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Theorem coesub 22632
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 22575 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
2 simpl 457 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  S ) )
31, 2sseldi 3487 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  CC ) )
4 ssid 3508 . . . . . 6  |-  CC  C_  CC
5 neg1cn 10646 . . . . . 6  |-  -u 1  e.  CC
6 plyconst 22581 . . . . . 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 3487 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  e.  (Poly `  CC ) )
10 plymulcl 22596 . . . . 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 22626 . . . 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 22630 . . . . . 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 6292 . . . . 5  |-  ( ( NN0  X.  { -u
1 } )  oF  x.  B )  =  ( ( NN0 
X.  { -u 1 } )  oF  x.  (coeff `  G
) )
2017, 19syl6eqr 2502 . . . 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 6297 . . 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 2484 . 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 9576 . . . . 5  |-  CC  e.  _V
2423a1i 11 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  CC  e.  _V )
25 plyf 22573 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
2625adantr 465 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F : CC
--> CC )
27 plyf 22573 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
2827adantl 466 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G : CC
--> CC )
29 ofnegsub 10541 . . . 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 1229 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  +  (
( CC  X.  { -u 1 } )  oF  x.  G ) )  =  ( F  oF  -  G
) )
3130fveq2d 5860 . 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 10808 . . . 4  |-  NN0  e.  _V
3332a1i 11 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  NN0  e.  _V )
3412coef3 22607 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
3534adantr 465 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A : NN0
--> CC )
3618coef3 22607 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  B : NN0
--> CC )
3736adantl 466 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B : NN0
--> CC )
38 ofnegsub 10541 . . 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 1229 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A  oF  +  (
( NN0  X.  { -u
1 } )  oF  x.  B ) )  =  ( A  oF  -  B
) )
4022, 31, 393eqtr3d 2492 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 1383    e. wcel 1804   _Vcvv 3095    C_ wss 3461   {csn 4014    X. cxp 4987   -->wf 5574   ` cfv 5578  (class class class)co 6281    oFcof 6523   CCcc 9493   1c1 9496    + caddc 9498    x. cmul 9500    - cmin 9810   -ucneg 9811   NN0cn0 10802  Polycply 22559  coeffccoe 22561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11093  df-rp 11232  df-fz 11684  df-fzo 11807  df-fl 11911  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-clim 13293  df-rlim 13294  df-sum 13491  df-0p 22055  df-ply 22563  df-coe 22565  df-dgr 22566
This theorem is referenced by:  dgrcolem2  22649  plydivlem4  22670  plydiveu  22672  vieta1lem2  22685  dgrsub2  31060
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