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Theorem plyremlem 20174
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 3327 . . . . 5  |-  CC  C_  CC
3 ax-1cn 9004 . . . . 5  |-  1  e.  CC
4 plyid 20081 . . . . 5  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  X p  e.  (Poly `  CC ) )
52, 3, 4mp2an 654 . . . 4  |-  X p  e.  (Poly `  CC )
6 plyconst 20078 . . . . 5  |-  ( ( CC  C_  CC  /\  A  e.  CC )  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
72, 6mpan 652 . . . 4  |-  ( A  e.  CC  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
8 plysubcl 20094 . . . 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 645 . . 3  |-  ( A  e.  CC  ->  (
X p  o F  -  ( CC  X.  { A } ) )  e.  (Poly `  CC ) )
101, 9syl5eqel 2488 . 2  |-  ( A  e.  CC  ->  G  e.  (Poly `  CC )
)
11 negcl 9262 . . . . . . . . 9  |-  ( A  e.  CC  ->  -u A  e.  CC )
12 addcom 9208 . . . . . . . . 9  |-  ( (
-u A  e.  CC  /\  z  e.  CC )  ->  ( -u A  +  z )  =  ( z  +  -u A ) )
1311, 12sylan 458 . . . . . . . 8  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( -u A  +  z )  =  ( z  +  -u A
) )
14 negsub 9305 . . . . . . . . 9  |-  ( ( z  e.  CC  /\  A  e.  CC )  ->  ( z  +  -u A )  =  ( z  -  A ) )
1514ancoms 440 . . . . . . . 8  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( z  +  -u A )  =  ( z  -  A ) )
1613, 15eqtrd 2436 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( -u A  +  z )  =  ( z  -  A ) )
1716mpteq2dva 4255 . . . . . 6  |-  ( A  e.  CC  ->  (
z  e.  CC  |->  (
-u A  +  z ) )  =  ( z  e.  CC  |->  ( z  -  A ) ) )
18 cnex 9027 . . . . . . . 8  |-  CC  e.  _V
1918a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  CC  e.  _V )
20 negex 9260 . . . . . . . 8  |-  -u A  e.  _V
2120a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  -> 
-u A  e.  _V )
22 simpr 448 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  z  e.  CC )
23 fconstmpt 4880 . . . . . . . 8  |-  ( CC 
X.  { -u A } )  =  ( z  e.  CC  |->  -u A )
2423a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  ( CC  X.  { -u A } )  =  ( z  e.  CC  |->  -u A ) )
25 df-idp 20061 . . . . . . . . 9  |-  X p  =  (  _I  |`  CC )
26 mptresid 5154 . . . . . . . . 9  |-  ( z  e.  CC  |->  z )  =  (  _I  |`  CC )
2725, 26eqtr4i 2427 . . . . . . . 8  |-  X p  =  ( z  e.  CC  |->  z )
2827a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  X p  =  ( z  e.  CC  |->  z ) )
2919, 21, 22, 24, 28offval2 6281 . . . . . 6  |-  ( A  e.  CC  ->  (
( CC  X.  { -u A } )  o F  +  X p )  =  ( z  e.  CC  |->  ( -u A  +  z )
) )
30 simpl 444 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  A  e.  CC )
31 fconstmpt 4880 . . . . . . . 8  |-  ( CC 
X.  { A }
)  =  ( z  e.  CC  |->  A )
3231a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  ( CC  X.  { A }
)  =  ( z  e.  CC  |->  A ) )
3319, 22, 30, 28, 32offval2 6281 . . . . . 6  |-  ( A  e.  CC  ->  (
X p  o F  -  ( CC  X.  { A } ) )  =  ( z  e.  CC  |->  ( z  -  A ) ) )
3417, 29, 333eqtr4d 2446 . . . . 5  |-  ( A  e.  CC  ->  (
( CC  X.  { -u A } )  o F  +  X p )  =  ( X p  o F  -  ( CC  X.  { A } ) ) )
3534, 1syl6eqr 2454 . . . 4  |-  ( A  e.  CC  ->  (
( CC  X.  { -u A } )  o F  +  X p )  =  G )
3635fveq2d 5691 . . 3  |-  ( A  e.  CC  ->  (deg `  ( ( CC  X.  { -u A } )  o F  +  X p ) )  =  (deg `  G )
)
37 plyconst 20078 . . . . 5  |-  ( ( CC  C_  CC  /\  -u A  e.  CC )  ->  ( CC  X.  { -u A } )  e.  (Poly `  CC ) )
382, 11, 37sylancr 645 . . . 4  |-  ( A  e.  CC  ->  ( CC  X.  { -u A } )  e.  (Poly `  CC ) )
395a1i 11 . . . 4  |-  ( A  e.  CC  ->  X p  e.  (Poly `  CC ) )
40 0dgr 20117 . . . . . 6  |-  ( -u A  e.  CC  ->  (deg
`  ( CC  X.  { -u A } ) )  =  0 )
4111, 40syl 16 . . . . 5  |-  ( A  e.  CC  ->  (deg `  ( CC  X.  { -u A } ) )  =  0 )
42 0lt1 9506 . . . . 5  |-  0  <  1
4341, 42syl6eqbr 4209 . . . 4  |-  ( A  e.  CC  ->  (deg `  ( CC  X.  { -u A } ) )  <  1 )
44 eqid 2404 . . . . 5  |-  (deg `  ( CC  X.  { -u A } ) )  =  (deg `  ( CC  X.  { -u A }
) )
45 dgrid 20135 . . . . . 6  |-  (deg `  X p )  =  1
4645eqcomi 2408 . . . . 5  |-  1  =  (deg `  X p
)
4744, 46dgradd2 20139 . . . 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 1184 . . 3  |-  ( A  e.  CC  ->  (deg `  ( ( CC  X.  { -u A } )  o F  +  X p ) )  =  1 )
4936, 48eqtr3d 2438 . 2  |-  ( A  e.  CC  ->  (deg `  G )  =  1 )
501, 33syl5eq 2448 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  G  =  ( z  e.  CC  |->  ( z  -  A ) ) )
5150fveq1d 5689 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( G `  z )  =  ( ( z  e.  CC  |->  ( z  -  A ) ) `
 z ) )
5251adantr 452 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( G `  z
)  =  ( ( z  e.  CC  |->  ( z  -  A ) ) `  z ) )
53 ovex 6065 . . . . . . . . . 10  |-  ( z  -  A )  e. 
_V
54 eqid 2404 . . . . . . . . . . 11  |-  ( z  e.  CC  |->  ( z  -  A ) )  =  ( z  e.  CC  |->  ( z  -  A ) )
5554fvmpt2 5771 . . . . . . . . . 10  |-  ( ( z  e.  CC  /\  ( z  -  A
)  e.  _V )  ->  ( ( z  e.  CC  |->  ( z  -  A ) ) `  z )  =  ( z  -  A ) )
5622, 53, 55sylancl 644 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( ( z  e.  CC  |->  ( z  -  A ) ) `  z )  =  ( z  -  A ) )
5752, 56eqtrd 2436 . . . . . . . 8  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( G `  z
)  =  ( z  -  A ) )
5857eqeq1d 2412 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( ( G `  z )  =  0  <-> 
( z  -  A
)  =  0 ) )
59 subeq0 9283 . . . . . . . 8  |-  ( ( z  e.  CC  /\  A  e.  CC )  ->  ( ( z  -  A )  =  0  <-> 
z  =  A ) )
6059ancoms 440 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( ( z  -  A )  =  0  <-> 
z  =  A ) )
6158, 60bitrd 245 . . . . . 6  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( ( G `  z )  =  0  <-> 
z  =  A ) )
6261pm5.32da 623 . . . . 5  |-  ( A  e.  CC  ->  (
( z  e.  CC  /\  ( G `  z
)  =  0 )  <-> 
( z  e.  CC  /\  z  =  A ) ) )
63 plyf 20070 . . . . . . 7  |-  ( G  e.  (Poly `  CC )  ->  G : CC --> CC )
64 ffn 5550 . . . . . . 7  |-  ( G : CC --> CC  ->  G  Fn  CC )
6510, 63, 643syl 19 . . . . . 6  |-  ( A  e.  CC  ->  G  Fn  CC )
66 fniniseg 5810 . . . . . 6  |-  ( G  Fn  CC  ->  (
z  e.  ( `' G " { 0 } )  <->  ( z  e.  CC  /\  ( G `
 z )  =  0 ) ) )
6765, 66syl 16 . . . . 5  |-  ( A  e.  CC  ->  (
z  e.  ( `' G " { 0 } )  <->  ( z  e.  CC  /\  ( G `
 z )  =  0 ) ) )
68 eleq1a 2473 . . . . . 6  |-  ( A  e.  CC  ->  (
z  =  A  -> 
z  e.  CC ) )
6968pm4.71rd 617 . . . . 5  |-  ( A  e.  CC  ->  (
z  =  A  <->  ( z  e.  CC  /\  z  =  A ) ) )
7062, 67, 693bitr4d 277 . . . 4  |-  ( A  e.  CC  ->  (
z  e.  ( `' G " { 0 } )  <->  z  =  A ) )
71 elsn 3789 . . . 4  |-  ( z  e.  { A }  <->  z  =  A )
7270, 71syl6bbr 255 . . 3  |-  ( A  e.  CC  ->  (
z  e.  ( `' G " { 0 } )  <->  z  e.  { A } ) )
7372eqrdv 2402 . 2  |-  ( A  e.  CC  ->  ( `' G " { 0 } )  =  { A } )
7410, 49, 733jca 1134 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916    C_ wss 3280   {csn 3774   class class class wbr 4172    e. cmpt 4226    _I cid 4453    X. cxp 4835   `'ccnv 4836    |` cres 4839   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    < clt 9076    - cmin 9247   -ucneg 9248  Polycply 20056   X pcidp 20057  degcdgr 20059
This theorem is referenced by:  plyrem  20175  facth  20176  fta1lem  20177  vieta1lem1  20180  vieta1lem2  20181  taylply2  20237  ftalem7  20814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-0p 19515  df-ply 20060  df-idp 20061  df-coe 20062  df-dgr 20063
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