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Theorem divgcdodd 13893
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 n2dvds1 13670 . . . 4  |-  -.  2  ||  1
2 nnz 10755 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  ZZ )
3 nnz 10755 . . . . . . . . . 10  |-  ( B  e.  NN  ->  B  e.  ZZ )
4 gcddvds 13787 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
52, 3, 4syl2an 477 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
65simpld 459 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  A )
72, 3anim12i 566 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  e.  ZZ  /\  B  e.  ZZ ) )
8 nnne0 10441 . . . . . . . . . . . . . 14  |-  ( A  e.  NN  ->  A  =/=  0 )
98neneqd 2648 . . . . . . . . . . . . 13  |-  ( A  e.  NN  ->  -.  A  =  0 )
109intnanrd 908 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  -.  ( A  =  0  /\  B  =  0
) )
1110adantr 465 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  -.  ( A  =  0  /\  B  =  0 ) )
12 gcdn0cl 13786 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
137, 11, 12syl2anc 661 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  NN )
1413nnzd 10833 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  ZZ )
1513nnne0d 10453 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  =/=  0 )
162adantr 465 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  A  e.  ZZ )
17 dvdsval2 13626 . . . . . . . . 9  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  ( A  gcd  B )  =/=  0  /\  A  e.  ZZ )  ->  (
( A  gcd  B
)  ||  A  <->  ( A  /  ( A  gcd  B ) )  e.  ZZ ) )
1814, 15, 16, 17syl3anc 1219 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  A  <->  ( A  /  ( A  gcd  B ) )  e.  ZZ ) )
196, 18mpbid 210 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  /  ( A  gcd  B ) )  e.  ZZ )
205simprd 463 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  B )
213adantl 466 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  B  e.  ZZ )
22 dvdsval2 13626 . . . . . . . . 9  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  ( A  gcd  B )  =/=  0  /\  B  e.  ZZ )  ->  (
( A  gcd  B
)  ||  B  <->  ( B  /  ( A  gcd  B ) )  e.  ZZ ) )
2314, 15, 21, 22syl3anc 1219 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  B  <->  ( B  /  ( A  gcd  B ) )  e.  ZZ ) )
2420, 23mpbid 210 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( B  /  ( A  gcd  B ) )  e.  ZZ )
25 2z 10765 . . . . . . . 8  |-  2  e.  ZZ
26 dvdsgcdb 13816 . . . . . . . 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
) ) ) ) )
2725, 26mp3an1 1302 . . . . . . 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
) ) ) ) )
2819, 24, 27syl2anc 661 . . . . . 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 ) ) ) ) )
29 gcddiv 13821 . . . . . . . . . 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
) ) ) )
3016, 21, 13, 5, 29syl31anc 1222 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  ( ( A  /  ( A  gcd  B ) )  gcd  ( B  /  ( A  gcd  B ) ) ) )
3113nncnd 10425 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  CC )
3231, 15dividd 10192 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  1 )
3330, 32eqtr3d 2492 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  / 
( A  gcd  B
) )  gcd  ( B  /  ( A  gcd  B ) ) )  =  1 )
3433breq2d 4388 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( 2  ||  (
( A  /  ( A  gcd  B ) )  gcd  ( B  / 
( A  gcd  B
) ) )  <->  2  ||  1 ) )
3534biimpd 207 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( 2  ||  (
( A  /  ( A  gcd  B ) )  gcd  ( B  / 
( A  gcd  B
) ) )  -> 
2  ||  1 ) )
3628, 35sylbid 215 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( 2  ||  ( A  /  ( A  gcd  B ) )  /\  2  ||  ( B  /  ( A  gcd  B ) ) )  -> 
2  ||  1 ) )
3736expdimp 437 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  2  ||  ( A  /  ( A  gcd  B ) ) )  -> 
( 2  ||  ( B  /  ( A  gcd  B ) )  ->  2  ||  1 ) )
381, 37mtoi 178 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  2  ||  ( A  /  ( A  gcd  B ) ) )  ->  -.  2  ||  ( B  /  ( A  gcd  B ) ) )
3938ex 434 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( 2  ||  ( A  /  ( A  gcd  B ) )  ->  -.  2  ||  ( B  / 
( A  gcd  B
) ) ) )
40 imor 412 . 2  |-  ( ( 2  ||  ( A  /  ( A  gcd  B ) )  ->  -.  2  ||  ( B  / 
( A  gcd  B
) ) )  <->  ( -.  2  ||  ( A  / 
( A  gcd  B
) )  \/  -.  2  ||  ( B  / 
( A  gcd  B
) ) ) )
4139, 40sylib 196 1  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( -.  2  ||  ( A  /  ( A  gcd  B ) )  \/  -.  2  ||  ( B  /  ( A  gcd  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1757    =/= wne 2641   class class class wbr 4376  (class class class)co 6176   0cc0 9369   1c1 9370    / cdiv 10080   NNcn 10409   2c2 10458   ZZcz 10733    || cdivides 13623    gcd cgcd 13778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446  ax-pre-sup 9447
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-2nd 6664  df-recs 6918  df-rdg 6952  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-sup 7778  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-div 10081  df-nn 10410  df-2 10467  df-3 10468  df-n0 10667  df-z 10734  df-uz 10949  df-rp 11079  df-fl 11729  df-mod 11796  df-seq 11894  df-exp 11953  df-cj 12676  df-re 12677  df-im 12678  df-sqr 12812  df-abs 12813  df-dvds 13624  df-gcd 13779
This theorem is referenced by:  pythagtrip  13989
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