Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bezoutr1 Unicode version

Theorem bezoutr1 26743
Description: Converse of bezout 12970 for the gcd = 1 case, sufficient condition for relative primality. (Contributed by Stefan O'Rear, 23-Sep-2014.)
Assertion
Ref Expression
bezoutr1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( ( ( A  x.  X )  +  ( B  x.  Y
) )  =  1  ->  ( A  gcd  B )  =  1 ) )

Proof of Theorem bezoutr1
StepHypRef Expression
1 bezoutr 26742 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  ( ( A  x.  X )  +  ( B  x.  Y ) ) )
21adantr 452 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( ( A  x.  X )  +  ( B  x.  Y ) )  =  1 )  ->  ( A  gcd  B )  ||  ( ( A  x.  X )  +  ( B  x.  Y ) ) )
3 simpr 448 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( ( A  x.  X )  +  ( B  x.  Y ) )  =  1 )  ->  ( ( A  x.  X )  +  ( B  x.  Y
) )  =  1 )
42, 3breqtrd 4178 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( ( A  x.  X )  +  ( B  x.  Y ) )  =  1 )  ->  ( A  gcd  B )  ||  1 )
5 gcdcl 12945 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  NN0 )
65nn0zd 10306 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  ZZ )
76ad2antrr 707 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( ( A  x.  X )  +  ( B  x.  Y ) )  =  1 )  ->  ( A  gcd  B )  e.  ZZ )
8 1nn 9944 . . . . . 6  |-  1  e.  NN
98a1i 11 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( ( A  x.  X )  +  ( B  x.  Y ) )  =  1 )  ->  1  e.  NN )
10 dvdsle 12823 . . . . 5  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  1  e.  NN )  ->  ( ( A  gcd  B )  ||  1  -> 
( A  gcd  B
)  <_  1 ) )
117, 9, 10syl2anc 643 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( ( A  x.  X )  +  ( B  x.  Y ) )  =  1 )  ->  ( ( A  gcd  B )  ||  1  ->  ( A  gcd  B )  <_  1 ) )
124, 11mpd 15 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( ( A  x.  X )  +  ( B  x.  Y ) )  =  1 )  ->  ( A  gcd  B )  <_  1 )
13 simpll 731 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( ( A  x.  X )  +  ( B  x.  Y ) )  =  1 )  ->  ( A  e.  ZZ  /\  B  e.  ZZ ) )
14 oveq1 6028 . . . . . . . . . . . . 13  |-  ( A  =  0  ->  ( A  x.  X )  =  ( 0  x.  X ) )
15 oveq1 6028 . . . . . . . . . . . . 13  |-  ( B  =  0  ->  ( B  x.  Y )  =  ( 0  x.  Y ) )
1614, 15oveqan12d 6040 . . . . . . . . . . . 12  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A  x.  X )  +  ( B  x.  Y
) )  =  ( ( 0  x.  X
)  +  ( 0  x.  Y ) ) )
17 zcn 10220 . . . . . . . . . . . . . 14  |-  ( X  e.  ZZ  ->  X  e.  CC )
1817mul02d 9197 . . . . . . . . . . . . 13  |-  ( X  e.  ZZ  ->  (
0  x.  X )  =  0 )
19 zcn 10220 . . . . . . . . . . . . . 14  |-  ( Y  e.  ZZ  ->  Y  e.  CC )
2019mul02d 9197 . . . . . . . . . . . . 13  |-  ( Y  e.  ZZ  ->  (
0  x.  Y )  =  0 )
2118, 20oveqan12d 6040 . . . . . . . . . . . 12  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( ( 0  x.  X )  +  ( 0  x.  Y ) )  =  ( 0  +  0 ) )
2216, 21sylan9eqr 2442 . . . . . . . . . . 11  |-  ( ( ( X  e.  ZZ  /\  Y  e.  ZZ )  /\  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A  x.  X )  +  ( B  x.  Y ) )  =  ( 0  +  0 ) )
23 00id 9174 . . . . . . . . . . 11  |-  ( 0  +  0 )  =  0
2422, 23syl6eq 2436 . . . . . . . . . 10  |-  ( ( ( X  e.  ZZ  /\  Y  e.  ZZ )  /\  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A  x.  X )  +  ( B  x.  Y ) )  =  0 )
2524adantll 695 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( A  =  0  /\  B  =  0
) )  ->  (
( A  x.  X
)  +  ( B  x.  Y ) )  =  0 )
26 ax-1ne0 8993 . . . . . . . . . . 11  |-  1  =/=  0
2726necomi 2633 . . . . . . . . . 10  |-  0  =/=  1
2827a1i 11 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( A  =  0  /\  B  =  0
) )  ->  0  =/=  1 )
2925, 28eqnetrd 2569 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( A  =  0  /\  B  =  0
) )  ->  (
( A  x.  X
)  +  ( B  x.  Y ) )  =/=  1 )
3029ex 424 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( ( A  =  0  /\  B  =  0 )  ->  (
( A  x.  X
)  +  ( B  x.  Y ) )  =/=  1 ) )
3130necon2bd 2600 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( ( ( A  x.  X )  +  ( B  x.  Y
) )  =  1  ->  -.  ( A  =  0  /\  B  =  0 ) ) )
3231imp 419 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( ( A  x.  X )  +  ( B  x.  Y ) )  =  1 )  ->  -.  ( A  =  0  /\  B  =  0 ) )
33 gcdn0cl 12942 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
3413, 32, 33syl2anc 643 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( ( A  x.  X )  +  ( B  x.  Y ) )  =  1 )  ->  ( A  gcd  B )  e.  NN )
35 nnle1eq1 9961 . . . 4  |-  ( ( A  gcd  B )  e.  NN  ->  (
( A  gcd  B
)  <_  1  <->  ( A  gcd  B )  =  1 ) )
3634, 35syl 16 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( ( A  x.  X )  +  ( B  x.  Y ) )  =  1 )  ->  ( ( A  gcd  B )  <_ 
1  <->  ( A  gcd  B )  =  1 ) )
3712, 36mpbid 202 . 2  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( ( A  x.  X )  +  ( B  x.  Y ) )  =  1 )  ->  ( A  gcd  B )  =  1 )
3837ex 424 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( ( ( A  x.  X )  +  ( B  x.  Y
) )  =  1  ->  ( A  gcd  B )  =  1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   class class class wbr 4154  (class class class)co 6021   0cc0 8924   1c1 8925    + caddc 8927    x. cmul 8929    <_ cle 9055   NNcn 9933   ZZcz 10215    || cdivides 12780    gcd cgcd 12934
This theorem is referenced by:  jm2.19lem1  26752
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-dvds 12781  df-gcd 12935
  Copyright terms: Public domain W3C validator