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Theorem coe1add 16341
Description: The coefficient vector of an addition. (Contributed by Stefan O'Rear, 24-Mar-2015.)
Hypotheses
Ref Expression
coe1add.y  |-  Y  =  (Poly1 `  R )
coe1add.b  |-  B  =  ( Base `  Y
)
coe1add.p  |-  .+b  =  ( +g  `  Y )
coe1add.q  |-  .+  =  ( +g  `  R )
Assertion
Ref Expression
coe1add  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .+b  G ) )  =  ( (coe1 `  F )  o F 
.+  (coe1 `  G ) ) )

Proof of Theorem coe1add
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . . 5  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
2 coe1add.y . . . . . 6  |-  Y  =  (Poly1 `  R )
3 eqid 2283 . . . . . 6  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
4 coe1add.b . . . . . 6  |-  B  =  ( Base `  Y
)
52, 3, 4ply1bas 16274 . . . . 5  |-  B  =  ( Base `  ( 1o mPoly  R ) )
6 coe1add.q . . . . 5  |-  .+  =  ( +g  `  R )
7 coe1add.p . . . . . 6  |-  .+b  =  ( +g  `  Y )
82, 1, 7ply1plusg 16303 . . . . 5  |-  .+b  =  ( +g  `  ( 1o mPoly  R ) )
9 simp2 956 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  F  e.  B )
10 simp3 957 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  G  e.  B )
111, 5, 6, 8, 9, 10mpladd 16186 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .+b  G )  =  ( F  o F 
.+  G ) )
1211coeq1d 4845 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (
( F  .+b  G
)  o.  ( a  e.  NN0  |->  ( 1o 
X.  { a } ) ) )  =  ( ( F  o F  .+  G )  o.  ( a  e.  NN0  |->  ( 1o  X.  { a } ) ) ) )
13 eqid 2283 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
142, 4, 13ply1basf 16283 . . . . . 6  |-  ( F  e.  B  ->  F : ( NN0  ^m  1o ) --> ( Base `  R
) )
15 ffn 5389 . . . . . 6  |-  ( F : ( NN0  ^m  1o ) --> ( Base `  R
)  ->  F  Fn  ( NN0  ^m  1o ) )
1614, 15syl 15 . . . . 5  |-  ( F  e.  B  ->  F  Fn  ( NN0  ^m  1o ) )
17163ad2ant2 977 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  F  Fn  ( NN0  ^m  1o ) )
182, 4, 13ply1basf 16283 . . . . . 6  |-  ( G  e.  B  ->  G : ( NN0  ^m  1o ) --> ( Base `  R
) )
19 ffn 5389 . . . . . 6  |-  ( G : ( NN0  ^m  1o ) --> ( Base `  R
)  ->  G  Fn  ( NN0  ^m  1o ) )
2018, 19syl 15 . . . . 5  |-  ( G  e.  B  ->  G  Fn  ( NN0  ^m  1o ) )
21203ad2ant3 978 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  G  Fn  ( NN0  ^m  1o ) )
22 df1o2 6491 . . . . . 6  |-  1o  =  { (/) }
23 nn0ex 9971 . . . . . 6  |-  NN0  e.  _V
24 0ex 4150 . . . . . 6  |-  (/)  e.  _V
25 eqid 2283 . . . . . 6  |-  ( a  e.  NN0  |->  ( 1o 
X.  { a } ) )  =  ( a  e.  NN0  |->  ( 1o 
X.  { a } ) )
2622, 23, 24, 25mapsnf1o3 6816 . . . . 5  |-  ( a  e.  NN0  |->  ( 1o 
X.  { a } ) ) : NN0 -1-1-onto-> ( NN0  ^m  1o )
27 f1of 5472 . . . . 5  |-  ( ( a  e.  NN0  |->  ( 1o 
X.  { a } ) ) : NN0 -1-1-onto-> ( NN0  ^m  1o )  -> 
( a  e.  NN0  |->  ( 1o  X.  { a } ) ) : NN0 --> ( NN0  ^m  1o ) )
2826, 27mp1i 11 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (
a  e.  NN0  |->  ( 1o 
X.  { a } ) ) : NN0 --> ( NN0  ^m  1o ) )
29 ovex 5883 . . . . 5  |-  ( NN0 
^m  1o )  e. 
_V
3029a1i 10 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( NN0  ^m  1o )  e. 
_V )
3123a1i 10 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  NN0  e.  _V )
32 inidm 3378 . . . 4  |-  ( ( NN0  ^m  1o )  i^i  ( NN0  ^m  1o ) )  =  ( NN0  ^m  1o )
3317, 21, 28, 30, 30, 31, 32ofco 6097 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (
( F  o F 
.+  G )  o.  ( a  e.  NN0  |->  ( 1o  X.  { a } ) ) )  =  ( ( F  o.  ( a  e. 
NN0  |->  ( 1o  X.  { a } ) ) )  o F 
.+  ( G  o.  ( a  e.  NN0  |->  ( 1o  X.  { a } ) ) ) ) )
3412, 33eqtrd 2315 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (
( F  .+b  G
)  o.  ( a  e.  NN0  |->  ( 1o 
X.  { a } ) ) )  =  ( ( F  o.  ( a  e.  NN0  |->  ( 1o  X.  { a } ) ) )  o F  .+  ( G  o.  ( a  e.  NN0  |->  ( 1o  X.  { a } ) ) ) ) )
352ply1rng 16326 . . . 4  |-  ( R  e.  Ring  ->  Y  e. 
Ring )
364, 7rngacl 15368 . . . 4  |-  ( ( Y  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .+b  G )  e.  B )
3735, 36syl3an1 1215 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .+b  G )  e.  B )
38 eqid 2283 . . . 4  |-  (coe1 `  ( F  .+b  G ) )  =  (coe1 `  ( F  .+b  G ) )
3938, 4, 2, 25coe1fval2 16291 . . 3  |-  ( ( F  .+b  G )  e.  B  ->  (coe1 `  ( F  .+b  G ) )  =  ( ( F 
.+b  G )  o.  ( a  e.  NN0  |->  ( 1o  X.  { a } ) ) ) )
4037, 39syl 15 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .+b  G ) )  =  ( ( F  .+b  G )  o.  ( a  e.  NN0  |->  ( 1o  X.  { a } ) ) ) )
41 eqid 2283 . . . . 5  |-  (coe1 `  F
)  =  (coe1 `  F
)
4241, 4, 2, 25coe1fval2 16291 . . . 4  |-  ( F  e.  B  ->  (coe1 `  F )  =  ( F  o.  ( a  e.  NN0  |->  ( 1o 
X.  { a } ) ) ) )
43423ad2ant2 977 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  F )  =  ( F  o.  ( a  e.  NN0  |->  ( 1o 
X.  { a } ) ) ) )
44 eqid 2283 . . . . 5  |-  (coe1 `  G
)  =  (coe1 `  G
)
4544, 4, 2, 25coe1fval2 16291 . . . 4  |-  ( G  e.  B  ->  (coe1 `  G )  =  ( G  o.  ( a  e.  NN0  |->  ( 1o 
X.  { a } ) ) ) )
46453ad2ant3 978 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  G )  =  ( G  o.  ( a  e.  NN0  |->  ( 1o 
X.  { a } ) ) ) )
4743, 46oveq12d 5876 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (
(coe1 `  F )  o F  .+  (coe1 `  G
) )  =  ( ( F  o.  (
a  e.  NN0  |->  ( 1o 
X.  { a } ) ) )  o F  .+  ( G  o.  ( a  e. 
NN0  |->  ( 1o  X.  { a } ) ) ) ) )
4834, 40, 473eqtr4d 2325 1  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .+b  G ) )  =  ( (coe1 `  F )  o F 
.+  (coe1 `  G ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   {csn 3640    e. cmpt 4077    X. cxp 4687    o. ccom 4693    Fn wfn 5250   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    o Fcof 6076   1oc1o 6472    ^m cmap 6772   NN0cn0 9965   Basecbs 13148   +g cplusg 13208   Ringcrg 15337   mPoly cmpl 16089  PwSer1cps1 16250  Poly1cpl1 16252  coe1cco1 16255
This theorem is referenced by:  coe1addfv  16342
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543  df-psr 16098  df-mpl 16100  df-opsr 16106  df-psr1 16257  df-ply1 16259  df-coe1 16262
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