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Theorem coesub 20175
Description: The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
coesub.1  |-  A  =  (coeff `  F )
coesub.2  |-  B  =  (coeff `  G )
Assertion
Ref Expression
coesub  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  o F  -  G
) )  =  ( A  o F  -  B ) )

Proof of Theorem coesub
StepHypRef Expression
1 plyssc 20119 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
2 simpl 444 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  S ) )
31, 2sseldi 3346 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  CC ) )
4 ssid 3367 . . . . . 6  |-  CC  C_  CC
5 neg1cn 10067 . . . . . 6  |-  -u 1  e.  CC
6 plyconst 20125 . . . . . 6  |-  ( ( CC  C_  CC  /\  -u 1  e.  CC )  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  CC ) )
74, 5, 6mp2an 654 . . . . 5  |-  ( CC 
X.  { -u 1 } )  e.  (Poly `  CC )
8 simpr 448 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  e.  (Poly `  S ) )
91, 8sseldi 3346 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  e.  (Poly `  CC ) )
10 plymulcl 20140 . . . . 5  |-  ( ( ( CC  X.  { -u 1 } )  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC ) )  -> 
( ( CC  X.  { -u 1 } )  o F  x.  G
)  e.  (Poly `  CC ) )
117, 9, 10sylancr 645 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( CC  X.  { -u 1 } )  o F  x.  G )  e.  (Poly `  CC )
)
12 coesub.1 . . . . 5  |-  A  =  (coeff `  F )
13 eqid 2436 . . . . 5  |-  (coeff `  ( ( CC  X.  { -u 1 } )  o F  x.  G
) )  =  (coeff `  ( ( CC  X.  { -u 1 } )  o F  x.  G
) )
1412, 13coeadd 20169 . . . 4  |-  ( ( F  e.  (Poly `  CC )  /\  (
( CC  X.  { -u 1 } )  o F  x.  G )  e.  (Poly `  CC ) )  ->  (coeff `  ( F  o F  +  ( ( CC 
X.  { -u 1 } )  o F  x.  G ) ) )  =  ( A  o F  +  (coeff `  ( ( CC  X.  { -u 1 } )  o F  x.  G
) ) ) )
153, 11, 14syl2anc 643 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  o F  +  ( ( CC  X.  { -u 1 } )  o F  x.  G ) ) )  =  ( A  o F  +  (coeff `  ( ( CC 
X.  { -u 1 } )  o F  x.  G ) ) ) )
16 coemulc 20173 . . . . . 6  |-  ( (
-u 1  e.  CC  /\  G  e.  (Poly `  CC ) )  ->  (coeff `  ( ( CC  X.  { -u 1 } )  o F  x.  G
) )  =  ( ( NN0  X.  { -u 1 } )  o F  x.  (coeff `  G ) ) )
175, 9, 16sylancr 645 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  (
( CC  X.  { -u 1 } )  o F  x.  G ) )  =  ( ( NN0  X.  { -u
1 } )  o F  x.  (coeff `  G ) ) )
18 coesub.2 . . . . . 6  |-  B  =  (coeff `  G )
1918oveq2i 6092 . . . . 5  |-  ( ( NN0  X.  { -u
1 } )  o F  x.  B )  =  ( ( NN0 
X.  { -u 1 } )  o F  x.  (coeff `  G
) )
2017, 19syl6eqr 2486 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  (
( CC  X.  { -u 1 } )  o F  x.  G ) )  =  ( ( NN0  X.  { -u
1 } )  o F  x.  B ) )
2120oveq2d 6097 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A  o F  +  (coeff `  ( ( CC  X.  { -u 1 } )  o F  x.  G
) ) )  =  ( A  o F  +  ( ( NN0 
X.  { -u 1 } )  o F  x.  B ) ) )
2215, 21eqtrd 2468 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  o F  +  ( ( CC  X.  { -u 1 } )  o F  x.  G ) ) )  =  ( A  o F  +  ( ( NN0  X.  { -u 1 } )  o F  x.  B
) ) )
23 cnex 9071 . . . . 5  |-  CC  e.  _V
2423a1i 11 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  CC  e.  _V )
25 plyf 20117 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
2625adantr 452 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F : CC
--> CC )
27 plyf 20117 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
2827adantl 453 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G : CC
--> CC )
29 ofnegsub 9998 . . . 4  |-  ( ( CC  e.  _V  /\  F : CC --> CC  /\  G : CC --> CC )  ->  ( F  o F  +  ( ( CC  X.  { -u 1 } )  o F  x.  G ) )  =  ( F  o F  -  G )
)
3024, 26, 28, 29syl3anc 1184 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  o F  +  (
( CC  X.  { -u 1 } )  o F  x.  G ) )  =  ( F  o F  -  G
) )
3130fveq2d 5732 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  o F  +  ( ( CC  X.  { -u 1 } )  o F  x.  G ) ) )  =  (coeff `  ( F  o F  -  G ) ) )
32 nn0ex 10227 . . . 4  |-  NN0  e.  _V
3332a1i 11 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  NN0  e.  _V )
3412coef3 20151 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
3534adantr 452 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A : NN0
--> CC )
3618coef3 20151 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  B : NN0
--> CC )
3736adantl 453 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B : NN0
--> CC )
38 ofnegsub 9998 . . 3  |-  ( ( NN0  e.  _V  /\  A : NN0 --> CC  /\  B : NN0 --> CC )  ->  ( A  o F  +  ( ( NN0  X.  { -u 1 } )  o F  x.  B ) )  =  ( A  o F  -  B )
)
3933, 35, 37, 38syl3anc 1184 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A  o F  +  (
( NN0  X.  { -u
1 } )  o F  x.  B ) )  =  ( A  o F  -  B
) )
4022, 31, 393eqtr3d 2476 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  o F  -  G
) )  =  ( A  o F  -  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    C_ wss 3320   {csn 3814    X. cxp 4876   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303   CCcc 8988   1c1 8991    + caddc 8993    x. cmul 8995    - cmin 9291   -ucneg 9292   NN0cn0 10221  Polycply 20103  coeffccoe 20105
This theorem is referenced by:  dgrcolem2  20192  plydivlem4  20213  plydiveu  20215  vieta1lem2  20228  dgrsub2  27316
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-rlim 12283  df-sum 12480  df-0p 19562  df-ply 20107  df-coe 20109  df-dgr 20110
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