MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bezoutlem1 Structured version   Unicode version

Theorem bezoutlem1 13038
Description: Lemma for bezout 13042. (Contributed by Mario Carneiro, 15-Mar-2014.)
Hypotheses
Ref Expression
bezout.1  |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y
) ) }
bezout.3  |-  ( ph  ->  A  e.  ZZ )
bezout.4  |-  ( ph  ->  B  e.  ZZ )
Assertion
Ref Expression
bezoutlem1  |-  ( ph  ->  ( A  =/=  0  ->  ( abs `  A
)  e.  M ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    ph, x, y, z
Allowed substitution hints:    M( x, y, z)

Proof of Theorem bezoutlem1
StepHypRef Expression
1 bezout.3 . . . 4  |-  ( ph  ->  A  e.  ZZ )
2 fveq2 5728 . . . . . . 7  |-  ( z  =  A  ->  ( abs `  z )  =  ( abs `  A
) )
3 oveq1 6088 . . . . . . 7  |-  ( z  =  A  ->  (
z  x.  x )  =  ( A  x.  x ) )
42, 3eqeq12d 2450 . . . . . 6  |-  ( z  =  A  ->  (
( abs `  z
)  =  ( z  x.  x )  <->  ( abs `  A )  =  ( A  x.  x ) ) )
54rexbidv 2726 . . . . 5  |-  ( z  =  A  ->  ( E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x )  <->  E. x  e.  ZZ  ( abs `  A
)  =  ( A  x.  x ) ) )
6 zre 10286 . . . . . 6  |-  ( z  e.  ZZ  ->  z  e.  RR )
7 1z 10311 . . . . . . . . 9  |-  1  e.  ZZ
8 ax-1rid 9060 . . . . . . . . . 10  |-  ( z  e.  RR  ->  (
z  x.  1 )  =  z )
98eqcomd 2441 . . . . . . . . 9  |-  ( z  e.  RR  ->  z  =  ( z  x.  1 ) )
10 oveq2 6089 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
z  x.  x )  =  ( z  x.  1 ) )
1110eqeq2d 2447 . . . . . . . . . 10  |-  ( x  =  1  ->  (
z  =  ( z  x.  x )  <->  z  =  ( z  x.  1 ) ) )
1211rspcev 3052 . . . . . . . . 9  |-  ( ( 1  e.  ZZ  /\  z  =  ( z  x.  1 ) )  ->  E. x  e.  ZZ  z  =  ( z  x.  x ) )
137, 9, 12sylancr 645 . . . . . . . 8  |-  ( z  e.  RR  ->  E. x  e.  ZZ  z  =  ( z  x.  x ) )
14 eqeq1 2442 . . . . . . . . 9  |-  ( ( abs `  z )  =  z  ->  (
( abs `  z
)  =  ( z  x.  x )  <->  z  =  ( z  x.  x
) ) )
1514rexbidv 2726 . . . . . . . 8  |-  ( ( abs `  z )  =  z  ->  ( E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x )  <->  E. x  e.  ZZ  z  =  ( z  x.  x ) ) )
1613, 15syl5ibrcom 214 . . . . . . 7  |-  ( z  e.  RR  ->  (
( abs `  z
)  =  z  ->  E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x ) ) )
17 znegcl 10313 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
187, 17ax-mp 8 . . . . . . . . 9  |-  -u 1  e.  ZZ
19 recn 9080 . . . . . . . . . . 11  |-  ( z  e.  RR  ->  z  e.  CC )
2019mulm1d 9485 . . . . . . . . . 10  |-  ( z  e.  RR  ->  ( -u 1  x.  z )  =  -u z )
21 neg1cn 10067 . . . . . . . . . . 11  |-  -u 1  e.  CC
22 mulcom 9076 . . . . . . . . . . 11  |-  ( (
-u 1  e.  CC  /\  z  e.  CC )  ->  ( -u 1  x.  z )  =  ( z  x.  -u 1
) )
2321, 19, 22sylancr 645 . . . . . . . . . 10  |-  ( z  e.  RR  ->  ( -u 1  x.  z )  =  ( z  x.  -u 1 ) )
2420, 23eqtr3d 2470 . . . . . . . . 9  |-  ( z  e.  RR  ->  -u z  =  ( z  x.  -u 1 ) )
25 oveq2 6089 . . . . . . . . . . 11  |-  ( x  =  -u 1  ->  (
z  x.  x )  =  ( z  x.  -u 1 ) )
2625eqeq2d 2447 . . . . . . . . . 10  |-  ( x  =  -u 1  ->  ( -u z  =  ( z  x.  x )  <->  -u z  =  ( z  x.  -u 1
) ) )
2726rspcev 3052 . . . . . . . . 9  |-  ( (
-u 1  e.  ZZ  /\  -u z  =  (
z  x.  -u 1
) )  ->  E. x  e.  ZZ  -u z  =  ( z  x.  x ) )
2818, 24, 27sylancr 645 . . . . . . . 8  |-  ( z  e.  RR  ->  E. x  e.  ZZ  -u z  =  ( z  x.  x ) )
29 eqeq1 2442 . . . . . . . . 9  |-  ( ( abs `  z )  =  -u z  ->  (
( abs `  z
)  =  ( z  x.  x )  <->  -u z  =  ( z  x.  x
) ) )
3029rexbidv 2726 . . . . . . . 8  |-  ( ( abs `  z )  =  -u z  ->  ( E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x )  <->  E. x  e.  ZZ  -u z  =  ( z  x.  x ) ) )
3128, 30syl5ibrcom 214 . . . . . . 7  |-  ( z  e.  RR  ->  (
( abs `  z
)  =  -u z  ->  E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x ) ) )
32 absor 12105 . . . . . . 7  |-  ( z  e.  RR  ->  (
( abs `  z
)  =  z  \/  ( abs `  z
)  =  -u z
) )
3316, 31, 32mpjaod 371 . . . . . 6  |-  ( z  e.  RR  ->  E. x  e.  ZZ  ( abs `  z
)  =  ( z  x.  x ) )
346, 33syl 16 . . . . 5  |-  ( z  e.  ZZ  ->  E. x  e.  ZZ  ( abs `  z
)  =  ( z  x.  x ) )
355, 34vtoclga 3017 . . . 4  |-  ( A  e.  ZZ  ->  E. x  e.  ZZ  ( abs `  A
)  =  ( A  x.  x ) )
361, 35syl 16 . . 3  |-  ( ph  ->  E. x  e.  ZZ  ( abs `  A )  =  ( A  x.  x ) )
37 bezout.4 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  ZZ )
3837zcnd 10376 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
3938adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ZZ )  ->  B  e.  CC )
4039mul01d 9265 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( B  x.  0 )  =  0 )
4140oveq2d 6097 . . . . . . 7  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( A  x.  x )  +  ( B  x.  0 ) )  =  ( ( A  x.  x )  +  0 ) )
421zcnd 10376 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
43 zcn 10287 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  x  e.  CC )
44 mulcl 9074 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  x.  x
)  e.  CC )
4542, 43, 44syl2an 464 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( A  x.  x )  e.  CC )
4645addid1d 9266 . . . . . . 7  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( A  x.  x )  +  0 )  =  ( A  x.  x
) )
4741, 46eqtrd 2468 . . . . . 6  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( A  x.  x )  +  ( B  x.  0 ) )  =  ( A  x.  x
) )
4847eqeq2d 2447 . . . . 5  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  0 ) )  <->  ( abs `  A )  =  ( A  x.  x ) ) )
49 0z 10293 . . . . . 6  |-  0  e.  ZZ
50 oveq2 6089 . . . . . . . . 9  |-  ( y  =  0  ->  ( B  x.  y )  =  ( B  x.  0 ) )
5150oveq2d 6097 . . . . . . . 8  |-  ( y  =  0  ->  (
( A  x.  x
)  +  ( B  x.  y ) )  =  ( ( A  x.  x )  +  ( B  x.  0 ) ) )
5251eqeq2d 2447 . . . . . . 7  |-  ( y  =  0  ->  (
( abs `  A
)  =  ( ( A  x.  x )  +  ( B  x.  y ) )  <->  ( abs `  A )  =  ( ( A  x.  x
)  +  ( B  x.  0 ) ) ) )
5352rspcev 3052 . . . . . 6  |-  ( ( 0  e.  ZZ  /\  ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  0 ) ) )  ->  E. y  e.  ZZ  ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  y
) ) )
5449, 53mpan 652 . . . . 5  |-  ( ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  0 ) )  ->  E. y  e.  ZZ  ( abs `  A
)  =  ( ( A  x.  x )  +  ( B  x.  y ) ) )
5548, 54syl6bir 221 . . . 4  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( abs `  A )  =  ( A  x.  x )  ->  E. y  e.  ZZ  ( abs `  A
)  =  ( ( A  x.  x )  +  ( B  x.  y ) ) ) )
5655reximdva 2818 . . 3  |-  ( ph  ->  ( E. x  e.  ZZ  ( abs `  A
)  =  ( A  x.  x )  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  y ) ) ) )
5736, 56mpd 15 . 2  |-  ( ph  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  y ) ) )
58 nnabscl 12129 . . . 4  |-  ( ( A  e.  ZZ  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  NN )
5958ex 424 . . 3  |-  ( A  e.  ZZ  ->  ( A  =/=  0  ->  ( abs `  A )  e.  NN ) )
601, 59syl 16 . 2  |-  ( ph  ->  ( A  =/=  0  ->  ( abs `  A
)  e.  NN ) )
61 eqeq1 2442 . . . . 5  |-  ( z  =  ( abs `  A
)  ->  ( z  =  ( ( A  x.  x )  +  ( B  x.  y
) )  <->  ( abs `  A )  =  ( ( A  x.  x
)  +  ( B  x.  y ) ) ) )
62612rexbidv 2748 . . . 4  |-  ( z  =  ( abs `  A
)  ->  ( E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( A  x.  x
)  +  ( B  x.  y ) )  <->  E. x  e.  ZZ  E. y  e.  ZZ  ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  y ) ) ) )
63 bezout.1 . . . 4  |-  M  =  { z  e.  NN  |  E. x  e.  ZZ  E. y  e.  ZZ  z  =  ( ( A  x.  x )  +  ( B  x.  y
) ) }
6462, 63elrab2 3094 . . 3  |-  ( ( abs `  A )  e.  M  <->  ( ( abs `  A )  e.  NN  /\  E. x  e.  ZZ  E. y  e.  ZZ  ( abs `  A
)  =  ( ( A  x.  x )  +  ( B  x.  y ) ) ) )
6564simplbi2com 1383 . 2  |-  ( E. x  e.  ZZ  E. y  e.  ZZ  ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  y ) )  ->  ( ( abs `  A )  e.  NN  ->  ( abs `  A )  e.  M
) )
6657, 60, 65sylsyld 54 1  |-  ( ph  ->  ( A  =/=  0  ->  ( abs `  A
)  e.  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   {crab 2709   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995   -ucneg 9292   NNcn 10000   ZZcz 10282   abscabs 12039
This theorem is referenced by:  bezoutlem2  13039  bezoutlem4  13041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041
  Copyright terms: Public domain W3C validator