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Theorem divgcdodd 12798
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 12774 . . . . 5  |-  2  e.  Prime
2 nprmdvds1 12790 . . . . 5  |-  ( 2  e.  Prime  ->  -.  2  ||  1 )
31, 2ax-mp 8 . . . 4  |-  -.  2  ||  1
4 nnz 10045 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  ZZ )
5 nnz 10045 . . . . . . . . . 10  |-  ( B  e.  NN  ->  B  e.  ZZ )
6 gcddvds 12694 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
74, 5, 6syl2an 463 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
87simpld 445 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  A )
94, 5anim12i 549 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  e.  ZZ  /\  B  e.  ZZ ) )
10 nnne0 9778 . . . . . . . . . . . . . 14  |-  ( A  e.  NN  ->  A  =/=  0 )
1110neneqd 2462 . . . . . . . . . . . . 13  |-  ( A  e.  NN  ->  -.  A  =  0 )
1211intnanrd 883 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  -.  ( A  =  0  /\  B  =  0
) )
1312adantr 451 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  -.  ( A  =  0  /\  B  =  0 ) )
14 gcdn0cl 12693 . . . . . . . . . . 11  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
159, 13, 14syl2anc 642 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  NN )
1615nnzd 10116 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  ZZ )
17 nnne0 9778 . . . . . . . . . 10  |-  ( ( A  gcd  B )  e.  NN  ->  ( A  gcd  B )  =/=  0 )
1815, 17syl 15 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  =/=  0 )
194adantr 451 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  A  e.  ZZ )
20 dvdsval2 12534 . . . . . . . . 9  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  ( A  gcd  B )  =/=  0  /\  A  e.  ZZ )  ->  (
( A  gcd  B
)  ||  A  <->  ( A  /  ( A  gcd  B ) )  e.  ZZ ) )
2116, 18, 19, 20syl3anc 1182 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  A  <->  ( A  /  ( A  gcd  B ) )  e.  ZZ ) )
228, 21mpbid 201 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  /  ( A  gcd  B ) )  e.  ZZ )
237simprd 449 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  B )
245adantl 452 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  B  e.  ZZ )
25 dvdsval2 12534 . . . . . . . . 9  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  ( A  gcd  B )  =/=  0  /\  B  e.  ZZ )  ->  (
( A  gcd  B
)  ||  B  <->  ( B  /  ( A  gcd  B ) )  e.  ZZ ) )
2616, 18, 24, 25syl3anc 1182 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  B  <->  ( B  /  ( A  gcd  B ) )  e.  ZZ ) )
2723, 26mpbid 201 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( B  /  ( A  gcd  B ) )  e.  ZZ )
28 2z 10054 . . . . . . . 8  |-  2  e.  ZZ
29 dvdsgcdb 12723 . . . . . . . 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
) ) ) ) )
3028, 29mp3an1 1264 . . . . . . 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
) ) ) ) )
3122, 27, 30syl2anc 642 . . . . . 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 ) ) ) ) )
32 gcddiv 12728 . . . . . . . . . 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
) ) ) )
3319, 24, 15, 7, 32syl31anc 1185 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  ( ( A  /  ( A  gcd  B ) )  gcd  ( B  /  ( A  gcd  B ) ) ) )
3415nncnd 9762 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  CC )
3534, 18dividd 9534 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  1 )
3633, 35eqtr3d 2317 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  / 
( A  gcd  B
) )  gcd  ( B  /  ( A  gcd  B ) ) )  =  1 )
3736breq2d 4035 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( 2  ||  (
( A  /  ( A  gcd  B ) )  gcd  ( B  / 
( A  gcd  B
) ) )  <->  2  ||  1 ) )
3837biimpd 198 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( 2  ||  (
( A  /  ( A  gcd  B ) )  gcd  ( B  / 
( A  gcd  B
) ) )  -> 
2  ||  1 ) )
3931, 38sylbid 206 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( 2  ||  ( A  /  ( A  gcd  B ) )  /\  2  ||  ( B  /  ( A  gcd  B ) ) )  -> 
2  ||  1 ) )
4039expdimp 426 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  2  ||  ( A  /  ( A  gcd  B ) ) )  -> 
( 2  ||  ( B  /  ( A  gcd  B ) )  ->  2  ||  1 ) )
413, 40mtoi 169 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  2  ||  ( A  /  ( A  gcd  B ) ) )  ->  -.  2  ||  ( B  /  ( A  gcd  B ) ) )
4241ex 423 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( 2  ||  ( A  /  ( A  gcd  B ) )  ->  -.  2  ||  ( B  / 
( A  gcd  B
) ) ) )
43 imor 401 . 2  |-  ( ( 2  ||  ( A  /  ( A  gcd  B ) )  ->  -.  2  ||  ( B  / 
( A  gcd  B
) ) )  <->  ( -.  2  ||  ( A  / 
( A  gcd  B
) )  \/  -.  2  ||  ( B  / 
( A  gcd  B
) ) ) )
4442, 43sylib 188 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 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023  (class class class)co 5858   0cc0 8737   1c1 8738    / cdiv 9423   NNcn 9746   2c2 9795   ZZcz 10024    || cdivides 12531    gcd cgcd 12685   Primecprime 12758
This theorem is referenced by:  pythagtrip  12887
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759
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