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Theorem divgcdodd 14347
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 14122 . . . 4  |-  -.  2  ||  1
2 nnz 10882 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  ZZ )
3 nnz 10882 . . . . . . . . . 10  |-  ( B  e.  NN  ->  B  e.  ZZ )
4 gcddvds 14240 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
52, 3, 4syl2an 475 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
65simpld 457 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  A )
72, 3anim12i 564 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  e.  ZZ  /\  B  e.  ZZ ) )
8 nnne0 10564 . . . . . . . . . . . . . 14  |-  ( A  e.  NN  ->  A  =/=  0 )
98neneqd 2656 . . . . . . . . . . . . 13  |-  ( A  e.  NN  ->  -.  A  =  0 )
109intnanrd 915 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  -.  ( A  =  0  /\  B  =  0
) )
1110adantr 463 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  -.  ( A  =  0  /\  B  =  0 ) )
12 gcdn0cl 14239 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
137, 11, 12syl2anc 659 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  NN )
1413nnzd 10964 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  ZZ )
1513nnne0d 10576 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  =/=  0 )
162adantr 463 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  A  e.  ZZ )
17 dvdsval2 14076 . . . . . . . . 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 1226 . . . . . . . 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 461 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  B )
213adantl 464 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  B  e.  ZZ )
22 dvdsval2 14076 . . . . . . . . 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 1226 . . . . . . . 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 10892 . . . . . . . 8  |-  2  e.  ZZ
26 dvdsgcdb 14269 . . . . . . . 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 1309 . . . . . . 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 659 . . . . . 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 14274 . . . . . . . . . 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 1229 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  ( ( A  /  ( A  gcd  B ) )  gcd  ( B  /  ( A  gcd  B ) ) ) )
3113nncnd 10547 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  CC )
3231, 15dividd 10314 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  1 )
3330, 32eqtr3d 2497 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  / 
( A  gcd  B
) )  gcd  ( B  /  ( A  gcd  B ) ) )  =  1 )
3433breq2d 4451 . . . . . . 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 435 . . . 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 432 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( 2  ||  ( A  /  ( A  gcd  B ) )  ->  -.  2  ||  ( B  / 
( A  gcd  B
) ) ) )
40 imor 410 . 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 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439  (class class class)co 6270   0cc0 9481   1c1 9482    / cdiv 10202   NNcn 10531   2c2 10581   ZZcz 10860    || cdvds 14073    gcd cgcd 14231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fl 11910  df-mod 11979  df-seq 12093  df-exp 12152  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-dvds 14074  df-gcd 14232
This theorem is referenced by:  pythagtrip  14445  divgcdoddALTV  32596
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