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Theorem bezoutlemsup 11686
Description: Lemma for Bézout's identity. The number satisfying the greatest common divisor condition is the supremum of divisors of both  A and  B. (Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.)
Hypotheses
Ref Expression
bezoutlemgcd.1  |-  ( ph  ->  A  e.  ZZ )
bezoutlemgcd.2  |-  ( ph  ->  B  e.  ZZ )
bezoutlemgcd.3  |-  ( ph  ->  D  e.  NN0 )
bezoutlemgcd.4  |-  ( ph  ->  A. z  e.  ZZ  ( z  ||  D  <->  ( z  ||  A  /\  z  ||  B ) ) )
bezoutlemgcd.5  |-  ( ph  ->  -.  ( A  =  0  /\  B  =  0 ) )
Assertion
Ref Expression
bezoutlemsup  |-  ( ph  ->  D  =  sup ( { z  e.  ZZ  |  ( z  ||  A  /\  z  ||  B
) } ,  RR ,  <  ) )
Distinct variable groups:    z, D    z, A    z, B    ph, z

Proof of Theorem bezoutlemsup
Dummy variables  w  f  g  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bezoutlemgcd.3 . . . 4  |-  ( ph  ->  D  e.  NN0 )
21nn0red 9024 . . 3  |-  ( ph  ->  D  e.  RR )
3 elrabi 2832 . . . . . . 7  |-  ( w  e.  { z  e.  ZZ  |  ( z 
||  A  /\  z  ||  B ) }  ->  w  e.  ZZ )
43adantl 275 . . . . . 6  |-  ( (
ph  /\  w  e.  { z  e.  ZZ  | 
( z  ||  A  /\  z  ||  B ) } )  ->  w  e.  ZZ )
54zred 9166 . . . . 5  |-  ( (
ph  /\  w  e.  { z  e.  ZZ  | 
( z  ||  A  /\  z  ||  B ) } )  ->  w  e.  RR )
62adantr 274 . . . . 5  |-  ( (
ph  /\  w  e.  { z  e.  ZZ  | 
( z  ||  A  /\  z  ||  B ) } )  ->  D  e.  RR )
7 breq1 3927 . . . . . . . . . 10  |-  ( z  =  w  ->  (
z  ||  A  <->  w  ||  A
) )
8 breq1 3927 . . . . . . . . . 10  |-  ( z  =  w  ->  (
z  ||  B  <->  w  ||  B
) )
97, 8anbi12d 464 . . . . . . . . 9  |-  ( z  =  w  ->  (
( z  ||  A  /\  z  ||  B )  <-> 
( w  ||  A  /\  w  ||  B ) ) )
109elrab 2835 . . . . . . . 8  |-  ( w  e.  { z  e.  ZZ  |  ( z 
||  A  /\  z  ||  B ) }  <->  ( w  e.  ZZ  /\  ( w 
||  A  /\  w  ||  B ) ) )
1110simprbi 273 . . . . . . 7  |-  ( w  e.  { z  e.  ZZ  |  ( z 
||  A  /\  z  ||  B ) }  ->  ( w  ||  A  /\  w  ||  B ) )
1211adantl 275 . . . . . 6  |-  ( (
ph  /\  w  e.  { z  e.  ZZ  | 
( z  ||  A  /\  z  ||  B ) } )  ->  (
w  ||  A  /\  w  ||  B ) )
13 breq1 3927 . . . . . . . . 9  |-  ( z  =  w  ->  (
z  <_  D  <->  w  <_  D ) )
149, 13imbi12d 233 . . . . . . . 8  |-  ( z  =  w  ->  (
( ( z  ||  A  /\  z  ||  B
)  ->  z  <_  D )  <->  ( ( w 
||  A  /\  w  ||  B )  ->  w  <_  D ) ) )
15 bezoutlemgcd.1 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ZZ )
16 bezoutlemgcd.2 . . . . . . . . . 10  |-  ( ph  ->  B  e.  ZZ )
17 bezoutlemgcd.4 . . . . . . . . . 10  |-  ( ph  ->  A. z  e.  ZZ  ( z  ||  D  <->  ( z  ||  A  /\  z  ||  B ) ) )
18 bezoutlemgcd.5 . . . . . . . . . 10  |-  ( ph  ->  -.  ( A  =  0  /\  B  =  0 ) )
1915, 16, 1, 17, 18bezoutlemle 11685 . . . . . . . . 9  |-  ( ph  ->  A. z  e.  ZZ  ( ( z  ||  A  /\  z  ||  B
)  ->  z  <_  D ) )
2019adantr 274 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ZZ )  ->  A. z  e.  ZZ  ( ( z 
||  A  /\  z  ||  B )  ->  z  <_  D ) )
21 simpr 109 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ZZ )  ->  w  e.  ZZ )
2214, 20, 21rspcdva 2789 . . . . . . 7  |-  ( (
ph  /\  w  e.  ZZ )  ->  ( ( w  ||  A  /\  w  ||  B )  ->  w  <_  D ) )
233, 22sylan2 284 . . . . . 6  |-  ( (
ph  /\  w  e.  { z  e.  ZZ  | 
( z  ||  A  /\  z  ||  B ) } )  ->  (
( w  ||  A  /\  w  ||  B )  ->  w  <_  D
) )
2412, 23mpd 13 . . . . 5  |-  ( (
ph  /\  w  e.  { z  e.  ZZ  | 
( z  ||  A  /\  z  ||  B ) } )  ->  w  <_  D )
255, 6, 24lensymd 7877 . . . 4  |-  ( (
ph  /\  w  e.  { z  e.  ZZ  | 
( z  ||  A  /\  z  ||  B ) } )  ->  -.  D  <  w )
2625ralrimiva 2503 . . 3  |-  ( ph  ->  A. w  e.  {
z  e.  ZZ  | 
( z  ||  A  /\  z  ||  B ) }  -.  D  < 
w )
271nn0zd 9164 . . . . . . . . . 10  |-  ( ph  ->  D  e.  ZZ )
28 iddvds 11495 . . . . . . . . . 10  |-  ( D  e.  ZZ  ->  D  ||  D )
2927, 28syl 14 . . . . . . . . 9  |-  ( ph  ->  D  ||  D )
30 breq1 3927 . . . . . . . . . . 11  |-  ( z  =  D  ->  (
z  ||  D  <->  D  ||  D
) )
31 breq1 3927 . . . . . . . . . . . 12  |-  ( z  =  D  ->  (
z  ||  A  <->  D  ||  A
) )
32 breq1 3927 . . . . . . . . . . . 12  |-  ( z  =  D  ->  (
z  ||  B  <->  D  ||  B
) )
3331, 32anbi12d 464 . . . . . . . . . . 11  |-  ( z  =  D  ->  (
( z  ||  A  /\  z  ||  B )  <-> 
( D  ||  A  /\  D  ||  B ) ) )
3430, 33bibi12d 234 . . . . . . . . . 10  |-  ( z  =  D  ->  (
( z  ||  D  <->  ( z  ||  A  /\  z  ||  B ) )  <-> 
( D  ||  D  <->  ( D  ||  A  /\  D  ||  B ) ) ) )
3534, 17, 27rspcdva 2789 . . . . . . . . 9  |-  ( ph  ->  ( D  ||  D  <->  ( D  ||  A  /\  D  ||  B ) ) )
3629, 35mpbid 146 . . . . . . . 8  |-  ( ph  ->  ( D  ||  A  /\  D  ||  B ) )
3736ad2antrr 479 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  RR )  /\  w  <  D )  ->  ( D  ||  A  /\  D  ||  B ) )
381ad2antrr 479 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  RR )  /\  w  <  D )  ->  D  e.  NN0 )
3938nn0zd 9164 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  RR )  /\  w  <  D )  ->  D  e.  ZZ )
4033elrab3 2836 . . . . . . . 8  |-  ( D  e.  ZZ  ->  ( D  e.  { z  e.  ZZ  |  ( z 
||  A  /\  z  ||  B ) }  <->  ( D  ||  A  /\  D  ||  B ) ) )
4139, 40syl 14 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  RR )  /\  w  <  D )  ->  ( D  e.  { z  e.  ZZ  |  ( z 
||  A  /\  z  ||  B ) }  <->  ( D  ||  A  /\  D  ||  B ) ) )
4237, 41mpbird 166 . . . . . 6  |-  ( ( ( ph  /\  w  e.  RR )  /\  w  <  D )  ->  D  e.  { z  e.  ZZ  |  ( z  ||  A  /\  z  ||  B
) } )
43 breq2 3928 . . . . . . 7  |-  ( u  =  D  ->  (
w  <  u  <->  w  <  D ) )
4443rspcev 2784 . . . . . 6  |-  ( ( D  e.  { z  e.  ZZ  |  ( z  ||  A  /\  z  ||  B ) }  /\  w  <  D
)  ->  E. u  e.  { z  e.  ZZ  |  ( z  ||  A  /\  z  ||  B
) } w  < 
u )
4542, 44sylancom 416 . . . . 5  |-  ( ( ( ph  /\  w  e.  RR )  /\  w  <  D )  ->  E. u  e.  { z  e.  ZZ  |  ( z  ||  A  /\  z  ||  B
) } w  < 
u )
4645ex 114 . . . 4  |-  ( (
ph  /\  w  e.  RR )  ->  ( w  <  D  ->  E. u  e.  { z  e.  ZZ  |  ( z  ||  A  /\  z  ||  B
) } w  < 
u ) )
4746ralrimiva 2503 . . 3  |-  ( ph  ->  A. w  e.  RR  ( w  <  D  ->  E. u  e.  { z  e.  ZZ  |  ( z  ||  A  /\  z  ||  B ) } w  <  u ) )
48 lttri3 7837 . . . . 5  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
4948adantl 275 . . . 4  |-  ( (
ph  /\  ( f  e.  RR  /\  g  e.  RR ) )  -> 
( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
5049eqsupti 6876 . . 3  |-  ( ph  ->  ( ( D  e.  RR  /\  A. w  e.  { z  e.  ZZ  |  ( z  ||  A  /\  z  ||  B
) }  -.  D  <  w  /\  A. w  e.  RR  ( w  < 
D  ->  E. u  e.  { z  e.  ZZ  |  ( z  ||  A  /\  z  ||  B
) } w  < 
u ) )  ->  sup ( { z  e.  ZZ  |  ( z 
||  A  /\  z  ||  B ) } ,  RR ,  <  )  =  D ) )
512, 26, 47, 50mp3and 1318 . 2  |-  ( ph  ->  sup ( { z  e.  ZZ  |  ( z  ||  A  /\  z  ||  B ) } ,  RR ,  <  )  =  D )
5251eqcomd 2143 1  |-  ( ph  ->  D  =  sup ( { z  e.  ZZ  |  ( z  ||  A  /\  z  ||  B
) } ,  RR ,  <  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   A.wral 2414   E.wrex 2415   {crab 2418   class class class wbr 3924   supcsup 6862   RRcr 7612   0cc0 7613    < clt 7793    <_ cle 7794   NN0cn0 8970   ZZcz 9047    || cdvds 11482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-sup 6864  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-n0 8971  df-z 9048  df-q 9405  df-dvds 11483
This theorem is referenced by:  dfgcd3  11687
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