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Theorem bezoutlemle 11685
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 3927 . . . . . . . 8  |-  ( z  =  w  ->  (
z  ||  D  <->  w  ||  D
) )
3 breq1 3927 . . . . . . . . 9  |-  ( z  =  w  ->  (
z  ||  A  <->  w  ||  A
) )
4 breq1 3927 . . . . . . . . 9  |-  ( z  =  w  ->  (
z  ||  B  <->  w  ||  B
) )
53, 4anbi12d 464 . . . . . . . 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 1682 . . . . . . 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 2652 . . . . . . . 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 479 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  A. w  e.  ZZ  ( w  ||  D  <->  ( w  ||  A  /\  w  ||  B ) ) )
14 simplr 519 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  z  e.  ZZ )
159, 13, 14rspcdva 2789 . . . . 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 479 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  D  e.  NN0 )
19 bezoutlemgcd.5 . . . . . . . . 9  |-  ( ph  ->  -.  ( A  =  0  /\  B  =  0 ) )
2019ad2antrr 479 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  -.  ( A  =  0  /\  B  =  0 ) )
21 breq1 3927 . . . . . . . . . . . 12  |-  ( z  =  0  ->  (
z  ||  D  <->  0  ||  D ) )
22 breq1 3927 . . . . . . . . . . . . 13  |-  ( z  =  0  ->  (
z  ||  A  <->  0  ||  A ) )
23 breq1 3927 . . . . . . . . . . . . 13  |-  ( z  =  0  ->  (
z  ||  B  <->  0  ||  B ) )
2422, 23anbi12d 464 . . . . . . . . . . . 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 9059 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  ZZ )
2725, 10, 26rspcdva 2789 . . . . . . . . . 10  |-  ( ph  ->  ( 0  ||  D  <->  ( 0  ||  A  /\  0  ||  B ) ) )
2827ad2antrr 479 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( 0  ||  D 
<->  ( 0  ||  A  /\  0  ||  B ) ) )
2918nn0zd 9164 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  D  e.  ZZ )
30 0dvds 11502 . . . . . . . . . 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 479 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  A  e.  ZZ )
34 0dvds 11502 . . . . . . . . . . 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 479 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  B  e.  ZZ )
38 0dvds 11502 . . . . . . . . . . 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 464 . . . . . . . . 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 662 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  -.  D  = 
0 )
4342neqned 2313 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  D  =/=  0
)
44 elnnne0 8984 . . . . . 6  |-  ( D  e.  NN  <->  ( D  e.  NN0  /\  D  =/=  0 ) )
4518, 43, 44sylanbrc 413 . . . . 5  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  D  e.  NN )
46 dvdsle 11531 . . . . 5  |-  ( ( z  e.  ZZ  /\  D  e.  NN )  ->  ( z  ||  D  ->  z  <_  D )
)
4714, 45, 46syl2anc 408 . . . 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 2503 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 1331    e. wcel 1480    =/= wne 2306   A.wral 2414   class class class wbr 3924   0cc0 7613    <_ cle 7794   NNcn 8713   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-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:  bezoutlemsup  11686
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