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

Theorem plyremlem 19700
Description: Closure of a linear factor. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypothesis
Ref Expression
plyrem.1  |-  G  =  ( X p  o F  -  ( CC  X.  { A } ) )
Assertion
Ref Expression
plyremlem  |-  ( A  e.  CC  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )

Proof of Theorem plyremlem
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 plyrem.1 . . 3  |-  G  =  ( X p  o F  -  ( CC  X.  { A } ) )
2 ssid 3210 . . . . 5  |-  CC  C_  CC
3 ax-1cn 8811 . . . . 5  |-  1  e.  CC
4 plyid 19607 . . . . 5  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  X p  e.  (Poly `  CC ) )
52, 3, 4mp2an 653 . . . 4  |-  X p  e.  (Poly `  CC )
6 plyconst 19604 . . . . 5  |-  ( ( CC  C_  CC  /\  A  e.  CC )  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
72, 6mpan 651 . . . 4  |-  ( A  e.  CC  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
8 plysubcl 19620 . . . 4  |-  ( ( X p  e.  (Poly `  CC )  /\  ( CC  X.  { A }
)  e.  (Poly `  CC ) )  ->  (
X p  o F  -  ( CC  X.  { A } ) )  e.  (Poly `  CC ) )
95, 7, 8sylancr 644 . . 3  |-  ( A  e.  CC  ->  (
X p  o F  -  ( CC  X.  { A } ) )  e.  (Poly `  CC ) )
101, 9syl5eqel 2380 . 2  |-  ( A  e.  CC  ->  G  e.  (Poly `  CC )
)
11 negcl 9068 . . . . . . . . 9  |-  ( A  e.  CC  ->  -u A  e.  CC )
12 addcom 9014 . . . . . . . . 9  |-  ( (
-u A  e.  CC  /\  z  e.  CC )  ->  ( -u A  +  z )  =  ( z  +  -u A ) )
1311, 12sylan 457 . . . . . . . 8  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( -u A  +  z )  =  ( z  +  -u A
) )
14 negsub 9111 . . . . . . . . 9  |-  ( ( z  e.  CC  /\  A  e.  CC )  ->  ( z  +  -u A )  =  ( z  -  A ) )
1514ancoms 439 . . . . . . . 8  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( z  +  -u A )  =  ( z  -  A ) )
1613, 15eqtrd 2328 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( -u A  +  z )  =  ( z  -  A ) )
1716mpteq2dva 4122 . . . . . 6  |-  ( A  e.  CC  ->  (
z  e.  CC  |->  (
-u A  +  z ) )  =  ( z  e.  CC  |->  ( z  -  A ) ) )
18 cnex 8834 . . . . . . . 8  |-  CC  e.  _V
1918a1i 10 . . . . . . 7  |-  ( A  e.  CC  ->  CC  e.  _V )
20 negex 9066 . . . . . . . 8  |-  -u A  e.  _V
2120a1i 10 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  -> 
-u A  e.  _V )
22 simpr 447 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  z  e.  CC )
23 fconstmpt 4748 . . . . . . . 8  |-  ( CC 
X.  { -u A } )  =  ( z  e.  CC  |->  -u A )
2423a1i 10 . . . . . . 7  |-  ( A  e.  CC  ->  ( CC  X.  { -u A } )  =  ( z  e.  CC  |->  -u A ) )
25 df-idp 19587 . . . . . . . . 9  |-  X p  =  (  _I  |`  CC )
26 mptresid 5020 . . . . . . . . 9  |-  ( z  e.  CC  |->  z )  =  (  _I  |`  CC )
2725, 26eqtr4i 2319 . . . . . . . 8  |-  X p  =  ( z  e.  CC  |->  z )
2827a1i 10 . . . . . . 7  |-  ( A  e.  CC  ->  X p  =  ( z  e.  CC  |->  z ) )
2919, 21, 22, 24, 28offval2 6111 . . . . . 6  |-  ( A  e.  CC  ->  (
( CC  X.  { -u A } )  o F  +  X p )  =  ( z  e.  CC  |->  ( -u A  +  z )
) )
30 simpl 443 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  A  e.  CC )
31 fconstmpt 4748 . . . . . . . 8  |-  ( CC 
X.  { A }
)  =  ( z  e.  CC  |->  A )
3231a1i 10 . . . . . . 7  |-  ( A  e.  CC  ->  ( CC  X.  { A }
)  =  ( z  e.  CC  |->  A ) )
3319, 22, 30, 28, 32offval2 6111 . . . . . 6  |-  ( A  e.  CC  ->  (
X p  o F  -  ( CC  X.  { A } ) )  =  ( z  e.  CC  |->  ( z  -  A ) ) )
3417, 29, 333eqtr4d 2338 . . . . 5  |-  ( A  e.  CC  ->  (
( CC  X.  { -u A } )  o F  +  X p )  =  ( X p  o F  -  ( CC  X.  { A } ) ) )
3534, 1syl6eqr 2346 . . . 4  |-  ( A  e.  CC  ->  (
( CC  X.  { -u A } )  o F  +  X p )  =  G )
3635fveq2d 5545 . . 3  |-  ( A  e.  CC  ->  (deg `  ( ( CC  X.  { -u A } )  o F  +  X p ) )  =  (deg `  G )
)
37 plyconst 19604 . . . . 5  |-  ( ( CC  C_  CC  /\  -u A  e.  CC )  ->  ( CC  X.  { -u A } )  e.  (Poly `  CC ) )
382, 11, 37sylancr 644 . . . 4  |-  ( A  e.  CC  ->  ( CC  X.  { -u A } )  e.  (Poly `  CC ) )
395a1i 10 . . . 4  |-  ( A  e.  CC  ->  X p  e.  (Poly `  CC ) )
40 0dgr 19643 . . . . . 6  |-  ( -u A  e.  CC  ->  (deg
`  ( CC  X.  { -u A } ) )  =  0 )
4111, 40syl 15 . . . . 5  |-  ( A  e.  CC  ->  (deg `  ( CC  X.  { -u A } ) )  =  0 )
42 0lt1 9312 . . . . 5  |-  0  <  1
4341, 42syl6eqbr 4076 . . . 4  |-  ( A  e.  CC  ->  (deg `  ( CC  X.  { -u A } ) )  <  1 )
44 eqid 2296 . . . . 5  |-  (deg `  ( CC  X.  { -u A } ) )  =  (deg `  ( CC  X.  { -u A }
) )
45 dgrid 19661 . . . . . 6  |-  (deg `  X p )  =  1
4645eqcomi 2300 . . . . 5  |-  1  =  (deg `  X p
)
4744, 46dgradd2 19665 . . . 4  |-  ( ( ( CC  X.  { -u A } )  e.  (Poly `  CC )  /\  X p  e.  (Poly `  CC )  /\  (deg `  ( CC  X.  { -u A } ) )  <  1 )  -> 
(deg `  ( ( CC  X.  { -u A } )  o F  +  X p ) )  =  1 )
4838, 39, 43, 47syl3anc 1182 . . 3  |-  ( A  e.  CC  ->  (deg `  ( ( CC  X.  { -u A } )  o F  +  X p ) )  =  1 )
4936, 48eqtr3d 2330 . 2  |-  ( A  e.  CC  ->  (deg `  G )  =  1 )
501, 33syl5eq 2340 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  G  =  ( z  e.  CC  |->  ( z  -  A ) ) )
5150fveq1d 5543 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( G `  z )  =  ( ( z  e.  CC  |->  ( z  -  A ) ) `
 z ) )
5251adantr 451 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( G `  z
)  =  ( ( z  e.  CC  |->  ( z  -  A ) ) `  z ) )
53 ovex 5899 . . . . . . . . . 10  |-  ( z  -  A )  e. 
_V
54 eqid 2296 . . . . . . . . . . 11  |-  ( z  e.  CC  |->  ( z  -  A ) )  =  ( z  e.  CC  |->  ( z  -  A ) )
5554fvmpt2 5624 . . . . . . . . . 10  |-  ( ( z  e.  CC  /\  ( z  -  A
)  e.  _V )  ->  ( ( z  e.  CC  |->  ( z  -  A ) ) `  z )  =  ( z  -  A ) )
5622, 53, 55sylancl 643 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( ( z  e.  CC  |->  ( z  -  A ) ) `  z )  =  ( z  -  A ) )
5752, 56eqtrd 2328 . . . . . . . 8  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( G `  z
)  =  ( z  -  A ) )
5857eqeq1d 2304 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( ( G `  z )  =  0  <-> 
( z  -  A
)  =  0 ) )
59 subeq0 9089 . . . . . . . 8  |-  ( ( z  e.  CC  /\  A  e.  CC )  ->  ( ( z  -  A )  =  0  <-> 
z  =  A ) )
6059ancoms 439 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( ( z  -  A )  =  0  <-> 
z  =  A ) )
6158, 60bitrd 244 . . . . . 6  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( ( G `  z )  =  0  <-> 
z  =  A ) )
6261pm5.32da 622 . . . . 5  |-  ( A  e.  CC  ->  (
( z  e.  CC  /\  ( G `  z
)  =  0 )  <-> 
( z  e.  CC  /\  z  =  A ) ) )
63 plyf 19596 . . . . . . 7  |-  ( G  e.  (Poly `  CC )  ->  G : CC --> CC )
64 ffn 5405 . . . . . . 7  |-  ( G : CC --> CC  ->  G  Fn  CC )
6510, 63, 643syl 18 . . . . . 6  |-  ( A  e.  CC  ->  G  Fn  CC )
66 fniniseg 5662 . . . . . 6  |-  ( G  Fn  CC  ->  (
z  e.  ( `' G " { 0 } )  <->  ( z  e.  CC  /\  ( G `
 z )  =  0 ) ) )
6765, 66syl 15 . . . . 5  |-  ( A  e.  CC  ->  (
z  e.  ( `' G " { 0 } )  <->  ( z  e.  CC  /\  ( G `
 z )  =  0 ) ) )
68 eleq1a 2365 . . . . . 6  |-  ( A  e.  CC  ->  (
z  =  A  -> 
z  e.  CC ) )
6968pm4.71rd 616 . . . . 5  |-  ( A  e.  CC  ->  (
z  =  A  <->  ( z  e.  CC  /\  z  =  A ) ) )
7062, 67, 693bitr4d 276 . . . 4  |-  ( A  e.  CC  ->  (
z  e.  ( `' G " { 0 } )  <->  z  =  A ) )
71 elsn 3668 . . . 4  |-  ( z  e.  { A }  <->  z  =  A )
7270, 71syl6bbr 254 . . 3  |-  ( A  e.  CC  ->  (
z  e.  ( `' G " { 0 } )  <->  z  e.  { A } ) )
7372eqrdv 2294 . 2  |-  ( A  e.  CC  ->  ( `' G " { 0 } )  =  { A } )
7410, 49, 733jca 1132 1  |-  ( A  e.  CC  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   {csn 3653   class class class wbr 4039    e. cmpt 4093    _I cid 4320    X. cxp 4703   `'ccnv 4704    |` cres 4707   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    - cmin 9053   -ucneg 9054  Polycply 19582   X pcidp 19583  degcdgr 19585
This theorem is referenced by:  plyrem  19701  facth  19702  fta1lem  19703  vieta1lem1  19706  vieta1lem2  19707  taylply2  19763  ftalem7  20332
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  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  ax-addf 8832
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-int 3879  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-se 4369  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-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  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-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-idp 19587  df-coe 19588  df-dgr 19589
  Copyright terms: Public domain W3C validator