ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  divgcdodd Unicode version

Theorem divgcdodd 11667
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 11457 . . . 4  |-  -.  2  ||  1
2 2z 8986 . . . . . . 7  |-  2  e.  ZZ
3 nnz 8977 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  ZZ )
4 nnz 8977 . . . . . . . . . 10  |-  ( B  e.  NN  ->  B  e.  ZZ )
5 gcddvds 11500 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
63, 4, 5syl2an 285 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
76simpld 111 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  A )
8 gcdnncl 11504 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  NN )
98nnzd 9076 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  ZZ )
108nnne0d 8675 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  =/=  0 )
113adantr 272 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  A  e.  ZZ )
12 dvdsval2 11344 . . . . . . . . 9  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  ( A  gcd  B )  =/=  0  /\  A  e.  ZZ )  ->  (
( A  gcd  B
)  ||  A  <->  ( A  /  ( A  gcd  B ) )  e.  ZZ ) )
139, 10, 11, 12syl3anc 1199 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  A  <->  ( A  /  ( A  gcd  B ) )  e.  ZZ ) )
147, 13mpbid 146 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  /  ( A  gcd  B ) )  e.  ZZ )
156simprd 113 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  ||  B )
164adantl 273 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  B  e.  ZZ )
17 dvdsval2 11344 . . . . . . . . 9  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  ( A  gcd  B )  =/=  0  /\  B  e.  ZZ )  ->  (
( A  gcd  B
)  ||  B  <->  ( B  /  ( A  gcd  B ) )  e.  ZZ ) )
189, 10, 16, 17syl3anc 1199 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  ||  B  <->  ( B  /  ( A  gcd  B ) )  e.  ZZ ) )
1915, 18mpbid 146 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( B  /  ( A  gcd  B ) )  e.  ZZ )
20 dvdsgcdb 11547 . . . . . . 7  |-  ( ( 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
) ) ) ) )
212, 14, 19, 20mp3an2i 1303 . . . . . 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 ) ) ) ) )
22 gcddiv 11553 . . . . . . . . . 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
) ) ) )
2311, 16, 8, 6, 22syl31anc 1202 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  ( ( A  /  ( A  gcd  B ) )  gcd  ( B  /  ( A  gcd  B ) ) ) )
248nncnd 8644 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
)  e.  CC )
258nnap0d 8676 . . . . . . . . . 10  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  gcd  B
) #  0 )
2624, 25dividapd 8459 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  gcd  B )  /  ( A  gcd  B ) )  =  1 )
2723, 26eqtr3d 2149 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  / 
( A  gcd  B
) )  gcd  ( B  /  ( A  gcd  B ) ) )  =  1 )
2827breq2d 3907 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( 2  ||  (
( A  /  ( A  gcd  B ) )  gcd  ( B  / 
( A  gcd  B
) ) )  <->  2  ||  1 ) )
2928biimpd 143 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( 2  ||  (
( A  /  ( A  gcd  B ) )  gcd  ( B  / 
( A  gcd  B
) ) )  -> 
2  ||  1 ) )
3021, 29sylbid 149 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( 2  ||  ( A  /  ( A  gcd  B ) )  /\  2  ||  ( B  /  ( A  gcd  B ) ) )  -> 
2  ||  1 ) )
3130expdimp 257 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  2  ||  ( A  /  ( A  gcd  B ) ) )  -> 
( 2  ||  ( B  /  ( A  gcd  B ) )  ->  2  ||  1 ) )
321, 31mtoi 636 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN )  /\  2  ||  ( A  /  ( A  gcd  B ) ) )  ->  -.  2  ||  ( B  /  ( A  gcd  B ) ) )
3332ex 114 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( 2  ||  ( A  /  ( A  gcd  B ) )  ->  -.  2  ||  ( B  / 
( A  gcd  B
) ) ) )
34 2nn 8785 . . . 4  |-  2  e.  NN
35 dvdsdc 11349 . . . 4  |-  ( ( 2  e.  NN  /\  ( A  /  ( A  gcd  B ) )  e.  ZZ )  -> DECID  2  ||  ( A  /  ( A  gcd  B ) ) )
3634, 14, 35sylancr 408 . . 3  |-  ( ( A  e.  NN  /\  B  e.  NN )  -> DECID  2 
||  ( A  / 
( A  gcd  B
) ) )
37 imordc 865 . . 3  |-  (DECID  2  ||  ( A  /  ( A  gcd  B ) )  ->  ( ( 2 
||  ( A  / 
( A  gcd  B
) )  ->  -.  2  ||  ( B  / 
( A  gcd  B
) ) )  <->  ( -.  2  ||  ( A  / 
( A  gcd  B
) )  \/  -.  2  ||  ( B  / 
( A  gcd  B
) ) ) ) )
3836, 37syl 14 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( 2  ||  ( A  /  ( A  gcd  B ) )  ->  -.  2  ||  ( B  /  ( A  gcd  B ) ) )  <->  ( -.  2  ||  ( A  /  ( A  gcd  B ) )  \/  -.  2  ||  ( B  /  ( A  gcd  B ) ) ) ) )
3933, 38mpbid 146 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    /\ wa 103    <-> wb 104    \/ wo 680  DECID wdc 802    = wceq 1314    e. wcel 1463    =/= wne 2282   class class class wbr 3895  (class class class)co 5728   0cc0 7547   1c1 7548    / cdiv 8345   NNcn 8630   2c2 8681   ZZcz 8958    || cdvds 11341    gcd cgcd 11483
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664  ax-caucvg 7665
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-sup 6823  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-2 8689  df-3 8690  df-4 8691  df-n0 8882  df-z 8959  df-uz 9229  df-q 9314  df-rp 9344  df-fz 9684  df-fzo 9813  df-fl 9936  df-mod 9989  df-seqfrec 10112  df-exp 10186  df-cj 10507  df-re 10508  df-im 10509  df-rsqrt 10662  df-abs 10663  df-dvds 11342  df-gcd 11484
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator