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Theorem coesub 22739
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 22682 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
2 simpl 455 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  S ) )
31, 2sseldi 3415 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  CC ) )
4 ssid 3436 . . . . . 6  |-  CC  C_  CC
5 neg1cn 10556 . . . . . 6  |-  -u 1  e.  CC
6 plyconst 22688 . . . . . 6  |-  ( ( CC  C_  CC  /\  -u 1  e.  CC )  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  CC ) )
74, 5, 6mp2an 670 . . . . 5  |-  ( CC 
X.  { -u 1 } )  e.  (Poly `  CC )
8 simpr 459 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  e.  (Poly `  S ) )
91, 8sseldi 3415 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  e.  (Poly `  CC ) )
10 plymulcl 22703 . . . . 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 661 . . . 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 2382 . . . . 5  |-  (coeff `  ( ( CC  X.  { -u 1 } )  oF  x.  G
) )  =  (coeff `  ( ( CC  X.  { -u 1 } )  oF  x.  G
) )
1412, 13coeadd 22733 . . . 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 659 . . 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 22737 . . . . . 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 661 . . . . 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 6207 . . . . 5  |-  ( ( NN0  X.  { -u
1 } )  oF  x.  B )  =  ( ( NN0 
X.  { -u 1 } )  oF  x.  (coeff `  G
) )
2017, 19syl6eqr 2441 . . . 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 6212 . . 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 2423 . 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 9484 . . . . 5  |-  CC  e.  _V
2423a1i 11 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  CC  e.  _V )
25 plyf 22680 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
2625adantr 463 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F : CC
--> CC )
27 plyf 22680 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
2827adantl 464 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G : CC
--> CC )
29 ofnegsub 10450 . . . 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 1226 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  +  (
( CC  X.  { -u 1 } )  oF  x.  G ) )  =  ( F  oF  -  G
) )
3130fveq2d 5778 . 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 10718 . . . 4  |-  NN0  e.  _V
3332a1i 11 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  NN0  e.  _V )
3412coef3 22714 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
3534adantr 463 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A : NN0
--> CC )
3618coef3 22714 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  B : NN0
--> CC )
3736adantl 464 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B : NN0
--> CC )
38 ofnegsub 10450 . . 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 1226 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A  oF  +  (
( NN0  X.  { -u
1 } )  oF  x.  B ) )  =  ( A  oF  -  B
) )
4022, 31, 393eqtr3d 2431 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 367    = wceq 1399    e. wcel 1826   _Vcvv 3034    C_ wss 3389   {csn 3944    X. cxp 4911   -->wf 5492   ` cfv 5496  (class class class)co 6196    oFcof 6437   CCcc 9401   1c1 9404    + caddc 9406    x. cmul 9408    - cmin 9718   -ucneg 9719   NN0cn0 10712  Polycply 22666  coeffccoe 22668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-sup 7816  df-oi 7850  df-card 8233  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-fz 11594  df-fzo 11718  df-fl 11828  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-clim 13313  df-rlim 13314  df-sum 13511  df-0p 22162  df-ply 22670  df-coe 22672  df-dgr 22673
This theorem is referenced by:  dgrcolem2  22756  plydivlem4  22777  plydiveu  22779  vieta1lem2  22792  dgrsub2  31252
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