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Theorem bezoutr1 11449
Description: Converse of bezout 11427 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 11448 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  ( ( A  x.  X )  +  ( B  x.  Y ) ) )
21adantr 271 . . . . 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 3891 . . . 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 11385 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  NN0 )
65nn0zd 8965 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  ZZ )
76ad2antrr 473 . . . . 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 8531 . . . . . 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 11272 . . . . 5  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  1  e.  NN )  ->  ( ( A  gcd  B )  ||  1  -> 
( A  gcd  B
)  <_  1 ) )
117, 9, 10syl2anc 404 . . . 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 497 . . . . 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 5697 . . . . . . . . . . . . 13  |-  ( A  =  0  ->  ( A  x.  X )  =  ( 0  x.  X ) )
15 oveq1 5697 . . . . . . . . . . . . 13  |-  ( B  =  0  ->  ( B  x.  Y )  =  ( 0  x.  Y ) )
1614, 15oveqan12d 5709 . . . . . . . . . . . 12  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A  x.  X )  +  ( B  x.  Y
) )  =  ( ( 0  x.  X
)  +  ( 0  x.  Y ) ) )
17 zcn 8853 . . . . . . . . . . . . . 14  |-  ( X  e.  ZZ  ->  X  e.  CC )
1817mul02d 7967 . . . . . . . . . . . . 13  |-  ( X  e.  ZZ  ->  (
0  x.  X )  =  0 )
19 zcn 8853 . . . . . . . . . . . . . 14  |-  ( Y  e.  ZZ  ->  Y  e.  CC )
2019mul02d 7967 . . . . . . . . . . . . 13  |-  ( Y  e.  ZZ  ->  (
0  x.  Y )  =  0 )
2118, 20oveqan12d 5709 . . . . . . . . . . . 12  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( ( 0  x.  X )  +  ( 0  x.  Y ) )  =  ( 0  +  0 ) )
2216, 21sylan9eqr 2149 . . . . . . . . . . 11  |-  ( ( ( X  e.  ZZ  /\  Y  e.  ZZ )  /\  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A  x.  X )  +  ( B  x.  Y ) )  =  ( 0  +  0 ) )
23 00id 7720 . . . . . . . . . . 11  |-  ( 0  +  0 )  =  0
2422, 23syl6eq 2143 . . . . . . . . . 10  |-  ( ( ( X  e.  ZZ  /\  Y  e.  ZZ )  /\  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A  x.  X )  +  ( B  x.  Y ) )  =  0 )
2524adantll 461 . . . . . . . . 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 8587 . . . . . . . . . 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 2286 . . . . . . . 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 2320 . . . . . 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 11381 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
3313, 31, 32syl2anc 404 . . . 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 8544 . . . 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 1296    e. wcel 1445    =/= wne 2262   class class class wbr 3867  (class class class)co 5690   0cc0 7447   1c1 7448    + caddc 7450    x. cmul 7452    <_ cle 7620   NNcn 8520   ZZcz 8848    || cdvds 11223    gcd cgcd 11365
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 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560  ax-arch 7561  ax-caucvg 7562
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-ilim 4220  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-frec 6194  df-sup 6759  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-2 8579  df-3 8580  df-4 8581  df-n0 8772  df-z 8849  df-uz 9119  df-q 9204  df-rp 9234  df-fz 9574  df-fzo 9703  df-fl 9826  df-mod 9879  df-seqfrec 10001  df-exp 10070  df-cj 10391  df-re 10392  df-im 10393  df-rsqrt 10546  df-abs 10547  df-dvds 11224  df-gcd 11366
This theorem is referenced by:  divgcdcoprm0  11510
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