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Theorem gcddiv 11918
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 9191 . . . . . . 7  |-  ( C  e.  NN  ->  C  e.  ZZ )
213ad2ant3 1005 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  C  e.  ZZ )
3 simp1 982 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  A  e.  ZZ )
4 divides 11696 . . . . . 6  |-  ( ( C  e.  ZZ  /\  A  e.  ZZ )  ->  ( C  ||  A  <->  E. a  e.  ZZ  (
a  x.  C )  =  A ) )
52, 3, 4syl2anc 409 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( C  ||  A  <->  E. a  e.  ZZ  ( a  x.  C )  =  A ) )
6 simp2 983 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  B  e.  ZZ )
7 divides 11696 . . . . . 6  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  ||  B  <->  E. b  e.  ZZ  (
b  x.  C )  =  B ) )
82, 6, 7syl2anc 409 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  ( C  ||  B  <->  E. b  e.  ZZ  ( b  x.  C )  =  B ) )
95, 8anbi12d 465 . . . 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 2626 . . . 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, 10bitr4di 197 . . 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 11865 . . . . . . . . . . . 12  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  gcd  b
)  e.  NN0 )
1312nn0cnd 9150 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  gcd  b
)  e.  CC )
14133adant3 1002 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
a  gcd  b )  e.  CC )
15 nncn 8846 . . . . . . . . . . 11  |-  ( C  e.  NN  ->  C  e.  CC )
16153ad2ant3 1005 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  C  e.  CC )
17 simp3 984 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  C  e.  NN )
1817nnap0d 8884 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  C #  0 )
1914, 16, 18divcanap4d 8673 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( ( a  gcd  b )  x.  C
)  /  C )  =  ( a  gcd  b ) )
20 nnnn0 9102 . . . . . . . . . . 11  |-  ( C  e.  NN  ->  C  e.  NN0 )
21 mulgcdr 11917 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN0 )  ->  (
( a  x.  C
)  gcd  ( b  x.  C ) )  =  ( ( a  gcd  b )  x.  C
) )
2220, 21syl3an3 1255 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( a  x.  C
)  gcd  ( b  x.  C ) )  =  ( ( a  gcd  b )  x.  C
) )
2322oveq1d 5841 . . . . . . . . 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 9177 . . . . . . . . . . . 12  |-  ( a  e.  ZZ  ->  a  e.  CC )
25243ad2ant1 1003 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  a  e.  CC )
2625, 16, 18divcanap4d 8673 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( a  x.  C
)  /  C )  =  a )
27 zcn 9177 . . . . . . . . . . . 12  |-  ( b  e.  ZZ  ->  b  e.  CC )
28273ad2ant2 1004 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  b  e.  CC )
2928, 16, 18divcanap4d 8673 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ  /\  C  e.  NN )  ->  (
( b  x.  C
)  /  C )  =  b )
3026, 29oveq12d 5844 . . . . . . . . 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 2200 . . . . . . . 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 5835 . . . . . . . . . 10  |-  ( ( ( a  x.  C
)  =  A  /\  ( b  x.  C
)  =  B )  ->  ( ( a  x.  C )  gcd  ( b  x.  C
) )  =  ( A  gcd  B ) )
3332oveq1d 5841 . . . . . . . . 9  |-  ( ( ( a  x.  C
)  =  A  /\  ( b  x.  C
)  =  B )  ->  ( ( ( a  x.  C )  gcd  ( b  x.  C ) )  /  C )  =  ( ( A  gcd  B
)  /  C ) )
34 oveq1 5833 . . . . . . . . . 10  |-  ( ( a  x.  C )  =  A  ->  (
( a  x.  C
)  /  C )  =  ( A  /  C ) )
35 oveq1 5833 . . . . . . . . . 10  |-  ( ( b  x.  C )  =  B  ->  (
( b  x.  C
)  /  C )  =  ( B  /  C ) )
3634, 35oveqan12d 5845 . . . . . . . . 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 2172 . . . . . . . 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 154 . . . . . . 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 1185 . . . . . 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 115 . . . . 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 2581 . . . 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 1005 . . 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 149 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  NN )  ->  (
( C  ||  A  /\  C  ||  B )  ->  ( ( A  gcd  B )  /  C )  =  ( ( A  /  C
)  gcd  ( B  /  C ) ) ) )
4443imp 123 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 103    <-> wb 104    /\ w3a 963    = wceq 1335    e. wcel 2128   E.wrex 2436   class class class wbr 3967  (class class class)co 5826   CCcc 7732    x. cmul 7739    / cdiv 8549   NNcn 8838   NN0cn0 9095   ZZcz 9172    || cdvds 11694    gcd cgcd 11841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-mulrcl 7833  ax-addcom 7834  ax-mulcom 7835  ax-addass 7836  ax-mulass 7837  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-1rid 7841  ax-0id 7842  ax-rnegex 7843  ax-precex 7844  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-apti 7849  ax-pre-ltadd 7850  ax-pre-mulgt0 7851  ax-pre-mulext 7852  ax-arch 7853  ax-caucvg 7854
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-ilim 4331  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-frec 6340  df-sup 6930  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-reap 8454  df-ap 8461  df-div 8550  df-inn 8839  df-2 8897  df-3 8898  df-4 8899  df-n0 9096  df-z 9173  df-uz 9445  df-q 9535  df-rp 9567  df-fz 9919  df-fzo 10051  df-fl 10178  df-mod 10231  df-seqfrec 10354  df-exp 10428  df-cj 10753  df-re 10754  df-im 10755  df-rsqrt 10909  df-abs 10910  df-dvds 11695  df-gcd 11842
This theorem is referenced by:  sqgcd  11928  divgcdodd  12033  divnumden  12086  hashgcdlem  12128  pythagtriplem19  12172
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