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Theorem bezoutr1 11932
Description: Converse of bezout 11910 for when the greater common divisor is one (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 11931 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  ( ( A  x.  X )  +  ( B  x.  Y ) ) )
21adantr 274 . . . . 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 109 . . . . 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 3992 . . . 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 11865 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  NN0 )
65nn0zd 9289 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  ZZ )
76ad2antrr 480 . . . . 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 8849 . . . . . 6  |-  1  e.  NN
98a1i 9 . . . . 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 11748 . . . . 5  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  1  e.  NN )  ->  ( ( A  gcd  B )  ||  1  -> 
( A  gcd  B
)  <_  1 ) )
117, 9, 10syl2anc 409 . . . 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 13 . . 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 519 . . . . 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 5833 . . . . . . . . . . . . 13  |-  ( A  =  0  ->  ( A  x.  X )  =  ( 0  x.  X ) )
15 oveq1 5833 . . . . . . . . . . . . 13  |-  ( B  =  0  ->  ( B  x.  Y )  =  ( 0  x.  Y ) )
1614, 15oveqan12d 5845 . . . . . . . . . . . 12  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A  x.  X )  +  ( B  x.  Y
) )  =  ( ( 0  x.  X
)  +  ( 0  x.  Y ) ) )
17 zcn 9177 . . . . . . . . . . . . . 14  |-  ( X  e.  ZZ  ->  X  e.  CC )
1817mul02d 8271 . . . . . . . . . . . . 13  |-  ( X  e.  ZZ  ->  (
0  x.  X )  =  0 )
19 zcn 9177 . . . . . . . . . . . . . 14  |-  ( Y  e.  ZZ  ->  Y  e.  CC )
2019mul02d 8271 . . . . . . . . . . . . 13  |-  ( Y  e.  ZZ  ->  (
0  x.  Y )  =  0 )
2118, 20oveqan12d 5845 . . . . . . . . . . . 12  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( ( 0  x.  X )  +  ( 0  x.  Y ) )  =  ( 0  +  0 ) )
2216, 21sylan9eqr 2212 . . . . . . . . . . 11  |-  ( ( ( X  e.  ZZ  /\  Y  e.  ZZ )  /\  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A  x.  X )  +  ( B  x.  Y ) )  =  ( 0  +  0 ) )
23 00id 8020 . . . . . . . . . . 11  |-  ( 0  +  0 )  =  0
2422, 23eqtrdi 2206 . . . . . . . . . 10  |-  ( ( ( X  e.  ZZ  /\  Y  e.  ZZ )  /\  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A  x.  X )  +  ( B  x.  Y ) )  =  0 )
2524adantll 468 . . . . . . . . 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 0ne1 8905 . . . . . . . . . 10  |-  0  =/=  1
2726a1i 9 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( A  =  0  /\  B  =  0
) )  ->  0  =/=  1 )
2825, 27eqnetrd 2351 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( A  =  0  /\  B  =  0
) )  ->  (
( A  x.  X
)  +  ( B  x.  Y ) )  =/=  1 )
2928ex 114 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( ( A  =  0  /\  B  =  0 )  ->  (
( A  x.  X
)  +  ( B  x.  Y ) )  =/=  1 ) )
3029necon2bd 2385 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( ( ( A  x.  X )  +  ( B  x.  Y
) )  =  1  ->  -.  ( A  =  0  /\  B  =  0 ) ) )
3130imp 123 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( ( A  x.  X )  +  ( B  x.  Y ) )  =  1 )  ->  -.  ( A  =  0  /\  B  =  0 ) )
32 gcdn0cl 11861 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
3313, 31, 32syl2anc 409 . . . 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 )
34 nnle1eq1 8862 . . . 4  |-  ( ( A  gcd  B )  e.  NN  ->  (
( A  gcd  B
)  <_  1  <->  ( A  gcd  B )  =  1 ) )
3533, 34syl 14 . . 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 ) )
3612, 35mpbid 146 . 2  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( ( A  x.  X )  +  ( B  x.  Y ) )  =  1 )  ->  ( A  gcd  B )  =  1 )
3736ex 114 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    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128    =/= wne 2327   class class class wbr 3967  (class class class)co 5826   0cc0 7734   1c1 7735    + caddc 7737    x. cmul 7739    <_ cle 7915   NNcn 8838   ZZcz 9172    || cdvds 11694    gcd cgcd 11841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-mulrcl 7833  ax-addcom 7834  ax-mulcom 7835  ax-addass 7836  ax-mulass 7837  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-1rid 7841  ax-0id 7842  ax-rnegex 7843  ax-precex 7844  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-apti 7849  ax-pre-ltadd 7850  ax-pre-mulgt0 7851  ax-pre-mulext 7852  ax-arch 7853  ax-caucvg 7854
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-ilim 4331  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-frec 6340  df-sup 6930  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-reap 8454  df-ap 8461  df-div 8550  df-inn 8839  df-2 8897  df-3 8898  df-4 8899  df-n0 9096  df-z 9173  df-uz 9445  df-q 9535  df-rp 9567  df-fz 9919  df-fzo 10051  df-fl 10178  df-mod 10231  df-seqfrec 10354  df-exp 10428  df-cj 10753  df-re 10754  df-im 10755  df-rsqrt 10909  df-abs 10910  df-dvds 11695  df-gcd 11842
This theorem is referenced by:  divgcdcoprm0  11993
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