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

Theorem bezoutlemeu 11488
Description: Lemma for Bézout's identity. There is exactly one nonnegative integer meeting the greatest common divisor condition. (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 ) ) )
Assertion
Ref Expression
bezoutlemeu  |-  ( ph  ->  E! d  e.  NN0  A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) ) )
Distinct variable groups:    z, D    A, d, z    B, d, z    ph, d
Allowed substitution hints:    ph( z)    D( d)

Proof of Theorem bezoutlemeu
Dummy variables  e  w  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bezoutlemgcd.1 . . 3  |-  ( ph  ->  A  e.  ZZ )
2 bezoutlemgcd.2 . . 3  |-  ( ph  ->  B  e.  ZZ )
3 bezoutlembi 11486 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  E. d  e.  NN0  ( A. z  e.  ZZ  ( z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) )  /\  E. s  e.  ZZ  E. t  e.  ZZ  d  =  ( ( A  x.  s
)  +  ( B  x.  t ) ) ) )
4 simpl 108 . . . . 5  |-  ( ( A. z  e.  ZZ  ( z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) )  /\  E. s  e.  ZZ  E. t  e.  ZZ  d  =  ( ( A  x.  s
)  +  ( B  x.  t ) ) )  ->  A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) ) )
54reximi 2488 . . . 4  |-  ( E. d  e.  NN0  ( A. z  e.  ZZ  ( z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) )  /\  E. s  e.  ZZ  E. t  e.  ZZ  d  =  ( ( A  x.  s
)  +  ( B  x.  t ) ) )  ->  E. d  e.  NN0  A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) ) )
63, 5syl 14 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  E. d  e.  NN0  A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) ) )
71, 2, 6syl2anc 406 . 2  |-  ( ph  ->  E. d  e.  NN0  A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) ) )
81ad2antrr 475 . . . . . 6  |-  ( ( ( ph  /\  (
d  e.  NN0  /\  e  e.  NN0 ) )  /\  ( A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) )  /\  A. z  e.  ZZ  ( z  ||  e 
<->  ( z  ||  A  /\  z  ||  B ) ) ) )  ->  A  e.  ZZ )
92ad2antrr 475 . . . . . 6  |-  ( ( ( ph  /\  (
d  e.  NN0  /\  e  e.  NN0 ) )  /\  ( A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) )  /\  A. z  e.  ZZ  ( z  ||  e 
<->  ( z  ||  A  /\  z  ||  B ) ) ) )  ->  B  e.  ZZ )
10 simplrl 505 . . . . . 6  |-  ( ( ( ph  /\  (
d  e.  NN0  /\  e  e.  NN0 ) )  /\  ( A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) )  /\  A. z  e.  ZZ  ( z  ||  e 
<->  ( z  ||  A  /\  z  ||  B ) ) ) )  -> 
d  e.  NN0 )
11 simprl 501 . . . . . . 7  |-  ( ( ( ph  /\  (
d  e.  NN0  /\  e  e.  NN0 ) )  /\  ( A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) )  /\  A. z  e.  ZZ  ( z  ||  e 
<->  ( z  ||  A  /\  z  ||  B ) ) ) )  ->  A. z  e.  ZZ  ( z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) ) )
12 breq1 3878 . . . . . . . . 9  |-  ( z  =  w  ->  (
z  ||  d  <->  w  ||  d
) )
13 breq1 3878 . . . . . . . . . 10  |-  ( z  =  w  ->  (
z  ||  A  <->  w  ||  A
) )
14 breq1 3878 . . . . . . . . . 10  |-  ( z  =  w  ->  (
z  ||  B  <->  w  ||  B
) )
1513, 14anbi12d 460 . . . . . . . . 9  |-  ( z  =  w  ->  (
( z  ||  A  /\  z  ||  B )  <-> 
( w  ||  A  /\  w  ||  B ) ) )
1612, 15bibi12d 234 . . . . . . . 8  |-  ( z  =  w  ->  (
( z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) )  <-> 
( w  ||  d  <->  ( w  ||  A  /\  w  ||  B ) ) ) )
1716cbvralv 2612 . . . . . . 7  |-  ( A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) )  <->  A. w  e.  ZZ  ( w  ||  d 
<->  ( w  ||  A  /\  w  ||  B ) ) )
1811, 17sylib 121 . . . . . 6  |-  ( ( ( ph  /\  (
d  e.  NN0  /\  e  e.  NN0 ) )  /\  ( A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) )  /\  A. z  e.  ZZ  ( z  ||  e 
<->  ( z  ||  A  /\  z  ||  B ) ) ) )  ->  A. w  e.  ZZ  ( w  ||  d  <->  ( w  ||  A  /\  w  ||  B ) ) )
19 simplrr 506 . . . . . 6  |-  ( ( ( ph  /\  (
d  e.  NN0  /\  e  e.  NN0 ) )  /\  ( A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) )  /\  A. z  e.  ZZ  ( z  ||  e 
<->  ( z  ||  A  /\  z  ||  B ) ) ) )  -> 
e  e.  NN0 )
20 simprr 502 . . . . . . 7  |-  ( ( ( ph  /\  (
d  e.  NN0  /\  e  e.  NN0 ) )  /\  ( A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) )  /\  A. z  e.  ZZ  ( z  ||  e 
<->  ( z  ||  A  /\  z  ||  B ) ) ) )  ->  A. z  e.  ZZ  ( z  ||  e  <->  ( z  ||  A  /\  z  ||  B ) ) )
21 breq1 3878 . . . . . . . . 9  |-  ( z  =  w  ->  (
z  ||  e  <->  w  ||  e
) )
2221, 15bibi12d 234 . . . . . . . 8  |-  ( z  =  w  ->  (
( z  ||  e  <->  ( z  ||  A  /\  z  ||  B ) )  <-> 
( w  ||  e  <->  ( w  ||  A  /\  w  ||  B ) ) ) )
2322cbvralv 2612 . . . . . . 7  |-  ( A. z  e.  ZZ  (
z  ||  e  <->  ( z  ||  A  /\  z  ||  B ) )  <->  A. w  e.  ZZ  ( w  ||  e 
<->  ( w  ||  A  /\  w  ||  B ) ) )
2420, 23sylib 121 . . . . . 6  |-  ( ( ( ph  /\  (
d  e.  NN0  /\  e  e.  NN0 ) )  /\  ( A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) )  /\  A. z  e.  ZZ  ( z  ||  e 
<->  ( z  ||  A  /\  z  ||  B ) ) ) )  ->  A. w  e.  ZZ  ( w  ||  e  <->  ( w  ||  A  /\  w  ||  B ) ) )
258, 9, 10, 18, 19, 24bezoutlemmo 11487 . . . . 5  |-  ( ( ( ph  /\  (
d  e.  NN0  /\  e  e.  NN0 ) )  /\  ( A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) )  /\  A. z  e.  ZZ  ( z  ||  e 
<->  ( z  ||  A  /\  z  ||  B ) ) ) )  -> 
d  =  e )
2625ex 114 . . . 4  |-  ( (
ph  /\  ( d  e.  NN0  /\  e  e. 
NN0 ) )  -> 
( ( A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) )  /\  A. z  e.  ZZ  ( z  ||  e 
<->  ( z  ||  A  /\  z  ||  B ) ) )  ->  d  =  e ) )
2726ralrimivva 2473 . . 3  |-  ( ph  ->  A. d  e.  NN0  A. e  e.  NN0  (
( A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) )  /\  A. z  e.  ZZ  ( z  ||  e 
<->  ( z  ||  A  /\  z  ||  B ) ) )  ->  d  =  e ) )
28 breq2 3879 . . . . . 6  |-  ( d  =  e  ->  (
z  ||  d  <->  z  ||  e ) )
2928bibi1d 232 . . . . 5  |-  ( d  =  e  ->  (
( z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) )  <-> 
( z  ||  e  <->  ( z  ||  A  /\  z  ||  B ) ) ) )
3029ralbidv 2396 . . . 4  |-  ( d  =  e  ->  ( A. z  e.  ZZ  ( z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) )  <->  A. z  e.  ZZ  ( z  ||  e  <->  ( z  ||  A  /\  z  ||  B ) ) ) )
3130rmo4 2830 . . 3  |-  ( E* d  e.  NN0  A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) )  <->  A. d  e.  NN0  A. e  e.  NN0  (
( A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) )  /\  A. z  e.  ZZ  ( z  ||  e 
<->  ( z  ||  A  /\  z  ||  B ) ) )  ->  d  =  e ) )
3227, 31sylibr 133 . 2  |-  ( ph  ->  E* d  e.  NN0  A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) ) )
33 reu5 2601 . 2  |-  ( E! d  e.  NN0  A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) )  <->  ( E. d  e.  NN0  A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) )  /\  E* d  e.  NN0  A. z  e.  ZZ  ( z  ||  d 
<->  ( z  ||  A  /\  z  ||  B ) ) ) )
347, 32, 33sylanbrc 411 1  |-  ( ph  ->  E! d  e.  NN0  A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1299    e. wcel 1448   A.wral 2375   E.wrex 2376   E!wreu 2377   E*wrmo 2378   class class class wbr 3875  (class class class)co 5706    + caddc 7503    x. cmul 7505   NN0cn0 8829   ZZcz 8906    || cdvds 11288
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 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-q 9262  df-rp 9292  df-fz 9632  df-fl 9884  df-mod 9937  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-dvds 11289
This theorem is referenced by:  dfgcd3  11491  bezout  11492
  Copyright terms: Public domain W3C validator