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

Theorem bezoutlem1 12733
Description: Lemma for bezout 12737. (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 5541 . . . . . . 7  |-  ( z  =  A  ->  ( abs `  z )  =  ( abs `  A
) )
3 oveq1 5881 . . . . . . 7  |-  ( z  =  A  ->  (
z  x.  x )  =  ( A  x.  x ) )
42, 3eqeq12d 2310 . . . . . 6  |-  ( z  =  A  ->  (
( abs `  z
)  =  ( z  x.  x )  <->  ( abs `  A )  =  ( A  x.  x ) ) )
54rexbidv 2577 . . . . 5  |-  ( z  =  A  ->  ( E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x )  <->  E. x  e.  ZZ  ( abs `  A
)  =  ( A  x.  x ) ) )
6 zre 10044 . . . . . 6  |-  ( z  e.  ZZ  ->  z  e.  RR )
7 1z 10069 . . . . . . . . 9  |-  1  e.  ZZ
8 ax-1rid 8823 . . . . . . . . . 10  |-  ( z  e.  RR  ->  (
z  x.  1 )  =  z )
98eqcomd 2301 . . . . . . . . 9  |-  ( z  e.  RR  ->  z  =  ( z  x.  1 ) )
10 oveq2 5882 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
z  x.  x )  =  ( z  x.  1 ) )
1110eqeq2d 2307 . . . . . . . . . 10  |-  ( x  =  1  ->  (
z  =  ( z  x.  x )  <->  z  =  ( z  x.  1 ) ) )
1211rspcev 2897 . . . . . . . . 9  |-  ( ( 1  e.  ZZ  /\  z  =  ( z  x.  1 ) )  ->  E. x  e.  ZZ  z  =  ( z  x.  x ) )
137, 9, 12sylancr 644 . . . . . . . 8  |-  ( z  e.  RR  ->  E. x  e.  ZZ  z  =  ( z  x.  x ) )
14 eqeq1 2302 . . . . . . . . 9  |-  ( ( abs `  z )  =  z  ->  (
( abs `  z
)  =  ( z  x.  x )  <->  z  =  ( z  x.  x
) ) )
1514rexbidv 2577 . . . . . . . 8  |-  ( ( abs `  z )  =  z  ->  ( E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x )  <->  E. x  e.  ZZ  z  =  ( z  x.  x ) ) )
1613, 15syl5ibrcom 213 . . . . . . 7  |-  ( z  e.  RR  ->  (
( abs `  z
)  =  z  ->  E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x ) ) )
17 znegcl 10071 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
187, 17ax-mp 8 . . . . . . . . 9  |-  -u 1  e.  ZZ
19 recn 8843 . . . . . . . . . . 11  |-  ( z  e.  RR  ->  z  e.  CC )
2019mulm1d 9247 . . . . . . . . . 10  |-  ( z  e.  RR  ->  ( -u 1  x.  z )  =  -u z )
21 neg1cn 9829 . . . . . . . . . . 11  |-  -u 1  e.  CC
22 mulcom 8839 . . . . . . . . . . 11  |-  ( (
-u 1  e.  CC  /\  z  e.  CC )  ->  ( -u 1  x.  z )  =  ( z  x.  -u 1
) )
2321, 19, 22sylancr 644 . . . . . . . . . 10  |-  ( z  e.  RR  ->  ( -u 1  x.  z )  =  ( z  x.  -u 1 ) )
2420, 23eqtr3d 2330 . . . . . . . . 9  |-  ( z  e.  RR  ->  -u z  =  ( z  x.  -u 1 ) )
25 oveq2 5882 . . . . . . . . . . 11  |-  ( x  =  -u 1  ->  (
z  x.  x )  =  ( z  x.  -u 1 ) )
2625eqeq2d 2307 . . . . . . . . . 10  |-  ( x  =  -u 1  ->  ( -u z  =  ( z  x.  x )  <->  -u z  =  ( z  x.  -u 1
) ) )
2726rspcev 2897 . . . . . . . . 9  |-  ( (
-u 1  e.  ZZ  /\  -u z  =  (
z  x.  -u 1
) )  ->  E. x  e.  ZZ  -u z  =  ( z  x.  x ) )
2818, 24, 27sylancr 644 . . . . . . . 8  |-  ( z  e.  RR  ->  E. x  e.  ZZ  -u z  =  ( z  x.  x ) )
29 eqeq1 2302 . . . . . . . . 9  |-  ( ( abs `  z )  =  -u z  ->  (
( abs `  z
)  =  ( z  x.  x )  <->  -u z  =  ( z  x.  x
) ) )
3029rexbidv 2577 . . . . . . . 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 213 . . . . . . 7  |-  ( z  e.  RR  ->  (
( abs `  z
)  =  -u z  ->  E. x  e.  ZZ  ( abs `  z )  =  ( z  x.  x ) ) )
32 absor 11801 . . . . . . 7  |-  ( z  e.  RR  ->  (
( abs `  z
)  =  z  \/  ( abs `  z
)  =  -u z
) )
3316, 31, 32mpjaod 370 . . . . . 6  |-  ( z  e.  RR  ->  E. x  e.  ZZ  ( abs `  z
)  =  ( z  x.  x ) )
346, 33syl 15 . . . . 5  |-  ( z  e.  ZZ  ->  E. x  e.  ZZ  ( abs `  z
)  =  ( z  x.  x ) )
355, 34vtoclga 2862 . . . 4  |-  ( A  e.  ZZ  ->  E. x  e.  ZZ  ( abs `  A
)  =  ( A  x.  x ) )
361, 35syl 15 . . 3  |-  ( ph  ->  E. x  e.  ZZ  ( abs `  A )  =  ( A  x.  x ) )
37 bezout.4 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  ZZ )
3837zcnd 10134 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
3938adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ZZ )  ->  B  e.  CC )
4039mul01d 9027 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( B  x.  0 )  =  0 )
4140oveq2d 5890 . . . . . . 7  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( A  x.  x )  +  ( B  x.  0 ) )  =  ( ( A  x.  x )  +  0 ) )
421zcnd 10134 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
43 zcn 10045 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  x  e.  CC )
44 mulcl 8837 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( A  x.  x
)  e.  CC )
4542, 43, 44syl2an 463 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( A  x.  x )  e.  CC )
4645addid1d 9028 . . . . . . 7  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( A  x.  x )  +  0 )  =  ( A  x.  x
) )
4741, 46eqtrd 2328 . . . . . 6  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( A  x.  x )  +  ( B  x.  0 ) )  =  ( A  x.  x
) )
4847eqeq2d 2307 . . . . 5  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  0 ) )  <->  ( abs `  A )  =  ( A  x.  x ) ) )
49 0z 10051 . . . . . 6  |-  0  e.  ZZ
50 oveq2 5882 . . . . . . . . 9  |-  ( y  =  0  ->  ( B  x.  y )  =  ( B  x.  0 ) )
5150oveq2d 5890 . . . . . . . 8  |-  ( y  =  0  ->  (
( A  x.  x
)  +  ( B  x.  y ) )  =  ( ( A  x.  x )  +  ( B  x.  0 ) ) )
5251eqeq2d 2307 . . . . . . 7  |-  ( y  =  0  ->  (
( abs `  A
)  =  ( ( A  x.  x )  +  ( B  x.  y ) )  <->  ( abs `  A )  =  ( ( A  x.  x
)  +  ( B  x.  0 ) ) ) )
5352rspcev 2897 . . . . . 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 651 . . . . 5  |-  ( ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  0 ) )  ->  E. y  e.  ZZ  ( abs `  A
)  =  ( ( A  x.  x )  +  ( B  x.  y ) ) )
5548, 54syl6bir 220 . . . 4  |-  ( (
ph  /\  x  e.  ZZ )  ->  ( ( abs `  A )  =  ( A  x.  x )  ->  E. y  e.  ZZ  ( abs `  A
)  =  ( ( A  x.  x )  +  ( B  x.  y ) ) ) )
5655reximdva 2668 . . 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 14 . 2  |-  ( ph  ->  E. x  e.  ZZ  E. y  e.  ZZ  ( abs `  A )  =  ( ( A  x.  x )  +  ( B  x.  y ) ) )
58 nnabscl 11825 . . . 4  |-  ( ( A  e.  ZZ  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  NN )
5958ex 423 . . 3  |-  ( A  e.  ZZ  ->  ( A  =/=  0  ->  ( abs `  A )  e.  NN ) )
601, 59syl 15 . 2  |-  ( ph  ->  ( A  =/=  0  ->  ( abs `  A
)  e.  NN ) )
61 eqeq1 2302 . . . . 5  |-  ( z  =  ( abs `  A
)  ->  ( z  =  ( ( A  x.  x )  +  ( B  x.  y
) )  <->  ( abs `  A )  =  ( ( A  x.  x
)  +  ( B  x.  y ) ) ) )
62612rexbidv 2599 . . . 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 2938 . . 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 1364 . 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 52 1  |-  ( ph  ->  ( A  =/=  0  ->  ( abs `  A
)  e.  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   -ucneg 9054   NNcn 9762   ZZcz 10040   abscabs 11735
This theorem is referenced by:  bezoutlem2  12734  bezoutlem4  12736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737
  Copyright terms: Public domain W3C validator