ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  bezoutlemsup Unicode version

Theorem bezoutlemsup 12002
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 9226 . . 3  |-  ( ph  ->  D  e.  RR )
3 elrabi 2890 . . . . . . 7  |-  ( w  e.  { z  e.  ZZ  |  ( z 
||  A  /\  z  ||  B ) }  ->  w  e.  ZZ )
43adantl 277 . . . . . 6  |-  ( (
ph  /\  w  e.  { z  e.  ZZ  | 
( z  ||  A  /\  z  ||  B ) } )  ->  w  e.  ZZ )
54zred 9371 . . . . 5  |-  ( (
ph  /\  w  e.  { z  e.  ZZ  | 
( z  ||  A  /\  z  ||  B ) } )  ->  w  e.  RR )
62adantr 276 . . . . 5  |-  ( (
ph  /\  w  e.  { z  e.  ZZ  | 
( z  ||  A  /\  z  ||  B ) } )  ->  D  e.  RR )
7 breq1 4005 . . . . . . . . . 10  |-  ( z  =  w  ->  (
z  ||  A  <->  w  ||  A
) )
8 breq1 4005 . . . . . . . . . 10  |-  ( z  =  w  ->  (
z  ||  B  <->  w  ||  B
) )
97, 8anbi12d 473 . . . . . . . . 9  |-  ( z  =  w  ->  (
( z  ||  A  /\  z  ||  B )  <-> 
( w  ||  A  /\  w  ||  B ) ) )
109elrab 2893 . . . . . . . 8  |-  ( w  e.  { z  e.  ZZ  |  ( z 
||  A  /\  z  ||  B ) }  <->  ( w  e.  ZZ  /\  ( w 
||  A  /\  w  ||  B ) ) )
1110simprbi 275 . . . . . . 7  |-  ( w  e.  { z  e.  ZZ  |  ( z 
||  A  /\  z  ||  B ) }  ->  ( w  ||  A  /\  w  ||  B ) )
1211adantl 277 . . . . . 6  |-  ( (
ph  /\  w  e.  { z  e.  ZZ  | 
( z  ||  A  /\  z  ||  B ) } )  ->  (
w  ||  A  /\  w  ||  B ) )
13 breq1 4005 . . . . . . . . 9  |-  ( z  =  w  ->  (
z  <_  D  <->  w  <_  D ) )
149, 13imbi12d 234 . . . . . . . 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 12001 . . . . . . . . 9  |-  ( ph  ->  A. z  e.  ZZ  ( ( z  ||  A  /\  z  ||  B
)  ->  z  <_  D ) )
2019adantr 276 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ZZ )  ->  A. z  e.  ZZ  ( ( z 
||  A  /\  z  ||  B )  ->  z  <_  D ) )
21 simpr 110 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ZZ )  ->  w  e.  ZZ )
2214, 20, 21rspcdva 2846 . . . . . . 7  |-  ( (
ph  /\  w  e.  ZZ )  ->  ( ( w  ||  A  /\  w  ||  B )  ->  w  <_  D ) )
233, 22sylan2 286 . . . . . 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 8075 . . . 4  |-  ( (
ph  /\  w  e.  { z  e.  ZZ  | 
( z  ||  A  /\  z  ||  B ) } )  ->  -.  D  <  w )
2625ralrimiva 2550 . . 3  |-  ( ph  ->  A. w  e.  {
z  e.  ZZ  | 
( z  ||  A  /\  z  ||  B ) }  -.  D  < 
w )
271nn0zd 9369 . . . . . . . . . 10  |-  ( ph  ->  D  e.  ZZ )
28 iddvds 11804 . . . . . . . . . 10  |-  ( D  e.  ZZ  ->  D  ||  D )
2927, 28syl 14 . . . . . . . . 9  |-  ( ph  ->  D  ||  D )
30 breq1 4005 . . . . . . . . . . 11  |-  ( z  =  D  ->  (
z  ||  D  <->  D  ||  D
) )
31 breq1 4005 . . . . . . . . . . . 12  |-  ( z  =  D  ->  (
z  ||  A  <->  D  ||  A
) )
32 breq1 4005 . . . . . . . . . . . 12  |-  ( z  =  D  ->  (
z  ||  B  <->  D  ||  B
) )
3331, 32anbi12d 473 . . . . . . . . . . 11  |-  ( z  =  D  ->  (
( z  ||  A  /\  z  ||  B )  <-> 
( D  ||  A  /\  D  ||  B ) ) )
3430, 33bibi12d 235 . . . . . . . . . 10  |-  ( z  =  D  ->  (
( z  ||  D  <->  ( z  ||  A  /\  z  ||  B ) )  <-> 
( D  ||  D  <->  ( D  ||  A  /\  D  ||  B ) ) ) )
3534, 17, 27rspcdva 2846 . . . . . . . . 9  |-  ( ph  ->  ( D  ||  D  <->  ( D  ||  A  /\  D  ||  B ) ) )
3629, 35mpbid 147 . . . . . . . 8  |-  ( ph  ->  ( D  ||  A  /\  D  ||  B ) )
3736ad2antrr 488 . . . . . . 7  |-  ( ( ( ph  /\  w  e.  RR )  /\  w  <  D )  ->  ( D  ||  A  /\  D  ||  B ) )
381ad2antrr 488 . . . . . . . . 9  |-  ( ( ( ph  /\  w  e.  RR )  /\  w  <  D )  ->  D  e.  NN0 )
3938nn0zd 9369 . . . . . . . 8  |-  ( ( ( ph  /\  w  e.  RR )  /\  w  <  D )  ->  D  e.  ZZ )
4033elrab3 2894 . . . . . . . 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 167 . . . . . 6  |-  ( ( ( ph  /\  w  e.  RR )  /\  w  <  D )  ->  D  e.  { z  e.  ZZ  |  ( z  ||  A  /\  z  ||  B
) } )
43 breq2 4006 . . . . . . 7  |-  ( u  =  D  ->  (
w  <  u  <->  w  <  D ) )
4443rspcev 2841 . . . . . 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 420 . . . . 5  |-  ( ( ( ph  /\  w  e.  RR )  /\  w  <  D )  ->  E. u  e.  { z  e.  ZZ  |  ( z  ||  A  /\  z  ||  B
) } w  < 
u )
4645ex 115 . . . 4  |-  ( (
ph  /\  w  e.  RR )  ->  ( w  <  D  ->  E. u  e.  { z  e.  ZZ  |  ( z  ||  A  /\  z  ||  B
) } w  < 
u ) )
4746ralrimiva 2550 . . 3  |-  ( ph  ->  A. w  e.  RR  ( w  <  D  ->  E. u  e.  { z  e.  ZZ  |  ( z  ||  A  /\  z  ||  B ) } w  <  u ) )
48 lttri3 8033 . . . . 5  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
4948adantl 277 . . . 4  |-  ( (
ph  /\  ( f  e.  RR  /\  g  e.  RR ) )  -> 
( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
5049eqsupti 6992 . . 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 1340 . 2  |-  ( ph  ->  sup ( { z  e.  ZZ  |  ( z  ||  A  /\  z  ||  B ) } ,  RR ,  <  )  =  D )
5251eqcomd 2183 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   class class class wbr 4002   supcsup 6978   RRcr 7807   0cc0 7808    < clt 7988    <_ cle 7989   NN0cn0 9172   ZZcz 9249    || cdvds 11787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-mulrcl 7907  ax-addcom 7908  ax-mulcom 7909  ax-addass 7910  ax-mulass 7911  ax-distr 7912  ax-i2m1 7913  ax-0lt1 7914  ax-1rid 7915  ax-0id 7916  ax-rnegex 7917  ax-precex 7918  ax-cnre 7919  ax-pre-ltirr 7920  ax-pre-ltwlin 7921  ax-pre-lttrn 7922  ax-pre-apti 7923  ax-pre-ltadd 7924  ax-pre-mulgt0 7925  ax-pre-mulext 7926
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-po 4295  df-iso 4296  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-sup 6980  df-pnf 7990  df-mnf 7991  df-xr 7992  df-ltxr 7993  df-le 7994  df-sub 8126  df-neg 8127  df-reap 8528  df-ap 8535  df-div 8626  df-inn 8916  df-n0 9173  df-z 9250  df-q 9616  df-dvds 11788
This theorem is referenced by:  dfgcd3  12003
  Copyright terms: Public domain W3C validator