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Theorem bezoutr 26483
Description: Partial converse to bezout 12717. 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 12692 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  NN0 )
21nn0zd 10111 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  ZZ )
32adantr 451 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  e.  ZZ )
4 simpll 730 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  ->  A  e.  ZZ )
5 simprl 732 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  ->  X  e.  ZZ )
64, 5zmulcld 10119 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  x.  X
)  e.  ZZ )
7 simplr 731 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  ->  B  e.  ZZ )
8 simprr 733 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  ->  Y  e.  ZZ )
97, 8zmulcld 10119 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( B  x.  Y
)  e.  ZZ )
10 gcddvds 12690 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
1110adantr 451 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
1211simpld 445 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  A )
13 dvdsmultr1 12559 . . . 4  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  A  e.  ZZ  /\  X  e.  ZZ )  ->  (
( A  gcd  B
)  ||  A  ->  ( A  gcd  B ) 
||  ( A  x.  X ) ) )
1413imp 418 . . 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 1185 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  ( A  x.  X ) )
1611simprd 449 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  B )
17 dvdsmultr1 12559 . . . 4  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  B  e.  ZZ  /\  Y  e.  ZZ )  ->  (
( A  gcd  B
)  ||  B  ->  ( A  gcd  B ) 
||  ( B  x.  Y ) ) )
1817imp 418 . . 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 1185 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( A  gcd  B
)  ||  ( B  x.  Y ) )
20 dvds2add 12556 . . 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 418 . 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 1190 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 358    /\ w3a 934    e. wcel 1685   class class class wbr 4024  (class class class)co 5820    + caddc 8736    x. cmul 8738   ZZcz 10020    || cdivides 12527    gcd cgcd 12681
This theorem is referenced by:  bezoutr1  26484
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-sup 7190  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-seq 11043  df-exp 11101  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-dvds 12528  df-gcd 12682
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