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Theorem facth 21900
Description: The factor theorem. If a polynomial  F has a root at  A, then  G  =  x  -  A is a factor of  F (and the other factor is  F quot  G). This is part of Metamath 100 proof #89. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypothesis
Ref Expression
facth.1  |-  G  =  ( Xp  oF  -  ( CC 
X.  { A }
) )
Assertion
Ref Expression
facth  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  =  ( G  oF  x.  ( F quot  G ) ) )

Proof of Theorem facth
StepHypRef Expression
1 facth.1 . . . . 5  |-  G  =  ( Xp  oF  -  ( CC 
X.  { A }
) )
2 eqid 2452 . . . . 5  |-  ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) )  =  ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) )
31, 2plyrem 21899 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) )  =  ( CC  X.  { ( F `  A ) } ) )
433adant3 1008 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) )  =  ( CC  X.  { ( F `  A ) } ) )
5 simp3 990 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( F `  A )  =  0 )
65sneqd 3992 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  { ( F `  A ) }  =  { 0 } )
76xpeq2d 4967 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( CC  X.  { ( F `
 A ) } )  =  ( CC 
X.  { 0 } ) )
84, 7eqtrd 2493 . 2  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) )  =  ( CC  X.  { 0 } ) )
9 cnex 9469 . . . 4  |-  CC  e.  _V
109a1i 11 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  CC  e.  _V )
11 simp1 988 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  e.  (Poly `  S )
)
12 plyf 21794 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
1311, 12syl 16 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F : CC --> CC )
141plyremlem 21898 . . . . . . 7  |-  ( A  e.  CC  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
15143ad2ant2 1010 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
1615simp1d 1000 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  G  e.  (Poly `  CC )
)
17 plyssc 21796 . . . . . . 7  |-  (Poly `  S )  C_  (Poly `  CC )
1817, 11sseldi 3457 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  e.  (Poly `  CC )
)
1915simp2d 1001 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  (deg `  G )  =  1 )
20 ax-1ne0 9457 . . . . . . . . 9  |-  1  =/=  0
2120a1i 11 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  1  =/=  0 )
2219, 21eqnetrd 2742 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  (deg `  G )  =/=  0
)
23 fveq2 5794 . . . . . . . . 9  |-  ( G  =  0p  -> 
(deg `  G )  =  (deg `  0p
) )
24 dgr0 21857 . . . . . . . . 9  |-  (deg ` 
0p )  =  0
2523, 24syl6eq 2509 . . . . . . . 8  |-  ( G  =  0p  -> 
(deg `  G )  =  0 )
2625necon3i 2689 . . . . . . 7  |-  ( (deg
`  G )  =/=  0  ->  G  =/=  0p )
2722, 26syl 16 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  G  =/=  0p )
28 quotcl2 21896 . . . . . 6  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0p )  ->  ( F quot  G )  e.  (Poly `  CC ) )
2918, 16, 27, 28syl3anc 1219 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( F quot  G )  e.  (Poly `  CC ) )
30 plymulcl 21817 . . . . 5  |-  ( ( G  e.  (Poly `  CC )  /\  ( F quot  G )  e.  (Poly `  CC ) )  -> 
( G  oF  x.  ( F quot  G
) )  e.  (Poly `  CC ) )
3116, 29, 30syl2anc 661 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( G  oF  x.  ( F quot  G ) )  e.  (Poly `  CC )
)
32 plyf 21794 . . . 4  |-  ( ( G  oF  x.  ( F quot  G ) )  e.  (Poly `  CC )  ->  ( G  oF  x.  ( F quot  G ) ) : CC --> CC )
3331, 32syl 16 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( G  oF  x.  ( F quot  G ) ) : CC --> CC )
34 ofsubeq0 10425 . . 3  |-  ( ( CC  e.  _V  /\  F : CC --> CC  /\  ( G  oF  x.  ( F quot  G ) ) : CC --> CC )  ->  ( ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) )  =  ( CC  X.  { 0 } )  <-> 
F  =  ( G  oF  x.  ( F quot  G ) ) ) )
3510, 13, 33, 34syl3anc 1219 . 2  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  (
( F  oF  -  ( G  oF  x.  ( F quot  G ) ) )  =  ( CC  X.  {
0 } )  <->  F  =  ( G  oF  x.  ( F quot  G ) ) ) )
368, 35mpbid 210 1  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  =  ( G  oF  x.  ( F quot  G ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   _Vcvv 3072   {csn 3980    X. cxp 4941   `'ccnv 4942   "cima 4946   -->wf 5517   ` cfv 5521  (class class class)co 6195    oFcof 6423   CCcc 9386   0cc0 9388   1c1 9389    x. cmul 9393    - cmin 9701   0pc0p 21275  Polycply 21780   Xpcidp 21781  degcdgr 21783   quot cquot 21884
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466  ax-addf 9467
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-of 6425  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-map 7321  df-pm 7322  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-oi 7830  df-card 8215  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-fz 11550  df-fzo 11661  df-fl 11754  df-seq 11919  df-exp 11978  df-hash 12216  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-clim 13079  df-rlim 13080  df-sum 13277  df-0p 21276  df-ply 21784  df-idp 21785  df-coe 21786  df-dgr 21787  df-quot 21885
This theorem is referenced by:  fta1lem  21901  vieta1lem1  21904  vieta1lem2  21905
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