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

Theorem bezoutlemeu 11607
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 11605 . . . 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 2506 . . . 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 408 . 2  |-  ( ph  ->  E. d  e.  NN0  A. z  e.  ZZ  (
z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) ) )
81ad2antrr 479 . . . . . 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 479 . . . . . 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 509 . . . . . 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 505 . . . . . . 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 3902 . . . . . . . . 9  |-  ( z  =  w  ->  (
z  ||  d  <->  w  ||  d
) )
13 breq1 3902 . . . . . . . . . 10  |-  ( z  =  w  ->  (
z  ||  A  <->  w  ||  A
) )
14 breq1 3902 . . . . . . . . . 10  |-  ( z  =  w  ->  (
z  ||  B  <->  w  ||  B
) )
1513, 14anbi12d 464 . . . . . . . . 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 2631 . . . . . . 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 510 . . . . . 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 506 . . . . . . 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 3902 . . . . . . . . 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 2631 . . . . . . 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 11606 . . . . 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 2491 . . 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 3903 . . . . . 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 2414 . . . 4  |-  ( d  =  e  ->  ( A. z  e.  ZZ  ( z  ||  d  <->  ( z  ||  A  /\  z  ||  B ) )  <->  A. z  e.  ZZ  ( z  ||  e  <->  ( z  ||  A  /\  z  ||  B ) ) ) )
3130rmo4 2850 . . 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 2620 . 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 413 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 1316    e. wcel 1465   A.wral 2393   E.wrex 2394   E!wreu 2395   E*wrmo 2396   class class class wbr 3899  (class class class)co 5742    + caddc 7591    x. cmul 7593   NN0cn0 8935   ZZcz 9012    || cdvds 11405
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311  df-div 8400  df-inn 8685  df-2 8743  df-3 8744  df-4 8745  df-n0 8936  df-z 9013  df-uz 9283  df-q 9368  df-rp 9398  df-fz 9746  df-fl 9998  df-mod 10051  df-seqfrec 10174  df-exp 10248  df-cj 10569  df-re 10570  df-im 10571  df-rsqrt 10725  df-abs 10726  df-dvds 11406
This theorem is referenced by:  dfgcd3  11610  bezout  11611
  Copyright terms: Public domain W3C validator