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Theorem facth 20215
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). (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypothesis
Ref Expression
facth.1  |-  G  =  ( X p  o F  -  ( CC  X.  { A } ) )
Assertion
Ref Expression
facth  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  =  ( G  o F  x.  ( F quot  G ) ) )

Proof of Theorem facth
StepHypRef Expression
1 facth.1 . . . . 5  |-  G  =  ( X p  o F  -  ( CC  X.  { A } ) )
2 eqid 2435 . . . . 5  |-  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  =  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )
31, 2plyrem 20214 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  =  ( CC  X.  { ( F `  A ) } ) )
433adant3 977 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  =  ( CC  X.  { ( F `  A ) } ) )
5 simp3 959 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( F `  A )  =  0 )
65sneqd 3819 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  { ( F `  A ) }  =  { 0 } )
76xpeq2d 4894 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( CC  X.  { ( F `
 A ) } )  =  ( CC 
X.  { 0 } ) )
84, 7eqtrd 2467 . 2  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  =  ( CC  X.  { 0 } ) )
9 cnex 9063 . . . 4  |-  CC  e.  _V
109a1i 11 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  CC  e.  _V )
11 simp1 957 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  e.  (Poly `  S )
)
12 plyf 20109 . . . 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 20213 . . . . . . 7  |-  ( A  e.  CC  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
15143ad2ant2 979 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
1615simp1d 969 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  G  e.  (Poly `  CC )
)
17 plyssc 20111 . . . . . . 7  |-  (Poly `  S )  C_  (Poly `  CC )
1817, 11sseldi 3338 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  e.  (Poly `  CC )
)
1915simp2d 970 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  (deg `  G )  =  1 )
20 ax-1ne0 9051 . . . . . . . . 9  |-  1  =/=  0
2120a1i 11 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  1  =/=  0 )
2219, 21eqnetrd 2616 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  (deg `  G )  =/=  0
)
23 fveq2 5720 . . . . . . . . 9  |-  ( G  =  0 p  -> 
(deg `  G )  =  (deg `  0 p
) )
24 dgr0 20172 . . . . . . . . 9  |-  (deg ` 
0 p )  =  0
2523, 24syl6eq 2483 . . . . . . . 8  |-  ( G  =  0 p  -> 
(deg `  G )  =  0 )
2625necon3i 2637 . . . . . . 7  |-  ( (deg
`  G )  =/=  0  ->  G  =/=  0 p )
2722, 26syl 16 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  G  =/=  0 p )
28 quotcl2 20211 . . . . . 6  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0 p )  ->  ( F quot  G )  e.  (Poly `  CC ) )
2918, 16, 27, 28syl3anc 1184 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( F quot  G )  e.  (Poly `  CC ) )
30 plymulcl 20132 . . . . 5  |-  ( ( G  e.  (Poly `  CC )  /\  ( F quot  G )  e.  (Poly `  CC ) )  -> 
( G  o F  x.  ( F quot  G
) )  e.  (Poly `  CC ) )
3116, 29, 30syl2anc 643 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( G  o F  x.  ( F quot  G ) )  e.  (Poly `  CC )
)
32 plyf 20109 . . . 4  |-  ( ( G  o F  x.  ( F quot  G )
)  e.  (Poly `  CC )  ->  ( G  o F  x.  ( F quot  G ) ) : CC --> CC )
3331, 32syl 16 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( G  o F  x.  ( F quot  G ) ) : CC --> CC )
34 ofsubeq0 9989 . . 3  |-  ( ( CC  e.  _V  /\  F : CC --> CC  /\  ( G  o F  x.  ( F quot  G ) ) : CC --> CC )  ->  ( ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  =  ( CC  X.  { 0 } )  <-> 
F  =  ( G  o F  x.  ( F quot  G ) ) ) )
3510, 13, 33, 34syl3anc 1184 . 2  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  (
( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  =  ( CC  X.  {
0 } )  <->  F  =  ( G  o F  x.  ( F quot  G ) ) ) )
368, 35mpbid 202 1  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  =  ( G  o F  x.  ( F quot  G ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948   {csn 3806    X. cxp 4868   `'ccnv 4869   "cima 4873   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295   CCcc 8980   0cc0 8982   1c1 8983    x. cmul 8987    - cmin 9283   0 pc0p 19553  Polycply 20095   X pcidp 20096  degcdgr 20098   quot cquot 20199
This theorem is referenced by:  fta1lem  20216  vieta1lem1  20219  vieta1lem2  20220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-0p 19554  df-ply 20099  df-idp 20100  df-coe 20101  df-dgr 20102  df-quot 20200
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