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Theorem plyremlem 21745
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 3370 . . . . 5  |-  CC  C_  CC
3 ax-1cn 9332 . . . . 5  |-  1  e.  CC
4 plyid 21652 . . . . 5  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  Xp  e.  (Poly `  CC ) )
52, 3, 4mp2an 672 . . . 4  |-  Xp  e.  (Poly `  CC )
6 plyconst 21649 . . . . 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 21665 . . . 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 2522 . 2  |-  ( A  e.  CC  ->  G  e.  (Poly `  CC )
)
11 negcl 9602 . . . . . . . . 9  |-  ( A  e.  CC  ->  -u A  e.  CC )
12 addcom 9547 . . . . . . . . 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 9649 . . . . . . . . 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 2470 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( -u A  +  z )  =  ( z  -  A ) )
1716mpteq2dva 4373 . . . . . 6  |-  ( A  e.  CC  ->  (
z  e.  CC  |->  (
-u A  +  z ) )  =  ( z  e.  CC  |->  ( z  -  A ) ) )
18 cnex 9355 . . . . . . . 8  |-  CC  e.  _V
1918a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  CC  e.  _V )
20 negex 9600 . . . . . . . 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 4877 . . . . . . . 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 21632 . . . . . . . . 9  |-  Xp  =  (  _I  |`  CC )
26 mptresid 5155 . . . . . . . . 9  |-  ( z  e.  CC  |->  z )  =  (  _I  |`  CC )
2725, 26eqtr4i 2461 . . . . . . . 8  |-  Xp  =  ( z  e.  CC  |->  z )
2827a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  Xp  =  ( z  e.  CC  |->  z ) )
2919, 21, 22, 24, 28offval2 6331 . . . . . 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 4877 . . . . . . . 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 6331 . . . . . 6  |-  ( A  e.  CC  ->  (
Xp  oF  -  ( CC  X.  { A } ) )  =  ( z  e.  CC  |->  ( z  -  A ) ) )
3417, 29, 333eqtr4d 2480 . . . . 5  |-  ( A  e.  CC  ->  (
( CC  X.  { -u A } )  oF  +  Xp )  =  ( Xp  oF  -  ( CC  X.  { A } ) ) )
3534, 1syl6eqr 2488 . . . 4  |-  ( A  e.  CC  ->  (
( CC  X.  { -u A } )  oF  +  Xp )  =  G )
3635fveq2d 5690 . . 3  |-  ( A  e.  CC  ->  (deg `  ( ( CC  X.  { -u A } )  oF  +  Xp ) )  =  (deg `  G )
)
37 plyconst 21649 . . . . 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 21688 . . . . . 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 9854 . . . . 5  |-  0  <  1
4341, 42syl6eqbr 4324 . . . 4  |-  ( A  e.  CC  ->  (deg `  ( CC  X.  { -u A } ) )  <  1 )
44 eqid 2438 . . . . 5  |-  (deg `  ( CC  X.  { -u A } ) )  =  (deg `  ( CC  X.  { -u A }
) )
45 dgrid 21706 . . . . . 6  |-  (deg `  Xp )  =  1
4645eqcomi 2442 . . . . 5  |-  1  =  (deg `  Xp
)
4744, 46dgradd2 21710 . . . 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 1218 . . 3  |-  ( A  e.  CC  ->  (deg `  ( ( CC  X.  { -u A } )  oF  +  Xp ) )  =  1 )
4936, 48eqtr3d 2472 . 2  |-  ( A  e.  CC  ->  (deg `  G )  =  1 )
501, 33syl5eq 2482 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  G  =  ( z  e.  CC  |->  ( z  -  A ) ) )
5150fveq1d 5688 . . . . . . . . . 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 6111 . . . . . . . . . 10  |-  ( z  -  A )  e. 
_V
54 eqid 2438 . . . . . . . . . . 11  |-  ( z  e.  CC  |->  ( z  -  A ) )  =  ( z  e.  CC  |->  ( z  -  A ) )
5554fvmpt2 5776 . . . . . . . . . 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 2470 . . . . . . . 8  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( G `  z
)  =  ( z  -  A ) )
5857eqeq1d 2446 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( ( G `  z )  =  0  <-> 
( z  -  A
)  =  0 ) )
59 subeq0 9627 . . . . . . . 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 21641 . . . . . 6  |-  ( G  e.  (Poly `  CC )  ->  G : CC --> CC )
64 ffn 5554 . . . . . 6  |-  ( G : CC --> CC  ->  G  Fn  CC )
65 fniniseg 5819 . . . . . 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 2507 . . . . . 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 3886 . . . 4  |-  ( z  e.  { A }  <->  z  =  A )
7169, 70syl6bbr 263 . . 3  |-  ( A  e.  CC  ->  (
z  e.  ( `' G " { 0 } )  <->  z  e.  { A } ) )
7271eqrdv 2436 . 2  |-  ( A  e.  CC  ->  ( `' G " { 0 } )  =  { A } )
7310, 49, 723jca 1168 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 965    = wceq 1369    e. wcel 1756   _Vcvv 2967    C_ wss 3323   {csn 3872   class class class wbr 4287    e. cmpt 4345    _I cid 4626    X. cxp 4833   `'ccnv 4834    |` cres 4837   "cima 4838    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086    oFcof 6313   CCcc 9272   0cc0 9274   1c1 9275    + caddc 9277    < clt 9410    - cmin 9587   -ucneg 9588  Polycply 21627   Xpcidp 21628  degcdgr 21630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  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-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-0p 21123  df-ply 21631  df-idp 21632  df-coe 21633  df-dgr 21634
This theorem is referenced by:  plyrem  21746  facth  21747  fta1lem  21748  vieta1lem1  21751  vieta1lem2  21752  taylply2  21808  ftalem7  22391
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