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Theorem bezoutr 26987
Description: Partial converse to bezout 13030. 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 13005 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  NN0 )
21nn0zd 10362 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  ZZ )
32adantr 452 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  e.  ZZ )
4 simpll 731 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  ->  A  e.  ZZ )
5 simprl 733 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  ->  X  e.  ZZ )
64, 5zmulcld 10370 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  x.  X
)  e.  ZZ )
7 simplr 732 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  ->  B  e.  ZZ )
8 simprr 734 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  ->  Y  e.  ZZ )
97, 8zmulcld 10370 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( B  x.  Y
)  e.  ZZ )
10 gcddvds 13003 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
1110adantr 452 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
1211simpld 446 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  A )
13 dvdsmultr1 12872 . . . 4  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  A  e.  ZZ  /\  X  e.  ZZ )  ->  (
( A  gcd  B
)  ||  A  ->  ( A  gcd  B ) 
||  ( A  x.  X ) ) )
1413imp 419 . . 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 1187 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  ( A  x.  X ) )
1611simprd 450 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  B )
17 dvdsmultr1 12872 . . . 4  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  B  e.  ZZ  /\  Y  e.  ZZ )  ->  (
( A  gcd  B
)  ||  B  ->  ( A  gcd  B ) 
||  ( B  x.  Y ) ) )
1817imp 419 . . 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 1187 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  ( B  x.  Y ) )
20 dvds2add 12869 . . 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 419 . 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 1192 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 359    /\ w3a 936    e. wcel 1725   class class class wbr 4204  (class class class)co 6072    + caddc 8982    x. cmul 8984   ZZcz 10271    || cdivides 12840    gcd cgcd 12994
This theorem is referenced by:  bezoutr1  26988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-sup 7437  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-n0 10211  df-z 10272  df-uz 10478  df-rp 10602  df-seq 11312  df-exp 11371  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-dvds 12841  df-gcd 12995
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