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Theorem divgcdodd 13119
Description: Either  A  /  ( A  gcd  B ) is odd or  B  /  ( A  gcd  B ) is odd. (Contributed by Scott Fenton, 19-Apr-2014.)
Assertion
Ref Expression
divgcdodd  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( -.  2  ||  ( A  /  ( A  gcd  B ) )  \/  -.  2  ||  ( B  /  ( A  gcd  B ) ) ) )

Proof of Theorem divgcdodd
StepHypRef Expression
1 2prm 13095 . . . . 5  |-  2  e.  Prime
2 nprmdvds1 13111 . . . . 5  |-  ( 2  e.  Prime  ->  -.  2  ||  1 )
31, 2ax-mp 8 . . . 4  |-  -.  2  ||  1
4 nnz 10303 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  ZZ )
5 nnz 10303 . . . . . . . . . 10  |-  ( B  e.  NN  ->  B  e.  ZZ )
6 gcddvds 13015 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
74, 5, 6syl2an 464 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
87simpld 446 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  A )
94, 5anim12i 550 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  e.  ZZ  /\  B  e.  ZZ ) )
10 nnne0 10032 . . . . . . . . . . . . . 14  |-  ( A  e.  NN  ->  A  =/=  0 )
1110neneqd 2617 . . . . . . . . . . . . 13  |-  ( A  e.  NN  ->  -.  A  =  0 )
1211intnanrd 884 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  -.  ( A  =  0  /\  B  =  0
) )
1312adantr 452 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  -.  ( A  =  0  /\  B  =  0 ) )
14 gcdn0cl 13014 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
159, 13, 14syl2anc 643 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  NN )
1615nnzd 10374 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  ZZ )
1715nnne0d 10044 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  =/=  0 )
184adantr 452 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  A  e.  ZZ )
19 dvdsval2 12855 . . . . . . . . 9  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  ( A  gcd  B )  =/=  0  /\  A  e.  ZZ )  ->  (
( A  gcd  B
)  ||  A  <->  ( A  /  ( A  gcd  B ) )  e.  ZZ ) )
2016, 17, 18, 19syl3anc 1184 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  A  <->  ( A  /  ( A  gcd  B ) )  e.  ZZ ) )
218, 20mpbid 202 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  /  ( A  gcd  B ) )  e.  ZZ )
227simprd 450 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  B )
235adantl 453 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  B  e.  ZZ )
24 dvdsval2 12855 . . . . . . . . 9  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  ( A  gcd  B )  =/=  0  /\  B  e.  ZZ )  ->  (
( A  gcd  B
)  ||  B  <->  ( B  /  ( A  gcd  B ) )  e.  ZZ ) )
2516, 17, 23, 24syl3anc 1184 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  B  <->  ( B  /  ( A  gcd  B ) )  e.  ZZ ) )
2622, 25mpbid 202 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( B  /  ( A  gcd  B ) )  e.  ZZ )
27 2z 10312 . . . . . . . 8  |-  2  e.  ZZ
28 dvdsgcdb 13044 . . . . . . . 8  |-  ( ( 2  e.  ZZ  /\  ( A  /  ( A  gcd  B ) )  e.  ZZ  /\  ( B  /  ( A  gcd  B ) )  e.  ZZ )  ->  ( ( 2 
||  ( A  / 
( A  gcd  B
) )  /\  2  ||  ( B  /  ( A  gcd  B ) ) )  <->  2  ||  (
( A  /  ( A  gcd  B ) )  gcd  ( B  / 
( A  gcd  B
) ) ) ) )
2927, 28mp3an1 1266 . . . . . . 7  |-  ( ( ( A  /  ( A  gcd  B ) )  e.  ZZ  /\  ( B  /  ( A  gcd  B ) )  e.  ZZ )  ->  ( ( 2 
||  ( A  / 
( A  gcd  B
) )  /\  2  ||  ( B  /  ( A  gcd  B ) ) )  <->  2  ||  (
( A  /  ( A  gcd  B ) )  gcd  ( B  / 
( A  gcd  B
) ) ) ) )
3021, 26, 29syl2anc 643 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( 2  ||  ( A  /  ( A  gcd  B ) )  /\  2  ||  ( B  /  ( A  gcd  B ) ) )  <->  2  ||  ( ( A  / 
( A  gcd  B
) )  gcd  ( B  /  ( A  gcd  B ) ) ) ) )
31 gcddiv 13049 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( A  gcd  B )  e.  NN )  /\  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )  ->  ( ( A  gcd  B )  / 
( A  gcd  B
) )  =  ( ( A  /  ( A  gcd  B ) )  gcd  ( B  / 
( A  gcd  B
) ) ) )
3218, 23, 15, 7, 31syl31anc 1187 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  ( ( A  /  ( A  gcd  B ) )  gcd  ( B  /  ( A  gcd  B ) ) ) )
3315nncnd 10016 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  CC )
3433, 17dividd 9788 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  1 )
3532, 34eqtr3d 2470 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  / 
( A  gcd  B
) )  gcd  ( B  /  ( A  gcd  B ) ) )  =  1 )
3635breq2d 4224 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( 2  ||  (
( A  /  ( A  gcd  B ) )  gcd  ( B  / 
( A  gcd  B
) ) )  <->  2  ||  1 ) )
3736biimpd 199 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( 2  ||  (
( A  /  ( A  gcd  B ) )  gcd  ( B  / 
( A  gcd  B
) ) )  -> 
2  ||  1 ) )
3830, 37sylbid 207 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( 2  ||  ( A  /  ( A  gcd  B ) )  /\  2  ||  ( B  /  ( A  gcd  B ) ) )  -> 
2  ||  1 ) )
3938expdimp 427 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  2  ||  ( A  /  ( A  gcd  B ) ) )  -> 
( 2  ||  ( B  /  ( A  gcd  B ) )  ->  2  ||  1 ) )
403, 39mtoi 171 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  2  ||  ( A  /  ( A  gcd  B ) ) )  ->  -.  2  ||  ( B  /  ( A  gcd  B ) ) )
4140ex 424 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( 2  ||  ( A  /  ( A  gcd  B ) )  ->  -.  2  ||  ( B  / 
( A  gcd  B
) ) ) )
42 imor 402 . 2  |-  ( ( 2  ||  ( A  /  ( A  gcd  B ) )  ->  -.  2  ||  ( B  / 
( A  gcd  B
) ) )  <->  ( -.  2  ||  ( A  / 
( A  gcd  B
) )  \/  -.  2  ||  ( B  / 
( A  gcd  B
) ) ) )
4341, 42sylib 189 1  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( -.  2  ||  ( A  /  ( A  gcd  B ) )  \/  -.  2  ||  ( B  /  ( A  gcd  B ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212  (class class class)co 6081   0cc0 8990   1c1 8991    / cdiv 9677   NNcn 10000   2c2 10049   ZZcz 10282    || cdivides 12852    gcd cgcd 13006   Primecprime 13079
This theorem is referenced by:  pythagtrip  13208
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-prm 13080
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