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Theorem bezoutr 27088
Description: Partial converse to bezout 13073. Existence of a linear combination does not set the GCD, but it does upper bound it. (Contributed by Stefan O'Rear, 23-Sep-2014.)
Assertion
Ref Expression
bezoutr  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  ( ( A  x.  X )  +  ( B  x.  Y ) ) )

Proof of Theorem bezoutr
StepHypRef Expression
1 gcdcl 13048 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  NN0 )
21nn0zd 10404 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  ZZ )
32adantr 453 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  e.  ZZ )
4 simpll 732 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  ->  A  e.  ZZ )
5 simprl 734 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  ->  X  e.  ZZ )
64, 5zmulcld 10412 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  x.  X
)  e.  ZZ )
7 simplr 733 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  ->  B  e.  ZZ )
8 simprr 735 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  ->  Y  e.  ZZ )
97, 8zmulcld 10412 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( B  x.  Y
)  e.  ZZ )
10 gcddvds 13046 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
1110adantr 453 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
1211simpld 447 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  A )
13 dvdsmultr1 12915 . . . 4  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  A  e.  ZZ  /\  X  e.  ZZ )  ->  (
( A  gcd  B
)  ||  A  ->  ( A  gcd  B ) 
||  ( A  x.  X ) ) )
1413imp 420 . . 3  |-  ( ( ( ( A  gcd  B )  e.  ZZ  /\  A  e.  ZZ  /\  X  e.  ZZ )  /\  ( A  gcd  B )  ||  A )  ->  ( A  gcd  B )  ||  ( A  x.  X
) )
153, 4, 5, 12, 14syl31anc 1188 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  ( A  x.  X ) )
1611simprd 451 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  B )
17 dvdsmultr1 12915 . . . 4  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  B  e.  ZZ  /\  Y  e.  ZZ )  ->  (
( A  gcd  B
)  ||  B  ->  ( A  gcd  B ) 
||  ( B  x.  Y ) ) )
1817imp 420 . . 3  |-  ( ( ( ( A  gcd  B )  e.  ZZ  /\  B  e.  ZZ  /\  Y  e.  ZZ )  /\  ( A  gcd  B )  ||  B )  ->  ( A  gcd  B )  ||  ( B  x.  Y
) )
193, 7, 8, 16, 18syl31anc 1188 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  ( B  x.  Y ) )
20 dvds2add 12912 . . 3  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  ( A  x.  X
)  e.  ZZ  /\  ( B  x.  Y
)  e.  ZZ )  ->  ( ( ( A  gcd  B ) 
||  ( A  x.  X )  /\  ( A  gcd  B )  ||  ( B  x.  Y
) )  ->  ( A  gcd  B )  ||  ( ( A  x.  X )  +  ( B  x.  Y ) ) ) )
2120imp 420 . 2  |-  ( ( ( ( A  gcd  B )  e.  ZZ  /\  ( A  x.  X
)  e.  ZZ  /\  ( B  x.  Y
)  e.  ZZ )  /\  ( ( A  gcd  B )  ||  ( A  x.  X
)  /\  ( A  gcd  B )  ||  ( B  x.  Y )
) )  ->  ( A  gcd  B )  ||  ( ( A  x.  X )  +  ( B  x.  Y ) ) )
223, 6, 9, 15, 19, 21syl32anc 1193 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  ( ( A  x.  X )  +  ( B  x.  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    e. wcel 1727   class class class wbr 4237  (class class class)co 6110    + caddc 9024    x. cmul 9026   ZZcz 10313    || cdivides 12883    gcd cgcd 13037
This theorem is referenced by:  bezoutr1  27089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-sup 7475  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-2 10089  df-3 10090  df-n0 10253  df-z 10314  df-uz 10520  df-rp 10644  df-seq 11355  df-exp 11414  df-cj 11935  df-re 11936  df-im 11937  df-sqr 12071  df-abs 12072  df-dvds 12884  df-gcd 13038
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