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Theorem coesub 22381
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 22325 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
2 simpl 457 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  S ) )
31, 2sseldi 3495 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F  e.  (Poly `  CC ) )
4 ssid 3516 . . . . . 6  |-  CC  C_  CC
5 neg1cn 10628 . . . . . 6  |-  -u 1  e.  CC
6 plyconst 22331 . . . . . 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 3495 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G  e.  (Poly `  CC ) )
10 plymulcl 22346 . . . . 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 2460 . . . . 5  |-  (coeff `  ( ( CC  X.  { -u 1 } )  oF  x.  G
) )  =  (coeff `  ( ( CC  X.  { -u 1 } )  oF  x.  G
) )
1412, 13coeadd 22375 . . . 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 22379 . . . . . 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 6286 . . . . 5  |-  ( ( NN0  X.  { -u
1 } )  oF  x.  B )  =  ( ( NN0 
X.  { -u 1 } )  oF  x.  (coeff `  G
) )
2017, 19syl6eqr 2519 . . . 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 6291 . . 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 2501 . 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 9562 . . . . 5  |-  CC  e.  _V
2423a1i 11 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  CC  e.  _V )
25 plyf 22323 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
2625adantr 465 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F : CC
--> CC )
27 plyf 22323 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
2827adantl 466 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G : CC
--> CC )
29 ofnegsub 10523 . . . 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 1223 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  +  (
( CC  X.  { -u 1 } )  oF  x.  G ) )  =  ( F  oF  -  G
) )
3130fveq2d 5861 . 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 10790 . . . 4  |-  NN0  e.  _V
3332a1i 11 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  NN0  e.  _V )
3412coef3 22357 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
3534adantr 465 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  A : NN0
--> CC )
3618coef3 22357 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  B : NN0
--> CC )
3736adantl 466 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  B : NN0
--> CC )
38 ofnegsub 10523 . . 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 1223 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A  oF  +  (
( NN0  X.  { -u
1 } )  oF  x.  B ) )  =  ( A  oF  -  B
) )
4022, 31, 393eqtr3d 2509 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 1374    e. wcel 1762   _Vcvv 3106    C_ wss 3469   {csn 4020    X. cxp 4990   -->wf 5575   ` cfv 5579  (class class class)co 6275    oFcof 6513   CCcc 9479   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9794   -ucneg 9795   NN0cn0 10784  Polycply 22309  coeffccoe 22311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-fl 11886  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-rlim 13261  df-sum 13458  df-0p 21805  df-ply 22313  df-coe 22315  df-dgr 22316
This theorem is referenced by:  dgrcolem2  22398  plydivlem4  22419  plydiveu  22421  vieta1lem2  22434  dgrsub2  30677
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