3332oveq1d 6028 . . . . . . . . 9  |-  ( ( ( a  x.  C
)  =  A  /\  ( b  x.  C
)  =  B )  ->  ( ( ( a  x.  C )  gcd  ( b  x.  C ) )  /  C )  =  ( ( A  gcd  B
)  /  C ) )
34 oveq1 6020 . . . . . . . . . 10  |-  ( ( a  x.  C )  =  A  ->  (
( a  x.  C
)  /  C )  =  ( A  /  C ) )
35 oveq1 6020 . . . . . . . . . 10  |-  ( ( b  x.  C )  =  B  ->  (
( b  x.  C
)  /  C )  =  ( B  /  C ) )
3634, 35oveqan12d 6032 . . . . . . . . 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 2394 . . . . . . . 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 212 . . . . . . 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 1153 . . . . . 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 425 . . . . 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 2772 . . . 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 980 . . 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 207 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( C  ||  A  /\  C  ||  B )  ->  ( ( A  gcd  B )  /  C )  =  ( ( A  /  C
)  gcd  ( B  /  C ) ) ) )
4443imp 419 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   E.wrex 2643   class class class wbr 4146  (class class class)co 6013   CCcc 8914   0cc0 8916    x. cmul 8921    / cdiv 9602   NNcn 9925   NN0cn0 10146   ZZcz 10207    || cdivides 12772    gcd cgcd 12926
This theorem is referenced by:  sqgcd  12978  divgcdodd  13039  divnumden  13060  pythagtriplem19  13127  hashgcdlem  27178
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-fl 11122  df-mod 11171  df-seq 11244  df-exp 11303  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-dvds 12773  df-gcd 12927
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