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Theorem facth 19702
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 2296 . . . . 5  |-  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  =  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )
31, 2plyrem 19701 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  =  ( CC  X.  { ( F `  A ) } ) )
433adant3 975 . . 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 957 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( F `  A )  =  0 )
65sneqd 3666 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  { ( F `  A ) }  =  { 0 } )
76xpeq2d 4729 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( CC  X.  { ( F `
 A ) } )  =  ( CC 
X.  { 0 } ) )
84, 7eqtrd 2328 . 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 8834 . . . 4  |-  CC  e.  _V
109a1i 10 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  CC  e.  _V )
11 simp1 955 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  e.  (Poly `  S )
)
12 plyf 19596 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
1311, 12syl 15 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F : CC --> CC )
141plyremlem 19700 . . . . . . 7  |-  ( A  e.  CC  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
15143ad2ant2 977 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
1615simp1d 967 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  G  e.  (Poly `  CC )
)
17 plyssc 19598 . . . . . . 7  |-  (Poly `  S )  C_  (Poly `  CC )
1817, 11sseldi 3191 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  e.  (Poly `  CC )
)
1915simp2d 968 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  (deg `  G )  =  1 )
20 ax-1ne0 8822 . . . . . . . . 9  |-  1  =/=  0
2120a1i 10 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  1  =/=  0 )
2219, 21eqnetrd 2477 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  (deg `  G )  =/=  0
)
23 fveq2 5541 . . . . . . . . 9  |-  ( G  =  0 p  -> 
(deg `  G )  =  (deg `  0 p
) )
24 dgr0 19659 . . . . . . . . 9  |-  (deg ` 
0 p )  =  0
2523, 24syl6eq 2344 . . . . . . . 8  |-  ( G  =  0 p  -> 
(deg `  G )  =  0 )
2625necon3i 2498 . . . . . . 7  |-  ( (deg
`  G )  =/=  0  ->  G  =/=  0 p )
2722, 26syl 15 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  G  =/=  0 p )
28 quotcl2 19698 . . . . . 6  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0 p )  ->  ( F quot  G )  e.  (Poly `  CC ) )
2918, 16, 27, 28syl3anc 1182 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( F quot  G )  e.  (Poly `  CC ) )
30 plymulcl 19619 . . . . 5  |-  ( ( G  e.  (Poly `  CC )  /\  ( F quot  G )  e.  (Poly `  CC ) )  -> 
( G  o F  x.  ( F quot  G
) )  e.  (Poly `  CC ) )
3116, 29, 30syl2anc 642 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( G  o F  x.  ( F quot  G ) )  e.  (Poly `  CC )
)
32 plyf 19596 . . . 4  |-  ( ( G  o F  x.  ( F quot  G )
)  e.  (Poly `  CC )  ->  ( G  o F  x.  ( F quot  G ) ) : CC --> CC )
3331, 32syl 15 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( G  o F  x.  ( F quot  G ) ) : CC --> CC )
34 ofsubeq0 9759 . . 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 1182 . 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 201 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 176    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   {csn 3653    X. cxp 4703   `'ccnv 4704   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   0cc0 8753   1c1 8754    x. cmul 8758    - cmin 9053   0 pc0p 19040  Polycply 19582   X pcidp 19583  degcdgr 19585   quot cquot 19686
This theorem is referenced by:  fta1lem  19703  vieta1lem1  19706  vieta1lem2  19707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-idp 19587  df-coe 19588  df-dgr 19589  df-quot 19687
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