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Theorem plyremlem 22434
Description: Closure of a linear factor. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypothesis
Ref Expression
plyrem.1  |-  G  =  ( Xp  oF  -  ( 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  =  ( Xp  oF  -  ( CC 
X.  { A }
) )
2 ssid 3523 . . . . 5  |-  CC  C_  CC
3 ax-1cn 9546 . . . . 5  |-  1  e.  CC
4 plyid 22341 . . . . 5  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  Xp  e.  (Poly `  CC ) )
52, 3, 4mp2an 672 . . . 4  |-  Xp  e.  (Poly `  CC )
6 plyconst 22338 . . . . 5  |-  ( ( CC  C_  CC  /\  A  e.  CC )  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
72, 6mpan 670 . . . 4  |-  ( A  e.  CC  ->  ( CC  X.  { A }
)  e.  (Poly `  CC ) )
8 plysubcl 22354 . . . 4  |-  ( ( Xp  e.  (Poly `  CC )  /\  ( CC  X.  { A }
)  e.  (Poly `  CC ) )  ->  (
Xp  oF  -  ( CC  X.  { A } ) )  e.  (Poly `  CC ) )
95, 7, 8sylancr 663 . . 3  |-  ( A  e.  CC  ->  (
Xp  oF  -  ( CC  X.  { A } ) )  e.  (Poly `  CC ) )
101, 9syl5eqel 2559 . 2  |-  ( A  e.  CC  ->  G  e.  (Poly `  CC )
)
11 negcl 9816 . . . . . . . . 9  |-  ( A  e.  CC  ->  -u A  e.  CC )
12 addcom 9761 . . . . . . . . 9  |-  ( (
-u A  e.  CC  /\  z  e.  CC )  ->  ( -u A  +  z )  =  ( z  +  -u A ) )
1311, 12sylan 471 . . . . . . . 8  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( -u A  +  z )  =  ( z  +  -u A
) )
14 negsub 9863 . . . . . . . . 9  |-  ( ( z  e.  CC  /\  A  e.  CC )  ->  ( z  +  -u A )  =  ( z  -  A ) )
1514ancoms 453 . . . . . . . 8  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( z  +  -u A )  =  ( z  -  A ) )
1613, 15eqtrd 2508 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( -u A  +  z )  =  ( z  -  A ) )
1716mpteq2dva 4533 . . . . . 6  |-  ( A  e.  CC  ->  (
z  e.  CC  |->  (
-u A  +  z ) )  =  ( z  e.  CC  |->  ( z  -  A ) ) )
18 cnex 9569 . . . . . . . 8  |-  CC  e.  _V
1918a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  CC  e.  _V )
20 negex 9814 . . . . . . . 8  |-  -u A  e.  _V
2120a1i 11 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  -> 
-u A  e.  _V )
22 simpr 461 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  z  e.  CC )
23 fconstmpt 5042 . . . . . . . 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 22321 . . . . . . . . 9  |-  Xp  =  (  _I  |`  CC )
26 mptresid 5326 . . . . . . . . 9  |-  ( z  e.  CC  |->  z )  =  (  _I  |`  CC )
2725, 26eqtr4i 2499 . . . . . . . 8  |-  Xp  =  ( z  e.  CC  |->  z )
2827a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  Xp  =  ( z  e.  CC  |->  z ) )
2919, 21, 22, 24, 28offval2 6538 . . . . . 6  |-  ( A  e.  CC  ->  (
( CC  X.  { -u A } )  oF  +  Xp )  =  ( z  e.  CC  |->  ( -u A  +  z )
) )
30 simpl 457 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  A  e.  CC )
31 fconstmpt 5042 . . . . . . . 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 6538 . . . . . 6  |-  ( A  e.  CC  ->  (
Xp  oF  -  ( CC  X.  { A } ) )  =  ( z  e.  CC  |->  ( z  -  A ) ) )
3417, 29, 333eqtr4d 2518 . . . . 5  |-  ( A  e.  CC  ->  (
( CC  X.  { -u A } )  oF  +  Xp )  =  ( Xp  oF  -  ( CC  X.  { A } ) ) )
3534, 1syl6eqr 2526 . . . 4  |-  ( A  e.  CC  ->  (
( CC  X.  { -u A } )  oF  +  Xp )  =  G )
3635fveq2d 5868 . . 3  |-  ( A  e.  CC  ->  (deg `  ( ( CC  X.  { -u A } )  oF  +  Xp ) )  =  (deg `  G )
)
37 plyconst 22338 . . . . 5  |-  ( ( CC  C_  CC  /\  -u A  e.  CC )  ->  ( CC  X.  { -u A } )  e.  (Poly `  CC ) )
382, 11, 37sylancr 663 . . . 4  |-  ( A  e.  CC  ->  ( CC  X.  { -u A } )  e.  (Poly `  CC ) )
395a1i 11 . . . 4  |-  ( A  e.  CC  ->  Xp  e.  (Poly `  CC ) )
40 0dgr 22377 . . . . . 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 10071 . . . . 5  |-  0  <  1
4341, 42syl6eqbr 4484 . . . 4  |-  ( A  e.  CC  ->  (deg `  ( CC  X.  { -u A } ) )  <  1 )
44 eqid 2467 . . . . 5  |-  (deg `  ( CC  X.  { -u A } ) )  =  (deg `  ( CC  X.  { -u A }
) )
45 dgrid 22395 . . . . . 6  |-  (deg `  Xp )  =  1
4645eqcomi 2480 . . . . 5  |-  1  =  (deg `  Xp
)
4744, 46dgradd2 22399 . . . 4  |-  ( ( ( CC  X.  { -u A } )  e.  (Poly `  CC )  /\  Xp  e.  (Poly `  CC )  /\  (deg `  ( CC  X.  { -u A } ) )  <  1 )  -> 
(deg `  ( ( CC  X.  { -u A } )  oF  +  Xp ) )  =  1 )
4838, 39, 43, 47syl3anc 1228 . . 3  |-  ( A  e.  CC  ->  (deg `  ( ( CC  X.  { -u A } )  oF  +  Xp ) )  =  1 )
4936, 48eqtr3d 2510 . 2  |-  ( A  e.  CC  ->  (deg `  G )  =  1 )
501, 33syl5eq 2520 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  G  =  ( z  e.  CC  |->  ( z  -  A ) ) )
5150fveq1d 5866 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( G `  z )  =  ( ( z  e.  CC  |->  ( z  -  A ) ) `
 z ) )
5251adantr 465 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( G `  z
)  =  ( ( z  e.  CC  |->  ( z  -  A ) ) `  z ) )
53 ovex 6307 . . . . . . . . . 10  |-  ( z  -  A )  e. 
_V
54 eqid 2467 . . . . . . . . . . 11  |-  ( z  e.  CC  |->  ( z  -  A ) )  =  ( z  e.  CC  |->  ( z  -  A ) )
5554fvmpt2 5955 . . . . . . . . . 10  |-  ( ( z  e.  CC  /\  ( z  -  A
)  e.  _V )  ->  ( ( z  e.  CC  |->  ( z  -  A ) ) `  z )  =  ( z  -  A ) )
5622, 53, 55sylancl 662 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( ( z  e.  CC  |->  ( z  -  A ) ) `  z )  =  ( z  -  A ) )
5752, 56eqtrd 2508 . . . . . . . 8  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( G `  z
)  =  ( z  -  A ) )
5857eqeq1d 2469 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( ( G `  z )  =  0  <-> 
( z  -  A
)  =  0 ) )
59 subeq0 9841 . . . . . . . 8  |-  ( ( z  e.  CC  /\  A  e.  CC )  ->  ( ( z  -  A )  =  0  <-> 
z  =  A ) )
6059ancoms 453 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( ( z  -  A )  =  0  <-> 
z  =  A ) )
6158, 60bitrd 253 . . . . . 6  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( ( G `  z )  =  0  <-> 
z  =  A ) )
6261pm5.32da 641 . . . . 5  |-  ( A  e.  CC  ->  (
( z  e.  CC  /\  ( G `  z
)  =  0 )  <-> 
( z  e.  CC  /\  z  =  A ) ) )
63 plyf 22330 . . . . . 6  |-  ( G  e.  (Poly `  CC )  ->  G : CC --> CC )
64 ffn 5729 . . . . . 6  |-  ( G : CC --> CC  ->  G  Fn  CC )
65 fniniseg 6000 . . . . . 6  |-  ( G  Fn  CC  ->  (
z  e.  ( `' G " { 0 } )  <->  ( z  e.  CC  /\  ( G `
 z )  =  0 ) ) )
6610, 63, 64, 654syl 21 . . . . 5  |-  ( A  e.  CC  ->  (
z  e.  ( `' G " { 0 } )  <->  ( z  e.  CC  /\  ( G `
 z )  =  0 ) ) )
67 eleq1a 2550 . . . . . 6  |-  ( A  e.  CC  ->  (
z  =  A  -> 
z  e.  CC ) )
6867pm4.71rd 635 . . . . 5  |-  ( A  e.  CC  ->  (
z  =  A  <->  ( z  e.  CC  /\  z  =  A ) ) )
6962, 66, 683bitr4d 285 . . . 4  |-  ( A  e.  CC  ->  (
z  e.  ( `' G " { 0 } )  <->  z  =  A ) )
70 elsn 4041 . . . 4  |-  ( z  e.  { A }  <->  z  =  A )
7169, 70syl6bbr 263 . . 3  |-  ( A  e.  CC  ->  (
z  e.  ( `' G " { 0 } )  <->  z  e.  { A } ) )
7271eqrdv 2464 . 2  |-  ( A  e.  CC  ->  ( `' G " { 0 } )  =  { A } )
7310, 49, 723jca 1176 1  |-  ( A  e.  CC  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   {csn 4027   class class class wbr 4447    |-> cmpt 4505    _I cid 4790    X. cxp 4997   `'ccnv 4998    |` cres 5001   "cima 5002    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282    oFcof 6520   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491    < clt 9624    - cmin 9801   -ucneg 9802  Polycply 22316   Xpcidp 22317  degcdgr 22319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-rlim 13271  df-sum 13468  df-0p 21812  df-ply 22320  df-idp 22321  df-coe 22322  df-dgr 22323
This theorem is referenced by:  plyrem  22435  facth  22436  fta1lem  22437  vieta1lem1  22440  vieta1lem2  22441  taylply2  22497  ftalem7  23080
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