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Theorem bezoutlemle 11336
Description: Lemma for Bézout's identity. The number satisfying the greatest common divisor condition is the largest number which divides 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
bezoutlemle  |-  ( ph  ->  A. z  e.  ZZ  ( ( z  ||  A  /\  z  ||  B
)  ->  z  <_  D ) )
Distinct variable groups:    z, D    z, A    z, B    ph, z

Proof of Theorem bezoutlemle
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . . 5  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( z  ||  A  /\  z  ||  B
) )
2 breq1 3854 . . . . . . . 8  |-  ( z  =  w  ->  (
z  ||  D  <->  w  ||  D
) )
3 breq1 3854 . . . . . . . . 9  |-  ( z  =  w  ->  (
z  ||  A  <->  w  ||  A
) )
4 breq1 3854 . . . . . . . . 9  |-  ( z  =  w  ->  (
z  ||  B  <->  w  ||  B
) )
53, 4anbi12d 458 . . . . . . . 8  |-  ( z  =  w  ->  (
( z  ||  A  /\  z  ||  B )  <-> 
( w  ||  A  /\  w  ||  B ) ) )
62, 5bibi12d 234 . . . . . . 7  |-  ( z  =  w  ->  (
( z  ||  D  <->  ( z  ||  A  /\  z  ||  B ) )  <-> 
( w  ||  D  <->  ( w  ||  A  /\  w  ||  B ) ) ) )
7 equcom 1640 . . . . . . 7  |-  ( z  =  w  <->  w  =  z )
8 bicom 139 . . . . . . 7  |-  ( ( ( z  ||  D  <->  ( z  ||  A  /\  z  ||  B ) )  <-> 
( w  ||  D  <->  ( w  ||  A  /\  w  ||  B ) ) )  <->  ( ( w 
||  D  <->  ( w  ||  A  /\  w  ||  B ) )  <->  ( z  ||  D  <->  ( z  ||  A  /\  z  ||  B
) ) ) )
96, 7, 83imtr3i 199 . . . . . 6  |-  ( w  =  z  ->  (
( w  ||  D  <->  ( w  ||  A  /\  w  ||  B ) )  <-> 
( z  ||  D  <->  ( z  ||  A  /\  z  ||  B ) ) ) )
10 bezoutlemgcd.4 . . . . . . . 8  |-  ( ph  ->  A. z  e.  ZZ  ( z  ||  D  <->  ( z  ||  A  /\  z  ||  B ) ) )
116cbvralv 2591 . . . . . . . 8  |-  ( A. z  e.  ZZ  (
z  ||  D  <->  ( z  ||  A  /\  z  ||  B ) )  <->  A. w  e.  ZZ  ( w  ||  D 
<->  ( w  ||  A  /\  w  ||  B ) ) )
1210, 11sylib 121 . . . . . . 7  |-  ( ph  ->  A. w  e.  ZZ  ( w  ||  D  <->  ( w  ||  A  /\  w  ||  B ) ) )
1312ad2antrr 473 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  A. w  e.  ZZ  ( w  ||  D  <->  ( w  ||  A  /\  w  ||  B ) ) )
14 simplr 498 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  z  e.  ZZ )
159, 13, 14rspcdva 2728 . . . . 5  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( z  ||  D 
<->  ( z  ||  A  /\  z  ||  B ) ) )
161, 15mpbird 166 . . . 4  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  z  ||  D
)
17 bezoutlemgcd.3 . . . . . . 7  |-  ( ph  ->  D  e.  NN0 )
1817ad2antrr 473 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  D  e.  NN0 )
19 bezoutlemgcd.5 . . . . . . . . 9  |-  ( ph  ->  -.  ( A  =  0  /\  B  =  0 ) )
2019ad2antrr 473 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  -.  ( A  =  0  /\  B  =  0 ) )
21 breq1 3854 . . . . . . . . . . . 12  |-  ( z  =  0  ->  (
z  ||  D  <->  0  ||  D ) )
22 breq1 3854 . . . . . . . . . . . . 13  |-  ( z  =  0  ->  (
z  ||  A  <->  0  ||  A ) )
23 breq1 3854 . . . . . . . . . . . . 13  |-  ( z  =  0  ->  (
z  ||  B  <->  0  ||  B ) )
2422, 23anbi12d 458 . . . . . . . . . . . 12  |-  ( z  =  0  ->  (
( z  ||  A  /\  z  ||  B )  <-> 
( 0  ||  A  /\  0  ||  B ) ) )
2521, 24bibi12d 234 . . . . . . . . . . 11  |-  ( z  =  0  ->  (
( z  ||  D  <->  ( z  ||  A  /\  z  ||  B ) )  <-> 
( 0  ||  D  <->  ( 0  ||  A  /\  0  ||  B ) ) ) )
26 0zd 8823 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  ZZ )
2725, 10, 26rspcdva 2728 . . . . . . . . . 10  |-  ( ph  ->  ( 0  ||  D  <->  ( 0  ||  A  /\  0  ||  B ) ) )
2827ad2antrr 473 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( 0  ||  D 
<->  ( 0  ||  A  /\  0  ||  B ) ) )
2918nn0zd 8927 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  D  e.  ZZ )
30 0dvds 11155 . . . . . . . . . 10  |-  ( D  e.  ZZ  ->  (
0  ||  D  <->  D  = 
0 ) )
3129, 30syl 14 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( 0  ||  D 
<->  D  =  0 ) )
32 bezoutlemgcd.1 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ZZ )
3332ad2antrr 473 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  A  e.  ZZ )
34 0dvds 11155 . . . . . . . . . . 11  |-  ( A  e.  ZZ  ->  (
0  ||  A  <->  A  = 
0 ) )
3533, 34syl 14 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( 0  ||  A 
<->  A  =  0 ) )
36 bezoutlemgcd.2 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  ZZ )
3736ad2antrr 473 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  B  e.  ZZ )
38 0dvds 11155 . . . . . . . . . . 11  |-  ( B  e.  ZZ  ->  (
0  ||  B  <->  B  = 
0 ) )
3937, 38syl 14 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( 0  ||  B 
<->  B  =  0 ) )
4035, 39anbi12d 458 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( ( 0 
||  A  /\  0  ||  B )  <->  ( A  =  0  /\  B  =  0 ) ) )
4128, 31, 403bitr3d 217 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( D  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )
4220, 41mtbird 634 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  -.  D  = 
0 )
4342neqned 2263 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  D  =/=  0
)
44 elnnne0 8748 . . . . . 6  |-  ( D  e.  NN  <->  ( D  e.  NN0  /\  D  =/=  0 ) )
4518, 43, 44sylanbrc 409 . . . . 5  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  D  e.  NN )
46 dvdsle 11184 . . . . 5  |-  ( ( z  e.  ZZ  /\  D  e.  NN )  ->  ( z  ||  D  ->  z  <_  D )
)
4714, 45, 46syl2anc 404 . . . 4  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( z  ||  D  ->  z  <_  D
) )
4816, 47mpd 13 . . 3  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  z  <_  D
)
4948ex 114 . 2  |-  ( (
ph  /\  z  e.  ZZ )  ->  ( ( z  ||  A  /\  z  ||  B )  -> 
z  <_  D )
)
5049ralrimiva 2447 1  |-  ( ph  ->  A. z  e.  ZZ  ( ( z  ||  A  /\  z  ||  B
)  ->  z  <_  D ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439    =/= wne 2256   A.wral 2360   class class class wbr 3851   0cc0 7411    <_ cle 7584   NNcn 8483   NN0cn0 8734   ZZcz 8811    || cdvds 11135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524
This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-po 4132  df-iso 4133  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-inn 8484  df-n0 8735  df-z 8812  df-q 9166  df-dvds 11136
This theorem is referenced by:  bezoutlemsup  11337
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