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Theorem coesub 21724
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  oF  -  G
) )  =  ( A  oF  -  B ) )

Proof of Theorem coesub
StepHypRef Expression
1 plyssc 21668 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
2 simpl 457 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  S ) )
31, 2sseldi 3354 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  CC ) )
4 ssid 3375 . . . . . 6  |-  CC  C_  CC
5 neg1cn 10425 . . . . . 6  |-  -u 1  e.  CC
6 plyconst 21674 . . . . . 6  |-  ( ( CC  C_  CC  /\  -u 1  e.  CC )  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  CC ) )
74, 5, 6mp2an 672 . . . . 5  |-  ( CC 
X.  { -u 1 } )  e.  (Poly `  CC )
8 simpr 461 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  e.  (Poly `  S ) )
91, 8sseldi 3354 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  e.  (Poly `  CC ) )
10 plymulcl 21689 . . . . 5  |-  ( ( ( CC  X.  { -u 1 } )  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC ) )  -> 
( ( CC  X.  { -u 1 } )  oF  x.  G
)  e.  (Poly `  CC ) )
117, 9, 10sylancr 663 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( CC  X.  { -u 1 } )  oF  x.  G )  e.  (Poly `  CC )
)
12 coesub.1 . . . . 5  |-  A  =  (coeff `  F )
13 eqid 2443 . . . . 5  |-  (coeff `  ( ( CC  X.  { -u 1 } )  oF  x.  G
) )  =  (coeff `  ( ( CC  X.  { -u 1 } )  oF  x.  G
) )
1412, 13coeadd 21718 . . . 4  |-  ( ( F  e.  (Poly `  CC )  /\  (
( CC  X.  { -u 1 } )  oF  x.  G )  e.  (Poly `  CC ) )  ->  (coeff `  ( F  oF  +  ( ( CC 
X.  { -u 1 } )  oF  x.  G ) ) )  =  ( A  oF  +  (coeff `  ( ( CC  X.  { -u 1 } )  oF  x.  G
) ) ) )
153, 11, 14syl2anc 661 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  oF  +  ( ( CC  X.  { -u 1 } )  oF  x.  G ) ) )  =  ( A  oF  +  (coeff `  ( ( CC 
X.  { -u 1 } )  oF  x.  G ) ) ) )
16 coemulc 21722 . . . . . 6  |-  ( (
-u 1  e.  CC  /\  G  e.  (Poly `  CC ) )  ->  (coeff `  ( ( CC  X.  { -u 1 } )  oF  x.  G
) )  =  ( ( NN0  X.  { -u 1 } )  oF  x.  (coeff `  G ) ) )
175, 9, 16sylancr 663 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  (
( CC  X.  { -u 1 } )  oF  x.  G ) )  =  ( ( NN0  X.  { -u
1 } )  oF  x.  (coeff `  G ) ) )
18 coesub.2 . . . . . 6  |-  B  =  (coeff `  G )
1918oveq2i 6102 . . . . 5  |-  ( ( NN0  X.  { -u
1 } )  oF  x.  B )  =  ( ( NN0 
X.  { -u 1 } )  oF  x.  (coeff `  G
) )
2017, 19syl6eqr 2493 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  (
( CC  X.  { -u 1 } )  oF  x.  G ) )  =  ( ( NN0  X.  { -u
1 } )  oF  x.  B ) )
2120oveq2d 6107 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A  oF  +  (coeff `  ( ( CC  X.  { -u 1 } )  oF  x.  G
) ) )  =  ( A  oF  +  ( ( NN0 
X.  { -u 1 } )  oF  x.  B ) ) )
2215, 21eqtrd 2475 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  oF  +  ( ( CC  X.  { -u 1 } )  oF  x.  G ) ) )  =  ( A  oF  +  ( ( NN0  X.  { -u 1 } )  oF  x.  B
) ) )
23 cnex 9363 . . . . 5  |-  CC  e.  _V
2423a1i 11 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  CC  e.  _V )
25 plyf 21666 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
2625adantr 465 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F : CC
--> CC )
27 plyf 21666 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
2827adantl 466 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G : CC
--> CC )
29 ofnegsub 10320 . . . 4  |-  ( ( CC  e.  _V  /\  F : CC --> CC  /\  G : CC --> CC )  ->  ( F  oF  +  ( ( CC  X.  { -u 1 } )  oF  x.  G ) )  =  ( F  oF  -  G )
)
3024, 26, 28, 29syl3anc 1218 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  +  (
( CC  X.  { -u 1 } )  oF  x.  G ) )  =  ( F  oF  -  G
) )
3130fveq2d 5695 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  oF  +  ( ( CC  X.  { -u 1 } )  oF  x.  G ) ) )  =  (coeff `  ( F  oF  -  G ) ) )
32 nn0ex 10585 . . . 4  |-  NN0  e.  _V
3332a1i 11 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  NN0  e.  _V )
3412coef3 21700 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
3534adantr 465 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A : NN0
--> CC )
3618coef3 21700 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  B : NN0
--> CC )
3736adantl 466 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B : NN0
--> CC )
38 ofnegsub 10320 . . 3  |-  ( ( NN0  e.  _V  /\  A : NN0 --> CC  /\  B : NN0 --> CC )  ->  ( A  oF  +  ( ( NN0  X.  { -u 1 } )  oF  x.  B ) )  =  ( A  oF  -  B )
)
3933, 35, 37, 38syl3anc 1218 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A  oF  +  (
( NN0  X.  { -u
1 } )  oF  x.  B ) )  =  ( A  oF  -  B
) )
4022, 31, 393eqtr3d 2483 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  oF  -  G
) )  =  ( A  oF  -  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2972    C_ wss 3328   {csn 3877    X. cxp 4838   -->wf 5414   ` cfv 5418  (class class class)co 6091    oFcof 6318   CCcc 9280   1c1 9283    + caddc 9285    x. cmul 9287    - cmin 9595   -ucneg 9596   NN0cn0 10579  Polycply 21652  coeffccoe 21654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-rlim 12967  df-sum 13164  df-0p 21148  df-ply 21656  df-coe 21658  df-dgr 21659
This theorem is referenced by:  dgrcolem2  21741  plydivlem4  21762  plydiveu  21764  vieta1lem2  21777  dgrsub2  29491
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