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Theorem bezoutlemle 12023
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 110 . . . . 5  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( z  ||  A  /\  z  ||  B
) )
2 breq1 4018 . . . . . . . 8  |-  ( z  =  w  ->  (
z  ||  D  <->  w  ||  D
) )
3 breq1 4018 . . . . . . . . 9  |-  ( z  =  w  ->  (
z  ||  A  <->  w  ||  A
) )
4 breq1 4018 . . . . . . . . 9  |-  ( z  =  w  ->  (
z  ||  B  <->  w  ||  B
) )
53, 4anbi12d 473 . . . . . . . 8  |-  ( z  =  w  ->  (
( z  ||  A  /\  z  ||  B )  <-> 
( w  ||  A  /\  w  ||  B ) ) )
62, 5bibi12d 235 . . . . . . 7  |-  ( z  =  w  ->  (
( z  ||  D  <->  ( z  ||  A  /\  z  ||  B ) )  <-> 
( w  ||  D  <->  ( w  ||  A  /\  w  ||  B ) ) ) )
7 equcom 1716 . . . . . . 7  |-  ( z  =  w  <->  w  =  z )
8 bicom 140 . . . . . . 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 200 . . . . . 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 2715 . . . . . . . 8  |-  ( A. z  e.  ZZ  (
z  ||  D  <->  ( z  ||  A  /\  z  ||  B ) )  <->  A. w  e.  ZZ  ( w  ||  D 
<->  ( w  ||  A  /\  w  ||  B ) ) )
1210, 11sylib 122 . . . . . . 7  |-  ( ph  ->  A. w  e.  ZZ  ( w  ||  D  <->  ( w  ||  A  /\  w  ||  B ) ) )
1312ad2antrr 488 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  A. w  e.  ZZ  ( w  ||  D  <->  ( w  ||  A  /\  w  ||  B ) ) )
14 simplr 528 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  z  e.  ZZ )
159, 13, 14rspcdva 2858 . . . . 5  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( z  ||  D 
<->  ( z  ||  A  /\  z  ||  B ) ) )
161, 15mpbird 167 . . . 4  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  z  ||  D
)
17 bezoutlemgcd.3 . . . . . . 7  |-  ( ph  ->  D  e.  NN0 )
1817ad2antrr 488 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  D  e.  NN0 )
19 bezoutlemgcd.5 . . . . . . . . 9  |-  ( ph  ->  -.  ( A  =  0  /\  B  =  0 ) )
2019ad2antrr 488 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  -.  ( A  =  0  /\  B  =  0 ) )
21 breq1 4018 . . . . . . . . . . . 12  |-  ( z  =  0  ->  (
z  ||  D  <->  0  ||  D ) )
22 breq1 4018 . . . . . . . . . . . . 13  |-  ( z  =  0  ->  (
z  ||  A  <->  0  ||  A ) )
23 breq1 4018 . . . . . . . . . . . . 13  |-  ( z  =  0  ->  (
z  ||  B  <->  0  ||  B ) )
2422, 23anbi12d 473 . . . . . . . . . . . 12  |-  ( z  =  0  ->  (
( z  ||  A  /\  z  ||  B )  <-> 
( 0  ||  A  /\  0  ||  B ) ) )
2521, 24bibi12d 235 . . . . . . . . . . 11  |-  ( z  =  0  ->  (
( z  ||  D  <->  ( z  ||  A  /\  z  ||  B ) )  <-> 
( 0  ||  D  <->  ( 0  ||  A  /\  0  ||  B ) ) ) )
26 0zd 9279 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  ZZ )
2725, 10, 26rspcdva 2858 . . . . . . . . . 10  |-  ( ph  ->  ( 0  ||  D  <->  ( 0  ||  A  /\  0  ||  B ) ) )
2827ad2antrr 488 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( 0  ||  D 
<->  ( 0  ||  A  /\  0  ||  B ) ) )
2918nn0zd 9387 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  D  e.  ZZ )
30 0dvds 11832 . . . . . . . . . 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 488 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  A  e.  ZZ )
34 0dvds 11832 . . . . . . . . . . 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 488 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  B  e.  ZZ )
38 0dvds 11832 . . . . . . . . . . 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 473 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( ( 0 
||  A  /\  0  ||  B )  <->  ( A  =  0  /\  B  =  0 ) ) )
4128, 31, 403bitr3d 218 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  ( D  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )
4220, 41mtbird 674 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  -.  D  = 
0 )
4342neqned 2364 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  D  =/=  0
)
44 elnnne0 9204 . . . . . 6  |-  ( D  e.  NN  <->  ( D  e.  NN0  /\  D  =/=  0 ) )
4518, 43, 44sylanbrc 417 . . . . 5  |-  ( ( ( ph  /\  z  e.  ZZ )  /\  (
z  ||  A  /\  z  ||  B ) )  ->  D  e.  NN )
46 dvdsle 11864 . . . . 5  |-  ( ( z  e.  ZZ  /\  D  e.  NN )  ->  ( z  ||  D  ->  z  <_  D )
)
4714, 45, 46syl2anc 411 . . . 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 115 . 2  |-  ( (
ph  /\  z  e.  ZZ )  ->  ( ( z  ||  A  /\  z  ||  B )  -> 
z  <_  D )
)
5049ralrimiva 2560 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 104    <-> wb 105    = wceq 1363    e. wcel 2158    =/= wne 2357   A.wral 2465   class class class wbr 4015   0cc0 7825    <_ cle 8007   NNcn 8933   NN0cn0 9190   ZZcz 9267    || cdvds 11808
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-n0 9191  df-z 9268  df-q 9634  df-dvds 11809
This theorem is referenced by:  bezoutlemsup  12024
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