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Theorem plyremlem 21896
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 3476 . . . . 5  |-  CC  C_  CC
3 ax-1cn 9444 . . . . 5  |-  1  e.  CC
4 plyid 21803 . . . . 5  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  Xp  e.  (Poly `  CC ) )
52, 3, 4mp2an 672 . . . 4  |-  Xp  e.  (Poly `  CC )
6 plyconst 21800 . . . . 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 21816 . . . 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 2543 . 2  |-  ( A  e.  CC  ->  G  e.  (Poly `  CC )
)
11 negcl 9714 . . . . . . . . 9  |-  ( A  e.  CC  ->  -u A  e.  CC )
12 addcom 9659 . . . . . . . . 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 9761 . . . . . . . . 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 2492 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( -u A  +  z )  =  ( z  -  A ) )
1716mpteq2dva 4479 . . . . . 6  |-  ( A  e.  CC  ->  (
z  e.  CC  |->  (
-u A  +  z ) )  =  ( z  e.  CC  |->  ( z  -  A ) ) )
18 cnex 9467 . . . . . . . 8  |-  CC  e.  _V
1918a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  CC  e.  _V )
20 negex 9712 . . . . . . . 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 4983 . . . . . . . 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 21783 . . . . . . . . 9  |-  Xp  =  (  _I  |`  CC )
26 mptresid 5261 . . . . . . . . 9  |-  ( z  e.  CC  |->  z )  =  (  _I  |`  CC )
2725, 26eqtr4i 2483 . . . . . . . 8  |-  Xp  =  ( z  e.  CC  |->  z )
2827a1i 11 . . . . . . 7  |-  ( A  e.  CC  ->  Xp  =  ( z  e.  CC  |->  z ) )
2919, 21, 22, 24, 28offval2 6439 . . . . . 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 4983 . . . . . . . 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 6439 . . . . . 6  |-  ( A  e.  CC  ->  (
Xp  oF  -  ( CC  X.  { A } ) )  =  ( z  e.  CC  |->  ( z  -  A ) ) )
3417, 29, 333eqtr4d 2502 . . . . 5  |-  ( A  e.  CC  ->  (
( CC  X.  { -u A } )  oF  +  Xp )  =  ( Xp  oF  -  ( CC  X.  { A } ) ) )
3534, 1syl6eqr 2510 . . . 4  |-  ( A  e.  CC  ->  (
( CC  X.  { -u A } )  oF  +  Xp )  =  G )
3635fveq2d 5796 . . 3  |-  ( A  e.  CC  ->  (deg `  ( ( CC  X.  { -u A } )  oF  +  Xp ) )  =  (deg `  G )
)
37 plyconst 21800 . . . . 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 21839 . . . . . 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 9966 . . . . 5  |-  0  <  1
4341, 42syl6eqbr 4430 . . . 4  |-  ( A  e.  CC  ->  (deg `  ( CC  X.  { -u A } ) )  <  1 )
44 eqid 2451 . . . . 5  |-  (deg `  ( CC  X.  { -u A } ) )  =  (deg `  ( CC  X.  { -u A }
) )
45 dgrid 21857 . . . . . 6  |-  (deg `  Xp )  =  1
4645eqcomi 2464 . . . . 5  |-  1  =  (deg `  Xp
)
4744, 46dgradd2 21861 . . . 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 1219 . . 3  |-  ( A  e.  CC  ->  (deg `  ( ( CC  X.  { -u A } )  oF  +  Xp ) )  =  1 )
4936, 48eqtr3d 2494 . 2  |-  ( A  e.  CC  ->  (deg `  G )  =  1 )
501, 33syl5eq 2504 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  G  =  ( z  e.  CC  |->  ( z  -  A ) ) )
5150fveq1d 5794 . . . . . . . . . 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 6218 . . . . . . . . . 10  |-  ( z  -  A )  e. 
_V
54 eqid 2451 . . . . . . . . . . 11  |-  ( z  e.  CC  |->  ( z  -  A ) )  =  ( z  e.  CC  |->  ( z  -  A ) )
5554fvmpt2 5883 . . . . . . . . . 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 2492 . . . . . . . 8  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( G `  z
)  =  ( z  -  A ) )
5857eqeq1d 2453 . . . . . . 7  |-  ( ( A  e.  CC  /\  z  e.  CC )  ->  ( ( G `  z )  =  0  <-> 
( z  -  A
)  =  0 ) )
59 subeq0 9739 . . . . . . . 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 21792 . . . . . 6  |-  ( G  e.  (Poly `  CC )  ->  G : CC --> CC )
64 ffn 5660 . . . . . 6  |-  ( G : CC --> CC  ->  G  Fn  CC )
65 fniniseg 5926 . . . . . 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 2534 . . . . . 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 3992 . . . 4  |-  ( z  e.  { A }  <->  z  =  A )
7169, 70syl6bbr 263 . . 3  |-  ( A  e.  CC  ->  (
z  e.  ( `' G " { 0 } )  <->  z  e.  { A } ) )
7271eqrdv 2448 . 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 1370    e. wcel 1758   _Vcvv 3071    C_ wss 3429   {csn 3978   class class class wbr 4393    |-> cmpt 4451    _I cid 4732    X. cxp 4939   `'ccnv 4940    |` cres 4943   "cima 4944    Fn wfn 5514   -->wf 5515   ` cfv 5519  (class class class)co 6193    oFcof 6421   CCcc 9384   0cc0 9386   1c1 9387    + caddc 9389    < clt 9522    - cmin 9699   -ucneg 9700  Polycply 21778   Xpcidp 21779  degcdgr 21781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-fz 11548  df-fzo 11659  df-fl 11752  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077  df-rlim 13078  df-sum 13275  df-0p 21274  df-ply 21782  df-idp 21783  df-coe 21784  df-dgr 21785
This theorem is referenced by:  plyrem  21897  facth  21898  fta1lem  21899  vieta1lem1  21902  vieta1lem2  21903  taylply2  21959  ftalem7  22542
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