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Theorem bezoutr1 30755
Description: Converse of bezout 14042 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 30754 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  ( ( A  x.  X )  +  ( B  x.  Y ) ) )
21adantr 465 . . . . 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 461 . . . . 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 4471 . . . 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 14017 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  NN0 )
65nn0zd 10965 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  ZZ )
76ad2antrr 725 . . . . 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 10548 . . . . . 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 13893 . . . . 5  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  1  e.  NN )  ->  ( ( A  gcd  B )  ||  1  -> 
( A  gcd  B
)  <_  1 ) )
117, 9, 10syl2anc 661 . . . 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 753 . . . . 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 6292 . . . . . . . . . . . . 13  |-  ( A  =  0  ->  ( A  x.  X )  =  ( 0  x.  X ) )
15 oveq1 6292 . . . . . . . . . . . . 13  |-  ( B  =  0  ->  ( B  x.  Y )  =  ( 0  x.  Y ) )
1614, 15oveqan12d 6304 . . . . . . . . . . . 12  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A  x.  X )  +  ( B  x.  Y
) )  =  ( ( 0  x.  X
)  +  ( 0  x.  Y ) ) )
17 zcn 10870 . . . . . . . . . . . . . 14  |-  ( X  e.  ZZ  ->  X  e.  CC )
1817mul02d 9778 . . . . . . . . . . . . 13  |-  ( X  e.  ZZ  ->  (
0  x.  X )  =  0 )
19 zcn 10870 . . . . . . . . . . . . . 14  |-  ( Y  e.  ZZ  ->  Y  e.  CC )
2019mul02d 9778 . . . . . . . . . . . . 13  |-  ( Y  e.  ZZ  ->  (
0  x.  Y )  =  0 )
2118, 20oveqan12d 6304 . . . . . . . . . . . 12  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( ( 0  x.  X )  +  ( 0  x.  Y ) )  =  ( 0  +  0 ) )
2216, 21sylan9eqr 2530 . . . . . . . . . . 11  |-  ( ( ( X  e.  ZZ  /\  Y  e.  ZZ )  /\  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A  x.  X )  +  ( B  x.  Y ) )  =  ( 0  +  0 ) )
23 00id 9755 . . . . . . . . . . 11  |-  ( 0  +  0 )  =  0
2422, 23syl6eq 2524 . . . . . . . . . 10  |-  ( ( ( X  e.  ZZ  /\  Y  e.  ZZ )  /\  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A  x.  X )  +  ( B  x.  Y ) )  =  0 )
2524adantll 713 . . . . . . . . 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 10604 . . . . . . . . . 10  |-  0  =/=  1
2726a1i 11 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  /\  ( A  =  0  /\  B  =  0
) )  ->  0  =/=  1 )
2825, 27eqnetrd 2760 . . . . . . . 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 434 . . . . . . 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 2682 . . . . . 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 429 . . . . 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 14014 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
3313, 31, 32syl2anc 661 . . . 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 10565 . . . 4  |-  ( ( A  gcd  B )  e.  NN  ->  (
( A  gcd  B
)  <_  1  <->  ( A  gcd  B )  =  1 ) )
3533, 34syl 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 ) )
3612, 35mpbid 210 . 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 434 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447  (class class class)co 6285   0cc0 9493   1c1 9494    + caddc 9496    x. cmul 9498    <_ cle 9630   NNcn 10537   ZZcz 10865    || cdivides 13850    gcd cgcd 14006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-seq 12077  df-exp 12136  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-dvds 13851  df-gcd 14007
This theorem is referenced by:  jm2.19lem1  30762
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