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Theorem bezoutlemle 10604
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 108 . . . . 5  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( z  ||  A  /\  z  ||  B
) )
2 breq1 3808 . . . . . . . 8  |-  ( z  =  w  ->  (
z  ||  D  <->  w  ||  D
) )
3 breq1 3808 . . . . . . . . 9  |-  ( z  =  w  ->  (
z  ||  A  <->  w  ||  A
) )
4 breq1 3808 . . . . . . . . 9  |-  ( z  =  w  ->  (
z  ||  B  <->  w  ||  B
) )
53, 4anbi12d 457 . . . . . . . 8  |-  ( z  =  w  ->  (
( z  ||  A  /\  z  ||  B )  <-> 
( w  ||  A  /\  w  ||  B ) ) )
62, 5bibi12d 233 . . . . . . 7  |-  ( z  =  w  ->  (
( z  ||  D  <->  ( z  ||  A  /\  z  ||  B ) )  <-> 
( w  ||  D  <->  ( w  ||  A  /\  w  ||  B ) ) ) )
7 equcom 1635 . . . . . . 7  |-  ( z  =  w  <->  w  =  z )
8 bicom 138 . . . . . . 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 198 . . . . . 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 2582 . . . . . . . 8  |-  ( A. z  e.  ZZ  (
z  ||  D  <->  ( z  ||  A  /\  z  ||  B ) )  <->  A. w  e.  ZZ  ( w  ||  D 
<->  ( w  ||  A  /\  w  ||  B ) ) )
1210, 11sylib 120 . . . . . . 7  |-  ( ph  ->  A. w  e.  ZZ  ( w  ||  D  <->  ( w  ||  A  /\  w  ||  B ) ) )
1312ad2antrr 472 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  A. w  e.  ZZ  ( w  ||  D  <->  ( w  ||  A  /\  w  ||  B ) ) )
14 simplr 497 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  z  e.  ZZ )
159, 13, 14rspcdva 2715 . . . . 5  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( z  ||  D 
<->  ( z  ||  A  /\  z  ||  B ) ) )
161, 15mpbird 165 . . . 4  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  z  ||  D
)
17 bezoutlemgcd.3 . . . . . . 7  |-  ( ph  ->  D  e.  NN0 )
1817ad2antrr 472 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  D  e.  NN0 )
19 bezoutlemgcd.5 . . . . . . . . 9  |-  ( ph  ->  -.  ( A  =  0  /\  B  =  0 ) )
2019ad2antrr 472 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  -.  ( A  =  0  /\  B  =  0 ) )
21 breq1 3808 . . . . . . . . . . . 12  |-  ( z  =  0  ->  (
z  ||  D  <->  0  ||  D ) )
22 breq1 3808 . . . . . . . . . . . . 13  |-  ( z  =  0  ->  (
z  ||  A  <->  0  ||  A ) )
23 breq1 3808 . . . . . . . . . . . . 13  |-  ( z  =  0  ->  (
z  ||  B  <->  0  ||  B ) )
2422, 23anbi12d 457 . . . . . . . . . . . 12  |-  ( z  =  0  ->  (
( z  ||  A  /\  z  ||  B )  <-> 
( 0  ||  A  /\  0  ||  B ) ) )
2521, 24bibi12d 233 . . . . . . . . . . 11  |-  ( z  =  0  ->  (
( z  ||  D  <->  ( z  ||  A  /\  z  ||  B ) )  <-> 
( 0  ||  D  <->  ( 0  ||  A  /\  0  ||  B ) ) ) )
26 0zd 8496 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  ZZ )
2725, 10, 26rspcdva 2715 . . . . . . . . . 10  |-  ( ph  ->  ( 0  ||  D  <->  ( 0  ||  A  /\  0  ||  B ) ) )
2827ad2antrr 472 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( 0  ||  D 
<->  ( 0  ||  A  /\  0  ||  B ) ) )
2918nn0zd 8600 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  D  e.  ZZ )
30 0dvds 10423 . . . . . . . . . 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 472 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  A  e.  ZZ )
34 0dvds 10423 . . . . . . . . . . 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 472 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  B  e.  ZZ )
38 0dvds 10423 . . . . . . . . . . 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 457 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( ( 0 
||  A  /\  0  ||  B )  <->  ( A  =  0  /\  B  =  0 ) ) )
4128, 31, 403bitr3d 216 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( D  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )
4220, 41mtbird 631 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  -.  D  = 
0 )
4342neqned 2256 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  D  =/=  0
)
44 elnnne0 8421 . . . . . 6  |-  ( D  e.  NN  <->  ( D  e.  NN0  /\  D  =/=  0 ) )
4518, 43, 44sylanbrc 408 . . . . 5  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  D  e.  NN )
46 dvdsle 10452 . . . . 5  |-  ( ( z  e.  ZZ  /\  D  e.  NN )  ->  ( z  ||  D  ->  z  <_  D )
)
4714, 45, 46syl2anc 403 . . . 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 113 . 2  |-  ( (
ph  /\  z  e.  ZZ )  ->  ( ( z  ||  A  /\  z  ||  B )  -> 
z  <_  D )
)
5049ralrimiva 2439 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 102    <-> wb 103    = wceq 1285    e. wcel 1434    =/= wne 2249   A.wral 2353   class class class wbr 3805   0cc0 7095    <_ cle 7268   NNcn 8158   NN0cn0 8407   ZZcz 8484    || cdvds 10403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-po 4079  df-iso 4080  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-n0 8408  df-z 8485  df-q 8838  df-dvds 10404
This theorem is referenced by:  bezoutlemsup  10605
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