Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bezoutr Unicode version

Theorem bezoutr 26940
Description: Partial converse to bezout 12997. 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 12972 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  e.  NN0 )
21nn0zd 10329 . . 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 10337 . 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 10337 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( B  x.  Y
)  e.  ZZ )
10 gcddvds 12970 . . . . 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 12839 . . . 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 12839 . . . 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 12836 . . 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 1721   class class class wbr 4172  (class class class)co 6040    + caddc 8949    x. cmul 8951   ZZcz 10238    || cdivides 12807    gcd cgcd 12961
This theorem is referenced by:  bezoutr1  26941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962
  Copyright terms: Public domain W3C validator