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Theorem bezoutr1 10816
Description: Converse of bezout 10794 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 10815 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  ( ( A  x.  X )  +  ( B  x.  Y ) ) )
21adantr 270 . . . . 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 108 . . . . 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 3838 . . . 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 10752 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  NN0 )
65nn0zd 8776 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  ZZ )
76ad2antrr 472 . . . . 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 8345 . . . . . 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 10639 . . . . 5  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  1  e.  NN )  ->  ( ( A  gcd  B )  ||  1  -> 
( A  gcd  B
)  <_  1 ) )
117, 9, 10syl2anc 403 . . . 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 496 . . . . 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 5601 . . . . . . . . . . . . 13  |-  ( A  =  0  ->  ( A  x.  X )  =  ( 0  x.  X ) )
15 oveq1 5601 . . . . . . . . . . . . 13  |-  ( B  =  0  ->  ( B  x.  Y )  =  ( 0  x.  Y ) )
1614, 15oveqan12d 5613 . . . . . . . . . . . 12  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A  x.  X )  +  ( B  x.  Y
) )  =  ( ( 0  x.  X
)  +  ( 0  x.  Y ) ) )
17 zcn 8665 . . . . . . . . . . . . . 14  |-  ( X  e.  ZZ  ->  X  e.  CC )
1817mul02d 7791 . . . . . . . . . . . . 13  |-  ( X  e.  ZZ  ->  (
0  x.  X )  =  0 )
19 zcn 8665 . . . . . . . . . . . . . 14  |-  ( Y  e.  ZZ  ->  Y  e.  CC )
2019mul02d 7791 . . . . . . . . . . . . 13  |-  ( Y  e.  ZZ  ->  (
0  x.  Y )  =  0 )
2118, 20oveqan12d 5613 . . . . . . . . . . . 12  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( ( 0  x.  X )  +  ( 0  x.  Y ) )  =  ( 0  +  0 ) )
2216, 21sylan9eqr 2139 . . . . . . . . . . 11  |-  ( ( ( X  e.  ZZ  /\  Y  e.  ZZ )  /\  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A  x.  X )  +  ( B  x.  Y ) )  =  ( 0  +  0 ) )
23 00id 7544 . . . . . . . . . . 11  |-  ( 0  +  0 )  =  0
2422, 23syl6eq 2133 . . . . . . . . . 10  |-  ( ( ( X  e.  ZZ  /\  Y  e.  ZZ )  /\  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A  x.  X )  +  ( B  x.  Y ) )  =  0 )
2524adantll 460 . . . . . . . . 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 8401 . . . . . . . . . 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 2275 . . . . . . . 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 113 . . . . . . 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 2309 . . . . . 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 122 . . . . 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 10748 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
3313, 31, 32syl2anc 403 . . . 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 8358 . . . 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 145 . 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 113 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 102    <-> wb 103    = wceq 1287    e. wcel 1436    =/= wne 2251   class class class wbr 3814  (class class class)co 5594   0cc0 7271   1c1 7272    + caddc 7274    x. cmul 7276    <_ cle 7444   NNcn 8334   ZZcz 8660    || cdvds 10590    gcd cgcd 10732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-mulrcl 7365  ax-addcom 7366  ax-mulcom 7367  ax-addass 7368  ax-mulass 7369  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-1rid 7373  ax-0id 7374  ax-rnegex 7375  ax-precex 7376  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-apti 7381  ax-pre-ltadd 7382  ax-pre-mulgt0 7383  ax-pre-mulext 7384  ax-arch 7385  ax-caucvg 7386
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-if 3377  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-id 4087  df-po 4090  df-iso 4091  df-iord 4160  df-on 4162  df-ilim 4163  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-frec 6091  df-sup 6600  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-reap 7970  df-ap 7977  df-div 8056  df-inn 8335  df-2 8393  df-3 8394  df-4 8395  df-n0 8584  df-z 8661  df-uz 8929  df-q 9014  df-rp 9044  df-fz 9334  df-fzo 9458  df-fl 9580  df-mod 9633  df-iseq 9755  df-iexp 9806  df-cj 10117  df-re 10118  df-im 10119  df-rsqrt 10272  df-abs 10273  df-dvds 10591  df-gcd 10733
This theorem is referenced by:  divgcdcoprm0  10877
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