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Theorem plypf1 21683
Description: Write the set of complex polynomials in a subring in terms of the abstract polynomial construction. (Contributed by Mario Carneiro, 3-Jul-2015.) (Proof shortened by AV, 29-Sep-2019.)
Hypotheses
Ref Expression
plypf1.r  |-  R  =  (flds  S )
plypf1.p  |-  P  =  (Poly1 `  R )
plypf1.a  |-  A  =  ( Base `  P
)
plypf1.e  |-  E  =  (eval1 ` fld )
Assertion
Ref Expression
plypf1  |-  ( S  e.  (SubRing ` fld )  ->  (Poly `  S )  =  ( E " A ) )

Proof of Theorem plypf1
Dummy variables  f 
a  k  n  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elply 21666 . . . . 5  |-  ( f  e.  (Poly `  S
)  <->  ( S  C_  CC  /\  E. n  e. 
NN0  E. a  e.  ( ( S  u.  {
0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) ) )
21simprbi 464 . . . 4  |-  ( f  e.  (Poly `  S
)  ->  E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) )
3 eqid 2443 . . . . . . . . 9  |-  (fld  ^s  CC )  =  (fld 
^s 
CC )
4 cnfldbas 17825 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
5 eqid 2443 . . . . . . . . 9  |-  ( 0g
`  (fld 
^s 
CC ) )  =  ( 0g `  (fld  ^s  CC ) )
6 cnex 9366 . . . . . . . . . 10  |-  CC  e.  _V
76a1i 11 . . . . . . . . 9  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  ->  CC  e.  _V )
8 fzfid 11798 . . . . . . . . 9  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( 0 ... n
)  e.  Fin )
9 cnrng 17841 . . . . . . . . . 10  |-fld  e.  Ring
10 rngcmn 16678 . . . . . . . . . 10  |-  (fld  e.  Ring  ->fld  e. CMnd )
119, 10mp1i 12 . . . . . . . . 9  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  ->fld  e. CMnd )
124subrgss 16869 . . . . . . . . . . . . 13  |-  ( S  e.  (SubRing ` fld )  ->  S  C_  CC )
1312ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  S  C_  CC )
14 elmapi 7237 . . . . . . . . . . . . . . 15  |-  ( a  e.  ( ( S  u.  { 0 } )  ^m  NN0 )  ->  a : NN0 --> ( S  u.  { 0 } ) )
1514ad2antll 728 . . . . . . . . . . . . . 14  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
a : NN0 --> ( S  u.  { 0 } ) )
16 subrgsubg 16874 . . . . . . . . . . . . . . . . . . 19  |-  ( S  e.  (SubRing ` fld )  ->  S  e.  (SubGrp ` fld ) )
17 cnfld0 17843 . . . . . . . . . . . . . . . . . . . 20  |-  0  =  ( 0g ` fld )
1817subg0cl 15692 . . . . . . . . . . . . . . . . . . 19  |-  ( S  e.  (SubGrp ` fld )  ->  0  e.  S )
1916, 18syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( S  e.  (SubRing ` fld )  ->  0  e.  S )
2019adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
0  e.  S )
2120snssd 4021 . . . . . . . . . . . . . . . 16  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  ->  { 0 }  C_  S )
22 ssequn2 3532 . . . . . . . . . . . . . . . 16  |-  ( { 0 }  C_  S  <->  ( S  u.  { 0 } )  =  S )
2321, 22sylib 196 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( S  u.  {
0 } )  =  S )
24 feq3 5547 . . . . . . . . . . . . . . 15  |-  ( ( S  u.  { 0 } )  =  S  ->  ( a : NN0 --> ( S  u.  { 0 } )  <->  a : NN0
--> S ) )
2523, 24syl 16 . . . . . . . . . . . . . 14  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( a : NN0 --> ( S  u.  { 0 } )  <->  a : NN0
--> S ) )
2615, 25mpbid 210 . . . . . . . . . . . . 13  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
a : NN0 --> S )
27 elfznn0 11484 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0 ... n )  ->  k  e.  NN0 )
28 ffvelrn 5844 . . . . . . . . . . . . 13  |-  ( ( a : NN0 --> S  /\  k  e.  NN0 )  -> 
( a `  k
)  e.  S )
2926, 27, 28syl2an 477 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( a `  k )  e.  S
)
3013, 29sseldd 3360 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( a `  k )  e.  CC )
3130adantrl 715 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  ( z  e.  CC  /\  k  e.  ( 0 ... n ) ) )  ->  ( a `  k )  e.  CC )
32 simprl 755 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  ( z  e.  CC  /\  k  e.  ( 0 ... n ) ) )  ->  z  e.  CC )
3327ad2antll 728 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  ( z  e.  CC  /\  k  e.  ( 0 ... n ) ) )  ->  k  e.  NN0 )
34 expcl 11886 . . . . . . . . . . 11  |-  ( ( z  e.  CC  /\  k  e.  NN0 )  -> 
( z ^ k
)  e.  CC )
3532, 33, 34syl2anc 661 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  ( z  e.  CC  /\  k  e.  ( 0 ... n ) ) )  ->  ( z ^ k )  e.  CC )
3631, 35mulcld 9409 . . . . . . . . 9  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  ( z  e.  CC  /\  k  e.  ( 0 ... n ) ) )  ->  ( (
a `  k )  x.  ( z ^ k
) )  e.  CC )
37 eqid 2443 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... n )  |->  ( z  e.  CC  |->  ( ( a `  k )  x.  ( z ^
k ) ) ) )  =  ( k  e.  ( 0 ... n )  |->  ( z  e.  CC  |->  ( ( a `  k )  x.  ( z ^
k ) ) ) )
386mptex 5951 . . . . . . . . . . 11  |-  ( z  e.  CC  |->  ( ( a `  k )  x.  ( z ^
k ) ) )  e.  _V
3938a1i 11 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( z  e.  CC  |->  ( ( a `
 k )  x.  ( z ^ k
) ) )  e. 
_V )
40 fvex 5704 . . . . . . . . . . 11  |-  ( 0g
`  (fld 
^s 
CC ) )  e. 
_V
4140a1i 11 . . . . . . . . . 10  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( 0g `  (fld  ^s  CC ) )  e.  _V )
4237, 8, 39, 41fsuppmptdm 7634 . . . . . . . . 9  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( k  e.  ( 0 ... n ) 
|->  ( z  e.  CC  |->  ( ( a `  k )  x.  (
z ^ k ) ) ) ) finSupp  ( 0g `  (fld 
^s 
CC ) ) )
433, 4, 5, 7, 8, 11, 36, 42pwsgsum 16476 . . . . . . . 8  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( (fld 
^s 
CC )  gsumg  ( k  e.  ( 0 ... n ) 
|->  ( z  e.  CC  |->  ( ( a `  k )  x.  (
z ^ k ) ) ) ) )  =  ( z  e.  CC  |->  (fld 
gsumg  ( k  e.  ( 0 ... n ) 
|->  ( ( a `  k )  x.  (
z ^ k ) ) ) ) ) )
44 fzfid 11798 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  z  e.  CC )  ->  ( 0 ... n
)  e.  Fin )
4536anassrs 648 . . . . . . . . . 10  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  z  e.  CC )  /\  k  e.  (
0 ... n ) )  ->  ( ( a `
 k )  x.  ( z ^ k
) )  e.  CC )
4644, 45gsumfsum 17882 . . . . . . . . 9  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  z  e.  CC )  ->  (fld 
gsumg  ( k  e.  ( 0 ... n ) 
|->  ( ( a `  k )  x.  (
z ^ k ) ) ) )  = 
sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) )
4746mpteq2dva 4381 . . . . . . . 8  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( z  e.  CC  |->  (fld  gsumg  (
k  e.  ( 0 ... n )  |->  ( ( a `  k
)  x.  ( z ^ k ) ) ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) )
4843, 47eqtrd 2475 . . . . . . 7  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( (fld 
^s 
CC )  gsumg  ( k  e.  ( 0 ... n ) 
|->  ( z  e.  CC  |->  ( ( a `  k )  x.  (
z ^ k ) ) ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )
493pwsrng 16710 . . . . . . . . . 10  |-  ( (fld  e. 
Ring  /\  CC  e.  _V )  ->  (fld 
^s 
CC )  e.  Ring )
509, 6, 49mp2an 672 . . . . . . . . 9  |-  (fld  ^s  CC )  e.  Ring
51 rngcmn 16678 . . . . . . . . 9  |-  ( (fld  ^s  CC )  e.  Ring  ->  (fld  ^s  CC )  e. CMnd )
5250, 51mp1i 12 . . . . . . . 8  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
(fld  ^s  CC )  e. CMnd )
53 cncrng 17840 . . . . . . . . . . 11  |-fld  e.  CRing
54 plypf1.e . . . . . . . . . . . 12  |-  E  =  (eval1 ` fld )
55 eqid 2443 . . . . . . . . . . . 12  |-  (Poly1 ` fld )  =  (Poly1 ` fld )
5654, 55, 3, 4evl1rhm 17769 . . . . . . . . . . 11  |-  (fld  e.  CRing  ->  E  e.  ( (Poly1 ` fld ) RingHom  (fld  ^s  CC ) ) )
5753, 56ax-mp 5 . . . . . . . . . 10  |-  E  e.  ( (Poly1 ` fld ) RingHom  (fld 
^s 
CC ) )
58 plypf1.r . . . . . . . . . . . 12  |-  R  =  (flds  S )
59 plypf1.p . . . . . . . . . . . 12  |-  P  =  (Poly1 `  R )
60 plypf1.a . . . . . . . . . . . 12  |-  A  =  ( Base `  P
)
6155, 58, 59, 60subrgply1 17690 . . . . . . . . . . 11  |-  ( S  e.  (SubRing ` fld )  ->  A  e.  (SubRing `  (Poly1 ` fld ) ) )
6261adantr 465 . . . . . . . . . 10  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  ->  A  e.  (SubRing `  (Poly1 ` fld )
) )
63 rhmima 16899 . . . . . . . . . 10  |-  ( ( E  e.  ( (Poly1 ` fld ) RingHom 
(fld  ^s  CC ) )  /\  A  e.  (SubRing `  (Poly1 ` fld ) ) )  -> 
( E " A
)  e.  (SubRing `  (fld  ^s  CC ) ) )
6457, 62, 63sylancr 663 . . . . . . . . 9  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( E " A
)  e.  (SubRing `  (fld  ^s  CC ) ) )
65 subrgsubg 16874 . . . . . . . . 9  |-  ( ( E " A )  e.  (SubRing `  (fld  ^s  CC ) )  ->  ( E " A )  e.  (SubGrp `  (fld 
^s 
CC ) ) )
66 subgsubm 15706 . . . . . . . . 9  |-  ( ( E " A )  e.  (SubGrp `  (fld  ^s  CC ) )  ->  ( E " A )  e.  (SubMnd `  (fld 
^s 
CC ) ) )
6764, 65, 663syl 20 . . . . . . . 8  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( E " A
)  e.  (SubMnd `  (fld  ^s  CC ) ) )
68 eqid 2443 . . . . . . . . . . . 12  |-  ( Base `  (fld 
^s 
CC ) )  =  ( Base `  (fld  ^s  CC ) )
699a1i 11 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->fld 
e.  Ring )
706a1i 11 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  CC  e.  _V )
71 fconst6g 5602 . . . . . . . . . . . . . 14  |-  ( ( a `  k )  e.  CC  ->  ( CC  X.  { ( a `
 k ) } ) : CC --> CC )
7230, 71syl 16 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( CC  X.  { ( a `  k ) } ) : CC --> CC )
733, 4, 68pwselbasb 14429 . . . . . . . . . . . . . 14  |-  ( (fld  e. 
Ring  /\  CC  e.  _V )  ->  ( ( CC 
X.  { ( a `
 k ) } )  e.  ( Base `  (fld 
^s 
CC ) )  <->  ( CC  X.  { ( a `  k ) } ) : CC --> CC ) )
749, 6, 73mp2an 672 . . . . . . . . . . . . 13  |-  ( ( CC  X.  { ( a `  k ) } )  e.  (
Base `  (fld  ^s  CC )
)  <->  ( CC  X.  { ( a `  k ) } ) : CC --> CC )
7572, 74sylibr 212 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( CC  X.  { ( a `  k ) } )  e.  ( Base `  (fld  ^s  CC ) ) )
7635anass1rs 805 . . . . . . . . . . . . . 14  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  /\  z  e.  CC )  ->  ( z ^
k )  e.  CC )
77 eqid 2443 . . . . . . . . . . . . . 14  |-  ( z  e.  CC  |->  ( z ^ k ) )  =  ( z  e.  CC  |->  ( z ^
k ) )
7876, 77fmptd 5870 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( z  e.  CC  |->  ( z ^
k ) ) : CC --> CC )
793, 4, 68pwselbasb 14429 . . . . . . . . . . . . . 14  |-  ( (fld  e. 
Ring  /\  CC  e.  _V )  ->  ( ( z  e.  CC  |->  ( z ^ k ) )  e.  ( Base `  (fld  ^s  CC ) )  <->  ( z  e.  CC  |->  ( z ^
k ) ) : CC --> CC ) )
809, 6, 79mp2an 672 . . . . . . . . . . . . 13  |-  ( ( z  e.  CC  |->  ( z ^ k ) )  e.  ( Base `  (fld 
^s 
CC ) )  <->  ( z  e.  CC  |->  ( z ^
k ) ) : CC --> CC )
8178, 80sylibr 212 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( z  e.  CC  |->  ( z ^
k ) )  e.  ( Base `  (fld  ^s  CC ) ) )
82 cnfldmul 17827 . . . . . . . . . . . 12  |-  x.  =  ( .r ` fld )
83 eqid 2443 . . . . . . . . . . . 12  |-  ( .r
`  (fld 
^s 
CC ) )  =  ( .r `  (fld  ^s  CC ) )
843, 68, 69, 70, 75, 81, 82, 83pwsmulrval 14432 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( ( CC 
X.  { ( a `
 k ) } ) ( .r `  (fld  ^s  CC ) ) ( z  e.  CC  |->  ( z ^ k ) ) )  =  ( ( CC  X.  { ( a `  k ) } )  oF  x.  ( z  e.  CC  |->  ( z ^
k ) ) ) )
8530adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  /\  z  e.  CC )  ->  ( a `  k )  e.  CC )
86 fconstmpt 4885 . . . . . . . . . . . . 13  |-  ( CC 
X.  { ( a `
 k ) } )  =  ( z  e.  CC  |->  ( a `
 k ) )
8786a1i 11 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( CC  X.  { ( a `  k ) } )  =  ( z  e.  CC  |->  ( a `  k ) ) )
88 eqidd 2444 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( z  e.  CC  |->  ( z ^
k ) )  =  ( z  e.  CC  |->  ( z ^ k
) ) )
8970, 85, 76, 87, 88offval2 6339 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( ( CC 
X.  { ( a `
 k ) } )  oF  x.  ( z  e.  CC  |->  ( z ^ k
) ) )  =  ( z  e.  CC  |->  ( ( a `  k )  x.  (
z ^ k ) ) ) )
9084, 89eqtrd 2475 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( ( CC 
X.  { ( a `
 k ) } ) ( .r `  (fld  ^s  CC ) ) ( z  e.  CC  |->  ( z ^ k ) ) )  =  ( z  e.  CC  |->  ( ( a `  k )  x.  ( z ^
k ) ) ) )
9164adantr 465 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( E " A )  e.  (SubRing `  (fld 
^s 
CC ) ) )
92 eqid 2443 . . . . . . . . . . . . . 14  |-  (algSc `  (Poly1 ` fld ) )  =  (algSc `  (Poly1 ` fld ) )
9354, 55, 4, 92evl1sca 17771 . . . . . . . . . . . . 13  |-  ( (fld  e. 
CRing  /\  ( a `  k )  e.  CC )  ->  ( E `  ( (algSc `  (Poly1 ` fld ) ) `  (
a `  k )
) )  =  ( CC  X.  { ( a `  k ) } ) )
9453, 30, 93sylancr 663 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( E `  ( (algSc `  (Poly1 ` fld ) ) `  (
a `  k )
) )  =  ( CC  X.  { ( a `  k ) } ) )
95 eqid 2443 . . . . . . . . . . . . . . . 16  |-  ( Base `  (Poly1 ` fld ) )  =  (
Base `  (Poly1 ` fld ) )
9695, 68rhmf 16819 . . . . . . . . . . . . . . 15  |-  ( E  e.  ( (Poly1 ` fld ) RingHom  (fld 
^s 
CC ) )  ->  E : ( Base `  (Poly1 ` fld )
) --> ( Base `  (fld  ^s  CC ) ) )
9757, 96ax-mp 5 . . . . . . . . . . . . . 14  |-  E :
( Base `  (Poly1 ` fld ) ) --> ( Base `  (fld 
^s 
CC ) )
98 ffn 5562 . . . . . . . . . . . . . 14  |-  ( E : ( Base `  (Poly1 ` fld )
) --> ( Base `  (fld  ^s  CC ) )  ->  E  Fn  ( Base `  (Poly1 ` fld ) ) )
9997, 98mp1i 12 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  E  Fn  ( Base `  (Poly1 ` fld ) ) )
10095subrgss 16869 . . . . . . . . . . . . . . 15  |-  ( A  e.  (SubRing `  (Poly1 ` fld )
)  ->  A  C_  ( Base `  (Poly1 ` fld ) ) )
10161, 100syl 16 . . . . . . . . . . . . . 14  |-  ( S  e.  (SubRing ` fld )  ->  A  C_  ( Base `  (Poly1 ` fld ) ) )
102101ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  A  C_  ( Base `  (Poly1 ` fld ) ) )
103 simpll 753 . . . . . . . . . . . . . . 15  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  S  e.  (SubRing ` fld ) )
10455, 92, 58, 59, 103, 60, 4, 30subrg1asclcl 17717 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( ( (algSc `  (Poly1 ` fld ) ) `  (
a `  k )
)  e.  A  <->  ( a `  k )  e.  S
) )
10529, 104mpbird 232 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( (algSc `  (Poly1 ` fld ) ) `  (
a `  k )
)  e.  A )
106 fnfvima 5958 . . . . . . . . . . . . 13  |-  ( ( E  Fn  ( Base `  (Poly1 ` fld ) )  /\  A  C_  ( Base `  (Poly1 ` fld )
)  /\  ( (algSc `  (Poly1 ` fld ) ) `  (
a `  k )
)  e.  A )  ->  ( E `  ( (algSc `  (Poly1 ` fld ) ) `  (
a `  k )
) )  e.  ( E " A ) )
10799, 102, 105, 106syl3anc 1218 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( E `  ( (algSc `  (Poly1 ` fld ) ) `  (
a `  k )
) )  e.  ( E " A ) )
10894, 107eqeltrrd 2518 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( CC  X.  { ( a `  k ) } )  e.  ( E " A ) )
10968subrgss 16869 . . . . . . . . . . . . . . . . 17  |-  ( ( E " A )  e.  (SubRing `  (fld  ^s  CC ) )  ->  ( E " A )  C_  ( Base `  (fld 
^s 
CC ) ) )
11091, 109syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( E " A )  C_  ( Base `  (fld 
^s 
CC ) ) )
11161ad2antrr 725 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  A  e.  (SubRing `  (Poly1 ` fld ) ) )
112 eqid 2443 . . . . . . . . . . . . . . . . . . . 20  |-  (mulGrp `  (Poly1 ` fld ) )  =  (mulGrp `  (Poly1 ` fld ) )
113112subrgsubm 16881 . . . . . . . . . . . . . . . . . . 19  |-  ( A  e.  (SubRing `  (Poly1 ` fld )
)  ->  A  e.  (SubMnd `  (mulGrp `  (Poly1 ` fld )
) ) )
114111, 113syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  A  e.  (SubMnd `  (mulGrp `  (Poly1 ` fld ) ) ) )
11527adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  k  e.  NN0 )
116 eqid 2443 . . . . . . . . . . . . . . . . . . 19  |-  (var1 ` fld )  =  (var1 ` fld )
117116, 103, 58, 59, 60subrgvr1cl 17719 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  (var1 ` fld )  e.  A )
118 eqid 2443 . . . . . . . . . . . . . . . . . . 19  |-  (.g `  (mulGrp `  (Poly1 ` fld ) ) )  =  (.g `  (mulGrp `  (Poly1 ` fld )
) )
119118submmulgcl 15664 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  (SubMnd `  (mulGrp `  (Poly1 ` fld ) ) )  /\  k  e.  NN0  /\  (var1 ` fld )  e.  A )  ->  (
k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) )  e.  A
)
120114, 115, 117, 119syl3anc 1218 . . . . . . . . . . . . . . . . 17  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) )  e.  A )
121 fnfvima 5958 . . . . . . . . . . . . . . . . 17  |-  ( ( E  Fn  ( Base `  (Poly1 ` fld ) )  /\  A  C_  ( Base `  (Poly1 ` fld )
)  /\  ( k
(.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) )  e.  A
)  ->  ( E `  ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) )  e.  ( E " A ) )
12299, 102, 120, 121syl3anc 1218 . . . . . . . . . . . . . . . 16  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( E `  ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) )  e.  ( E " A ) )
123110, 122sseldd 3360 . . . . . . . . . . . . . . 15  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( E `  ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) )  e.  (
Base `  (fld  ^s  CC )
) )
1243, 4, 68, 69, 70, 123pwselbas 14430 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( E `  ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) : CC --> CC )
125124feqmptd 5747 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( E `  ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) )  =  ( z  e.  CC  |->  ( ( E `  (
k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) `  z ) ) )
12653a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  /\  z  e.  CC )  ->fld 
e.  CRing )
127 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  /\  z  e.  CC )  ->  z  e.  CC )
12854, 116, 4, 55, 95, 126, 127evl1vard 17774 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  /\  z  e.  CC )  ->  ( (var1 ` fld )  e.  ( Base `  (Poly1 ` fld ) )  /\  (
( E `  (var1 ` fld )
) `  z )  =  z ) )
129 eqid 2443 . . . . . . . . . . . . . . . . 17  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
130115adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  /\  z  e.  CC )  ->  k  e.  NN0 )
13154, 55, 4, 95, 126, 127, 128, 118, 129, 130evl1expd 17782 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  /\  z  e.  CC )  ->  ( ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) )  e.  (
Base `  (Poly1 ` fld ) )  /\  (
( E `  (
k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) `  z )  =  ( k (.g `  (mulGrp ` fld ) ) z ) ) )
132131simprd 463 . . . . . . . . . . . . . . 15  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  /\  z  e.  CC )  ->  ( ( E `
 ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) `  z
)  =  ( k (.g `  (mulGrp ` fld ) ) z ) )
133 cnfldexp 17852 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  CC  /\  k  e.  NN0 )  -> 
( k (.g `  (mulGrp ` fld ) ) z )  =  ( z ^ k
) )
134127, 130, 133syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  /\  z  e.  CC )  ->  ( k (.g `  (mulGrp ` fld ) ) z )  =  ( z ^
k ) )
135132, 134eqtrd 2475 . . . . . . . . . . . . . 14  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  /\  z  e.  CC )  ->  ( ( E `
 ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) `  z
)  =  ( z ^ k ) )
136135mpteq2dva 4381 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( z  e.  CC  |->  ( ( E `
 ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) `  z
) )  =  ( z  e.  CC  |->  ( z ^ k ) ) )
137125, 136eqtrd 2475 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( E `  ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) )  =  ( z  e.  CC  |->  ( z ^ k ) ) )
138137, 122eqeltrrd 2518 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( z  e.  CC  |->  ( z ^
k ) )  e.  ( E " A
) )
13983subrgmcl 16880 . . . . . . . . . . 11  |-  ( ( ( E " A
)  e.  (SubRing `  (fld  ^s  CC ) )  /\  ( CC 
X.  { ( a `
 k ) } )  e.  ( E
" A )  /\  ( z  e.  CC  |->  ( z ^ k
) )  e.  ( E " A ) )  ->  ( ( CC  X.  { ( a `
 k ) } ) ( .r `  (fld  ^s  CC ) ) ( z  e.  CC  |->  ( z ^ k ) ) )  e.  ( E
" A ) )
14091, 108, 138, 139syl3anc 1218 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( ( CC 
X.  { ( a `
 k ) } ) ( .r `  (fld  ^s  CC ) ) ( z  e.  CC  |->  ( z ^ k ) ) )  e.  ( E
" A ) )
14190, 140eqeltrrd 2518 . . . . . . . . 9  |-  ( ( ( S  e.  (SubRing ` fld )  /\  ( n  e. 
NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  k  e.  ( 0 ... n ) )  ->  ( z  e.  CC  |->  ( ( a `
 k )  x.  ( z ^ k
) ) )  e.  ( E " A
) )
142141, 37fmptd 5870 . . . . . . . 8  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( k  e.  ( 0 ... n ) 
|->  ( z  e.  CC  |->  ( ( a `  k )  x.  (
z ^ k ) ) ) ) : ( 0 ... n
) --> ( E " A ) )
14337, 8, 141, 41fsuppmptdm 7634 . . . . . . . 8  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( k  e.  ( 0 ... n ) 
|->  ( z  e.  CC  |->  ( ( a `  k )  x.  (
z ^ k ) ) ) ) finSupp  ( 0g `  (fld 
^s 
CC ) ) )
1445, 52, 8, 67, 142, 143gsumsubmcl 16407 . . . . . . 7  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( (fld 
^s 
CC )  gsumg  ( k  e.  ( 0 ... n ) 
|->  ( z  e.  CC  |->  ( ( a `  k )  x.  (
z ^ k ) ) ) ) )  e.  ( E " A ) )
14548, 144eqeltrrd 2518 . . . . . 6  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )  e.  ( E
" A ) )
146 eleq1 2503 . . . . . 6  |-  ( f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) )  ->  (
f  e.  ( E
" A )  <->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) )  e.  ( E " A ) ) )
147145, 146syl5ibrcom 222 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )  ->  f  e.  ( E " A ) ) )
148147rexlimdvva 2851 . . . 4  |-  ( S  e.  (SubRing ` fld )  ->  ( E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )  ->  f  e.  ( E " A ) ) )
1492, 148syl5 32 . . 3  |-  ( S  e.  (SubRing ` fld )  ->  ( f  e.  (Poly `  S
)  ->  f  e.  ( E " A ) ) )
150 ffun 5564 . . . . . 6  |-  ( E : ( Base `  (Poly1 ` fld )
) --> ( Base `  (fld  ^s  CC ) )  ->  Fun  E )
15197, 150ax-mp 5 . . . . 5  |-  Fun  E
152 fvelima 5746 . . . . 5  |-  ( ( Fun  E  /\  f  e.  ( E " A
) )  ->  E. a  e.  A  ( E `  a )  =  f )
153151, 152mpan 670 . . . 4  |-  ( f  e.  ( E " A )  ->  E. a  e.  A  ( E `  a )  =  f )
154101sselda 3359 . . . . . . . . . . 11  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  a  e.  ( Base `  (Poly1 ` fld )
) )
155 eqid 2443 . . . . . . . . . . . 12  |-  ( .s
`  (Poly1 ` fld ) )  =  ( .s `  (Poly1 ` fld ) )
156 eqid 2443 . . . . . . . . . . . 12  |-  (coe1 `  a
)  =  (coe1 `  a
)
15755, 116, 95, 155, 112, 118, 156ply1coe 17749 . . . . . . . . . . 11  |-  ( (fld  e. 
Ring  /\  a  e.  (
Base `  (Poly1 ` fld ) ) )  -> 
a  =  ( (Poly1 ` fld ) 
gsumg  ( k  e.  NN0  |->  ( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) ) ) )
1589, 154, 157sylancr 663 . . . . . . . . . 10  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  a  =  ( (Poly1 ` fld )  gsumg  ( k  e.  NN0  |->  ( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) ) ) )
159158fveq2d 5698 . . . . . . . . 9  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  ( E `  a )  =  ( E `  ( (Poly1 ` fld )  gsumg  ( k  e.  NN0  |->  ( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) ) ) ) )
160 eqid 2443 . . . . . . . . . 10  |-  ( 0g
`  (Poly1 ` fld ) )  =  ( 0g `  (Poly1 ` fld ) )
16155ply1rng 17706 . . . . . . . . . . . 12  |-  (fld  e.  Ring  -> 
(Poly1 `
fld )  e.  Ring )
1629, 161ax-mp 5 . . . . . . . . . . 11  |-  (Poly1 ` fld )  e.  Ring
163 rngcmn 16678 . . . . . . . . . . 11  |-  ( (Poly1 ` fld )  e.  Ring  ->  (Poly1 ` fld )  e. CMnd )
164162, 163mp1i 12 . . . . . . . . . 10  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (Poly1 ` fld )  e. CMnd )
165 rngmnd 16657 . . . . . . . . . . 11  |-  ( (fld  ^s  CC )  e.  Ring  ->  (fld  ^s  CC )  e.  Mnd )
16650, 165mp1i 12 . . . . . . . . . 10  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (fld  ^s  CC )  e.  Mnd )
167 nn0ex 10588 . . . . . . . . . . 11  |-  NN0  e.  _V
168167a1i 11 . . . . . . . . . 10  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  NN0  e.  _V )
169 rhmghm 16818 . . . . . . . . . . . 12  |-  ( E  e.  ( (Poly1 ` fld ) RingHom  (fld 
^s 
CC ) )  ->  E  e.  ( (Poly1 ` fld )  GrpHom  (fld 
^s 
CC ) ) )
17057, 169ax-mp 5 . . . . . . . . . . 11  |-  E  e.  ( (Poly1 ` fld )  GrpHom  (fld 
^s 
CC ) )
171 ghmmhm 15760 . . . . . . . . . . 11  |-  ( E  e.  ( (Poly1 ` fld )  GrpHom  (fld  ^s  CC ) )  ->  E  e.  ( (Poly1 ` fld ) MndHom  (fld 
^s 
CC ) ) )
172170, 171mp1i 12 . . . . . . . . . 10  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  E  e.  ( (Poly1 ` fld ) MndHom  (fld 
^s 
CC ) ) )
17355ply1lmod 17710 . . . . . . . . . . . . 13  |-  (fld  e.  Ring  -> 
(Poly1 `
fld )  e.  LMod )
1749, 173mp1i 12 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  NN0 )  ->  (Poly1 ` fld )  e.  LMod )
17512ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  NN0 )  ->  S  C_  CC )
176 eqid 2443 . . . . . . . . . . . . . . . . 17  |-  ( Base `  R )  =  (
Base `  R )
177156, 60, 59, 176coe1f 17670 . . . . . . . . . . . . . . . 16  |-  ( a  e.  A  ->  (coe1 `  a ) : NN0 --> (
Base `  R )
)
17858subrgbas 16877 . . . . . . . . . . . . . . . . 17  |-  ( S  e.  (SubRing ` fld )  ->  S  =  ( Base `  R
) )
179 feq3 5547 . . . . . . . . . . . . . . . . 17  |-  ( S  =  ( Base `  R
)  ->  ( (coe1 `  a ) : NN0 --> S  <-> 
(coe1 `  a ) : NN0 --> ( Base `  R
) ) )
180178, 179syl 16 . . . . . . . . . . . . . . . 16  |-  ( S  e.  (SubRing ` fld )  ->  ( (coe1 `  a ) : NN0 --> S  <-> 
(coe1 `  a ) : NN0 --> ( Base `  R
) ) )
181177, 180syl5ibr 221 . . . . . . . . . . . . . . 15  |-  ( S  e.  (SubRing ` fld )  ->  ( a  e.  A  ->  (coe1 `  a ) : NN0 --> S ) )
182181imp 429 . . . . . . . . . . . . . 14  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (coe1 `  a ) : NN0 --> S )
183182ffvelrnda 5846 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  NN0 )  ->  ( (coe1 `  a ) `  k
)  e.  S )
184175, 183sseldd 3360 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  NN0 )  ->  ( (coe1 `  a ) `  k
)  e.  CC )
185112rngmgp 16654 . . . . . . . . . . . . . 14  |-  ( (Poly1 ` fld )  e.  Ring  ->  (mulGrp `  (Poly1 ` fld ) )  e.  Mnd )
186162, 185mp1i 12 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  NN0 )  ->  (mulGrp `  (Poly1 ` fld )
)  e.  Mnd )
187 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  NN0 )  ->  k  e.  NN0 )
188116, 55, 95vr1cl 17674 . . . . . . . . . . . . . 14  |-  (fld  e.  Ring  -> 
(var1 ` fld )  e.  ( Base `  (Poly1 ` fld ) ) )
1899, 188mp1i 12 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  NN0 )  ->  (var1 ` fld )  e.  ( Base `  (Poly1 ` fld ) ) )
190112, 95mgpbas 16600 . . . . . . . . . . . . . 14  |-  ( Base `  (Poly1 ` fld ) )  =  (
Base `  (mulGrp `  (Poly1 ` fld )
) )
191190, 118mulgnn0cl 15646 . . . . . . . . . . . . 13  |-  ( ( (mulGrp `  (Poly1 ` fld ) )  e.  Mnd  /\  k  e.  NN0  /\  (var1 ` fld )  e.  ( Base `  (Poly1 ` fld ) ) )  -> 
( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) )  e.  ( Base `  (Poly1 ` fld ) ) )
192186, 187, 189, 191syl3anc 1218 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  NN0 )  ->  ( k
(.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) )  e.  (
Base `  (Poly1 ` fld ) ) )
19355ply1sca 17711 . . . . . . . . . . . . . 14  |-  (fld  e.  Ring  ->fld  =  (Scalar `  (Poly1 ` fld ) ) )
1949, 193ax-mp 5 . . . . . . . . . . . . 13  |-fld  =  (Scalar `  (Poly1 ` fld )
)
19595, 194, 155, 4lmodvscl 16968 . . . . . . . . . . . 12  |-  ( ( (Poly1 ` fld )  e.  LMod  /\  (
(coe1 `  a ) `  k )  e.  CC  /\  ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) )  e.  ( Base `  (Poly1 ` fld ) ) )  -> 
( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) )  e.  (
Base `  (Poly1 ` fld ) ) )
196174, 184, 192, 195syl3anc 1218 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  NN0 )  ->  ( (
(coe1 `  a ) `  k ) ( .s
`  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) )  e.  ( Base `  (Poly1 ` fld )
) )
197 eqid 2443 . . . . . . . . . . 11  |-  ( k  e.  NN0  |->  ( ( (coe1 `  a ) `  k ) ( .s
`  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) )  =  ( k  e. 
NN0  |->  ( ( (coe1 `  a ) `  k
) ( .s `  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) )
198196, 197fmptd 5870 . . . . . . . . . 10  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
k  e.  NN0  |->  ( ( (coe1 `  a ) `  k ) ( .s
`  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) ) : NN0 --> ( Base `  (Poly1 ` fld ) ) )
199167mptex 5951 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  |->  ( ( (coe1 `  a ) `  k ) ( .s
`  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) )  e.  _V
200 funmpt 5457 . . . . . . . . . . . . 13  |-  Fun  (
k  e.  NN0  |->  ( ( (coe1 `  a ) `  k ) ( .s
`  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) )
201 fvex 5704 . . . . . . . . . . . . 13  |-  ( 0g
`  (Poly1 ` fld ) )  e.  _V
202199, 200, 2013pm3.2i 1166 . . . . . . . . . . . 12  |-  ( ( k  e.  NN0  |->  ( ( (coe1 `  a ) `  k ) ( .s
`  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) )  e.  _V  /\  Fun  ( k  e.  NN0  |->  ( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) )  /\  ( 0g `  (Poly1 ` fld ) )  e.  _V )
203202a1i 11 . . . . . . . . . . 11  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
( k  e.  NN0  |->  ( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) )  e. 
_V  /\  Fun  ( k  e.  NN0  |->  ( ( (coe1 `  a ) `  k ) ( .s
`  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) )  /\  ( 0g `  (Poly1 ` fld ) )  e.  _V ) )
204156, 95, 55, 17coe1sfi 17671 . . . . . . . . . . . . 13  |-  ( a  e.  ( Base `  (Poly1 ` fld )
)  ->  (coe1 `  a
) finSupp  0 )
205154, 204syl 16 . . . . . . . . . . . 12  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (coe1 `  a ) finSupp  0 )
206205fsuppimpd 7630 . . . . . . . . . . 11  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
(coe1 `  a ) supp  0
)  e.  Fin )
207182feqmptd 5747 . . . . . . . . . . . . . 14  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (coe1 `  a )  =  ( k  e.  NN0  |->  ( (coe1 `  a ) `  k
) ) )
208207oveq1d 6109 . . . . . . . . . . . . 13  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
(coe1 `  a ) supp  0
)  =  ( ( k  e.  NN0  |->  ( (coe1 `  a ) `  k
) ) supp  0 ) )
209 eqimss2 3412 . . . . . . . . . . . . 13  |-  ( ( (coe1 `  a ) supp  0
)  =  ( ( k  e.  NN0  |->  ( (coe1 `  a ) `  k
) ) supp  0 )  ->  ( ( k  e.  NN0  |->  ( (coe1 `  a ) `  k
) ) supp  0 ) 
C_  ( (coe1 `  a
) supp  0 ) )
210208, 209syl 16 . . . . . . . . . . . 12  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
( k  e.  NN0  |->  ( (coe1 `  a ) `  k ) ) supp  0
)  C_  ( (coe1 `  a ) supp  0 ) )
2119, 173mp1i 12 . . . . . . . . . . . . 13  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (Poly1 ` fld )  e.  LMod )
21295, 194, 155, 17, 160lmod0vs 16984 . . . . . . . . . . . . 13  |-  ( ( (Poly1 ` fld )  e.  LMod  /\  x  e.  ( Base `  (Poly1 ` fld )
) )  ->  (
0 ( .s `  (Poly1 ` fld ) ) x )  =  ( 0g `  (Poly1 ` fld ) ) )
213211, 212sylan 471 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  x  e.  ( Base `  (Poly1 ` fld ) ) )  -> 
( 0 ( .s
`  (Poly1 ` fld ) ) x )  =  ( 0g `  (Poly1 ` fld ) ) )
214 c0ex 9383 . . . . . . . . . . . . 13  |-  0  e.  _V
215214a1i 11 . . . . . . . . . . . 12  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  0  e.  _V )
216210, 213, 183, 192, 215suppssov1 6724 . . . . . . . . . . 11  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
( k  e.  NN0  |->  ( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) ) supp  ( 0g `  (Poly1 ` fld ) ) )  C_  ( (coe1 `  a ) supp  0
) )
217 suppssfifsupp 7638 . . . . . . . . . . 11  |-  ( ( ( ( k  e. 
NN0  |->  ( ( (coe1 `  a ) `  k
) ( .s `  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) )  e.  _V  /\  Fun  ( k  e.  NN0  |->  ( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) )  /\  ( 0g `  (Poly1 ` fld ) )  e.  _V )  /\  ( ( (coe1 `  a ) supp  0 )  e.  Fin  /\  (
( k  e.  NN0  |->  ( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) ) supp  ( 0g `  (Poly1 ` fld ) ) )  C_  ( (coe1 `  a ) supp  0
) ) )  -> 
( k  e.  NN0  |->  ( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) ) finSupp  ( 0g `  (Poly1 ` fld ) ) )
218203, 206, 216, 217syl12anc 1216 . . . . . . . . . 10  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
k  e.  NN0  |->  ( ( (coe1 `  a ) `  k ) ( .s
`  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) ) finSupp 
( 0g `  (Poly1 ` fld )
) )
21995, 160, 164, 166, 168, 172, 198, 218gsummhm 16435 . . . . . . . . 9  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
(fld  ^s  CC )  gsumg  ( E  o.  (
k  e.  NN0  |->  ( ( (coe1 `  a ) `  k ) ( .s
`  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) ) ) )  =  ( E `  ( (Poly1 ` fld ) 
gsumg  ( k  e.  NN0  |->  ( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) ) ) ) )
220 eqidd 2444 . . . . . . . . . . . 12  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
k  e.  NN0  |->  ( ( (coe1 `  a ) `  k ) ( .s
`  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) )  =  ( k  e. 
NN0  |->  ( ( (coe1 `  a ) `  k
) ( .s `  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) ) )
22197a1i 11 . . . . . . . . . . . . 13  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  E : ( Base `  (Poly1 ` fld )
) --> ( Base `  (fld  ^s  CC ) ) )
222221feqmptd 5747 . . . . . . . . . . . 12  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  E  =  ( x  e.  ( Base `  (Poly1 ` fld )
)  |->  ( E `  x ) ) )
223 fveq2 5694 . . . . . . . . . . . 12  |-  ( x  =  ( ( (coe1 `  a ) `  k
) ( .s `  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) )  -> 
( E `  x
)  =  ( E `
 ( ( (coe1 `  a ) `  k
) ( .s `  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) ) )
224196, 220, 222, 223fmptco 5879 . . . . . . . . . . 11  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  ( E  o.  ( k  e.  NN0  |->  ( ( (coe1 `  a ) `  k
) ( .s `  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) ) )  =  ( k  e.  NN0  |->  ( E `
 ( ( (coe1 `  a ) `  k
) ( .s `  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) ) ) )
2259a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  NN0 )  ->fld  e.  Ring )
2266a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  NN0 )  ->  CC  e.  _V )
22797ffvelrni 5845 . . . . . . . . . . . . . . . 16  |-  ( ( ( (coe1 `  a ) `  k ) ( .s
`  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) )  e.  ( Base `  (Poly1 ` fld )
)  ->  ( E `  ( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) )  e.  ( Base `  (fld  ^s  CC ) ) )
228196, 227syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  NN0 )  ->  ( E `  ( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) )  e.  ( Base `  (fld  ^s  CC ) ) )
2293, 4, 68, 225, 226, 228pwselbas 14430 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  NN0 )  ->  ( E `  ( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) ) : CC --> CC )
230229feqmptd 5747 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  NN0 )  ->  ( E `  ( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) )  =  ( z  e.  CC  |->  ( ( E `  ( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) ) `  z ) ) )
23153a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  k  e.  NN0 )  /\  z  e.  CC )  ->fld  e.  CRing )
232 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  k  e.  NN0 )  /\  z  e.  CC )  ->  z  e.  CC )
23354, 116, 4, 55, 95, 231, 232evl1vard 17774 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  k  e.  NN0 )  /\  z  e.  CC )  ->  (
(var1 ` fld )  e.  ( Base `  (Poly1 ` fld ) )  /\  (
( E `  (var1 ` fld )
) `  z )  =  z ) )
234187adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  k  e.  NN0 )  /\  z  e.  CC )  ->  k  e.  NN0 )
23554, 55, 4, 95, 231, 232, 233, 118, 129, 234evl1expd 17782 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  k  e.  NN0 )  /\  z  e.  CC )  ->  (
( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) )  e.  ( Base `  (Poly1 ` fld ) )  /\  (
( E `  (
k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) `  z )  =  ( k (.g `  (mulGrp ` fld ) ) z ) ) )
236232, 234, 133syl2anc 661 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  k  e.  NN0 )  /\  z  e.  CC )  ->  (
k (.g `  (mulGrp ` fld ) ) z )  =  ( z ^
k ) )
237236eqeq2d 2454 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  k  e.  NN0 )  /\  z  e.  CC )  ->  (
( ( E `  ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) `  z
)  =  ( k (.g `  (mulGrp ` fld ) ) z )  <-> 
( ( E `  ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) `  z
)  =  ( z ^ k ) ) )
238237anbi2d 703 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  k  e.  NN0 )  /\  z  e.  CC )  ->  (
( ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) )  e.  ( Base `  (Poly1 ` fld ) )  /\  (
( E `  (
k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) `  z )  =  ( k (.g `  (mulGrp ` fld ) ) z ) )  <->  ( ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) )  e.  (
Base `  (Poly1 ` fld ) )  /\  (
( E `  (
k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) `  z )  =  ( z ^ k ) ) ) )
239235, 238mpbid 210 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  k  e.  NN0 )  /\  z  e.  CC )  ->  (
( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) )  e.  ( Base `  (Poly1 ` fld ) )  /\  (
( E `  (
k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) `  z )  =  ( z ^ k ) ) )
240184adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  k  e.  NN0 )  /\  z  e.  CC )  ->  (
(coe1 `  a ) `  k )  e.  CC )
24154, 55, 4, 95, 231, 232, 239, 240, 155, 82evl1vsd 17781 . . . . . . . . . . . . . . 15  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  k  e.  NN0 )  /\  z  e.  CC )  ->  (
( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) )  e.  (
Base `  (Poly1 ` fld ) )  /\  (
( E `  (
( (coe1 `  a ) `  k ) ( .s
`  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) ) `
 z )  =  ( ( (coe1 `  a
) `  k )  x.  ( z ^ k
) ) ) )
242241simprd 463 . . . . . . . . . . . . . 14  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  k  e.  NN0 )  /\  z  e.  CC )  ->  (
( E `  (
( (coe1 `  a ) `  k ) ( .s
`  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) ) `
 z )  =  ( ( (coe1 `  a
) `  k )  x.  ( z ^ k
) ) )
243242mpteq2dva 4381 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  NN0 )  ->  ( z  e.  CC  |->  ( ( E `
 ( ( (coe1 `  a ) `  k
) ( .s `  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) ) `
 z ) )  =  ( z  e.  CC  |->  ( ( (coe1 `  a ) `  k
)  x.  ( z ^ k ) ) ) )
244230, 243eqtrd 2475 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  NN0 )  ->  ( E `  ( ( (coe1 `  a
) `  k )
( .s `  (Poly1 ` fld )
) ( k (.g `  (mulGrp `  (Poly1 ` fld ) ) ) (var1 ` fld ) ) ) )  =  ( z  e.  CC  |->  ( ( (coe1 `  a
) `  k )  x.  ( z ^ k
) ) ) )
245244mpteq2dva 4381 . . . . . . . . . . 11  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
k  e.  NN0  |->  ( E `
 ( ( (coe1 `  a ) `  k
) ( .s `  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) ) )  =  ( k  e.  NN0  |->  ( z  e.  CC  |->  ( ( (coe1 `  a ) `  k )  x.  (
z ^ k ) ) ) ) )
246224, 245eqtrd 2475 . . . . . . . . . 10  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  ( E  o.  ( k  e.  NN0  |->  ( ( (coe1 `  a ) `  k
) ( .s `  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) ) )  =  ( k  e.  NN0  |->  ( z  e.  CC  |->  ( ( (coe1 `  a ) `  k )  x.  (
z ^ k ) ) ) ) )
247246oveq2d 6110 . . . . . . . . 9  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
(fld  ^s  CC )  gsumg  ( E  o.  (
k  e.  NN0  |->  ( ( (coe1 `  a ) `  k ) ( .s
`  (Poly1 ` fld ) ) ( k (.g `  (mulGrp `  (Poly1 ` fld )
) ) (var1 ` fld ) ) ) ) ) )  =  ( (fld 
^s 
CC )  gsumg  ( k  e.  NN0  |->  ( z  e.  CC  |->  ( ( (coe1 `  a
) `  k )  x.  ( z ^ k
) ) ) ) ) )
248159, 219, 2473eqtr2d 2481 . . . . . . . 8  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  ( E `  a )  =  ( (fld  ^s  CC ) 
gsumg  ( k  e.  NN0  |->  ( z  e.  CC  |->  ( ( (coe1 `  a
) `  k )  x.  ( z ^ k
) ) ) ) ) )
2496a1i 11 . . . . . . . . 9  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  CC  e.  _V )
2509, 10mp1i 12 . . . . . . . . 9  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->fld  e. CMnd )
251184adantlr 714 . . . . . . . . . . 11  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  NN0 )  ->  (
(coe1 `  a ) `  k )  e.  CC )
25234adantll 713 . . . . . . . . . . 11  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  NN0 )  ->  (
z ^ k )  e.  CC )
253251, 252mulcld 9409 . . . . . . . . . 10  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  NN0 )  ->  (
( (coe1 `  a ) `  k )  x.  (
z ^ k ) )  e.  CC )
254253anasss 647 . . . . . . . . 9  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  ( z  e.  CC  /\  k  e. 
NN0 ) )  -> 
( ( (coe1 `  a
) `  k )  x.  ( z ^ k
) )  e.  CC )
255167mptex 5951 . . . . . . . . . . . 12  |-  ( k  e.  NN0  |->  ( z  e.  CC  |->  ( ( (coe1 `  a ) `  k )  x.  (
z ^ k ) ) ) )  e. 
_V
256 funmpt 5457 . . . . . . . . . . . 12  |-  Fun  (
k  e.  NN0  |->  ( z  e.  CC  |->  ( ( (coe1 `  a ) `  k )  x.  (
z ^ k ) ) ) )
257255, 256, 403pm3.2i 1166 . . . . . . . . . . 11  |-  ( ( k  e.  NN0  |->  ( z  e.  CC  |->  ( ( (coe1 `  a ) `  k )  x.  (
z ^ k ) ) ) )  e. 
_V  /\  Fun  ( k  e.  NN0  |->  ( z  e.  CC  |->  ( ( (coe1 `  a ) `  k )  x.  (
z ^ k ) ) ) )  /\  ( 0g `  (fld  ^s  CC ) )  e.  _V )
258257a1i 11 . . . . . . . . . 10  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
( k  e.  NN0  |->  ( z  e.  CC  |->  ( ( (coe1 `  a
) `  k )  x.  ( z ^ k
) ) ) )  e.  _V  /\  Fun  ( k  e.  NN0  |->  ( z  e.  CC  |->  ( ( (coe1 `  a
) `  k )  x.  ( z ^ k
) ) ) )  /\  ( 0g `  (fld  ^s  CC ) )  e.  _V ) )
259 fzfid 11798 . . . . . . . . . 10  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) )  e.  Fin )
260 eldifn 3482 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) )  ->  -.  k  e.  ( 0 ... if ( 0  <_  (
( deg1  `
fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) )
261260adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  -.  k  e.  ( 0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) )
262154ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  a  e.  ( Base `  (Poly1 ` fld )
) )
263 eldifi 3481 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) )  ->  k  e.  NN0 )
264263adantl 466 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  k  e.  NN0 )
265 eqid 2443 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( deg1  ` fld )  =  ( deg1  ` fld )
266265, 55, 95, 17, 156deg1ge 21573 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( a  e.  ( Base `  (Poly1 ` fld ) )  /\  k  e.  NN0  /\  ( (coe1 `  a ) `  k
)  =/=  0 )  ->  k  <_  (
( deg1  `
fld ) `  a ) )
2672663expia 1189 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( a  e.  ( Base `  (Poly1 ` fld ) )  /\  k  e.  NN0 )  ->  (
( (coe1 `  a ) `  k )  =/=  0  ->  k  <_  ( ( deg1  ` fld ) `  a ) ) )
268262, 264, 267syl2anc 661 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  (
( (coe1 `  a ) `  k )  =/=  0  ->  k  <_  ( ( deg1  ` fld ) `  a ) ) )
269 0xr 9433 . . . . . . . . . . . . . . . . . . . . . . 23  |-  0  e.  RR*
270265, 55, 95deg1xrcl 21556 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( a  e.  ( Base `  (Poly1 ` fld )
)  ->  ( ( deg1  ` fld ) `  a )  e.  RR* )
271154, 270syl 16 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
( deg1  `
fld ) `  a )  e.  RR* )
272271ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  (
( deg1  `
fld ) `  a )  e.  RR* )
273 xrmax2 11151 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 0  e.  RR*  /\  (
( deg1  `
fld ) `  a )  e.  RR* )  ->  (
( deg1  `
fld ) `  a )  <_  if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) )
274269, 272, 273sylancr 663 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  (
( deg1  `
fld ) `  a )  <_  if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) )
275264nn0red 10640 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  k  e.  RR )
276275rexrd 9436 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  k  e.  RR* )
277 ifcl 3834 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( deg1  ` fld ) `  a )  e.  RR*  /\  0  e.  RR* )  ->  if ( 0  <_  (
( deg1  `
fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 )  e. 
RR* )
278272, 269, 277sylancl 662 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  if ( 0  <_  (
( deg1  `
fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 )  e. 
RR* )
279 xrletr 11135 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( k  e.  RR*  /\  (
( deg1  `
fld ) `  a )  e.  RR*  /\  if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 )  e. 
RR* )  ->  (
( k  <_  (
( deg1  `
fld ) `  a )  /\  ( ( deg1  ` fld ) `  a )  <_  if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) )  ->  k  <_  if ( 0  <_  (
( deg1  `
fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) )
280276, 272, 278, 279syl3anc 1218 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  (
( k  <_  (
( deg1  `
fld ) `  a )  /\  ( ( deg1  ` fld ) `  a )  <_  if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) )  ->  k  <_  if ( 0  <_  (
( deg1  `
fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) )
281274, 280mpan2d 674 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  (
k  <_  ( ( deg1  ` fld ) `  a )  ->  k  <_  if ( 0  <_ 
( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) )
282268, 281syld 44 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  (
( (coe1 `  a ) `  k )  =/=  0  ->  k  <_  if (
0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) )
283282, 264jctild 543 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  (
( (coe1 `  a ) `  k )  =/=  0  ->  ( k  e.  NN0  /\  k  <_  if (
0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )
284265, 55, 95deg1cl 21557 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( a  e.  ( Base `  (Poly1 ` fld )
)  ->  ( ( deg1  ` fld ) `  a )  e.  ( NN0  u.  { -oo } ) )
285154, 284syl 16 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
( deg1  `
fld ) `  a )  e.  ( NN0  u.  { -oo } ) )
286 elun 3500 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( deg1  ` fld ) `  a )  e.  ( NN0  u.  { -oo } )  <->  ( (
( deg1  `
fld ) `  a )  e.  NN0  \/  (
( deg1  `
fld ) `  a )  e.  { -oo }
) )
287285, 286sylib 196 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
( ( deg1  ` fld ) `  a )  e.  NN0  \/  (
( deg1  `
fld ) `  a )  e.  { -oo }
) )
288 nn0ge0 10608 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( deg1  ` fld ) `  a )  e.  NN0  ->  0  <_ 
( ( deg1  ` fld ) `  a ) )
289 iftrue 3800 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 0  <_  ( ( deg1  ` fld ) `  a )  ->  if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 )  =  ( ( deg1  ` fld ) `  a ) )
290288, 289syl 16 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( deg1  ` fld ) `  a )  e.  NN0  ->  if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 )  =  ( ( deg1  ` fld ) `  a ) )
291 id 22 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( deg1  ` fld ) `  a )  e.  NN0  ->  ( ( deg1  ` fld ) `
 a )  e. 
NN0 )
292290, 291eqeltrd 2517 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( deg1  ` fld ) `  a )  e.  NN0  ->  if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 )  e. 
NN0 )
293 mnflt0 11108 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |- -oo  <  0
294 mnfxr 11097 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |- -oo  e.  RR*
295 xrltnle 9446 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( -oo  e.  RR*  /\  0  e.  RR* )  ->  ( -oo  <  0  <->  -.  0  <_ -oo ) )
296294, 269, 295mp2an 672 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( -oo  <  0  <->  -.  0  <_ -oo )
297293, 296mpbi 208 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  -.  0  <_ -oo
298 elsni 3905 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( deg1  ` fld ) `  a )  e.  { -oo }  ->  ( ( deg1  ` fld ) `  a )  = -oo )
299298breq2d 4307 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( deg1  ` fld ) `  a )  e.  { -oo }  ->  ( 0  <_  (
( deg1  `
fld ) `  a )  <->  0  <_ -oo )
)
300297, 299mtbiri 303 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( deg1  ` fld ) `  a )  e.  { -oo }  ->  -.  0  <_  (
( deg1  `
fld ) `  a ) )
301 iffalse 3802 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( -.  0  <_  ( ( deg1  ` fld ) `  a )  ->  if ( 0  <_  (
( deg1  `
fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 )  =  0 )
302300, 301syl 16 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( deg1  ` fld ) `  a )  e.  { -oo }  ->  if ( 0  <_ 
( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 )  =  0 )
303 0nn0 10597 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  0  e.  NN0
304302, 303syl6eqel 2531 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( deg1  ` fld ) `  a )  e.  { -oo }  ->  if ( 0  <_ 
( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 )  e. 
NN0 )
305292, 304jaoi 379 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( deg1  ` fld ) `  a )  e.  NN0  \/  (
( deg1  `
fld ) `  a )  e.  { -oo }
)  ->  if (
0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 )  e. 
NN0 )
306287, 305syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  if ( 0  <_  (
( deg1  `
fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 )  e. 
NN0 )
307306ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  if ( 0  <_  (
( deg1  `
fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 )  e. 
NN0 )
308 fznn0 11527 . . . . . . . . . . . . . . . . . . . 20  |-  ( if ( 0  <_  (
( deg1  `
fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 )  e. 
NN0  ->  ( k  e.  ( 0 ... if ( 0  <_  (
( deg1  `
fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) )  <-> 
( k  e.  NN0  /\  k  <_  if (
0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )
309307, 308syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  (
k  e.  ( 0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) )  <-> 
( k  e.  NN0  /\  k  <_  if (
0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )
310283, 309sylibrd 234 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  (
( (coe1 `  a ) `  k )  =/=  0  ->  k  e.  ( 0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )
311310necon1bd 2682 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  ( -.  k  e.  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) )  ->  ( (coe1 `  a
) `  k )  =  0 ) )
312261, 311mpd 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  (
(coe1 `  a ) `  k )  =  0 )
313312oveq1d 6109 . . . . . . . . . . . . . . 15  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  (
( (coe1 `  a ) `  k )  x.  (
z ^ k ) )  =  ( 0  x.  ( z ^
k ) ) )
314263, 252sylan2 474 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  (
z ^ k )  e.  CC )
315314mul02d 9570 . . . . . . . . . . . . . . 15  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  (
0  x.  ( z ^ k ) )  =  0 )
316313, 315eqtrd 2475 . . . . . . . . . . . . . 14  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  z  e.  CC )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  (
( (coe1 `  a ) `  k )  x.  (
z ^ k ) )  =  0 )
317316an32s 802 . . . . . . . . . . . . 13  |-  ( ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  /\  k  e.  ( NN0  \  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  /\  z  e.  CC )  ->  (
( (coe1 `  a ) `  k )  x.  (
z ^ k ) )  =  0 )
318317mpteq2dva 4381 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  ( NN0  \  ( 0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  (
z  e.  CC  |->  ( ( (coe1 `  a ) `  k )  x.  (
z ^ k ) ) )  =  ( z  e.  CC  |->  0 ) )
319 fconstmpt 4885 . . . . . . . . . . . . 13  |-  ( CC 
X.  { 0 } )  =  ( z  e.  CC  |->  0 )
320 rngmnd 16657 . . . . . . . . . . . . . . 15  |-  (fld  e.  Ring  ->fld  e.  Mnd )
3219, 320ax-mp 5 . . . . . . . . . . . . . 14  |-fld  e.  Mnd
3223, 17pws0g 15460 . . . . . . . . . . . . . 14  |-  ( (fld  e. 
Mnd  /\  CC  e.  _V )  ->  ( CC 
X.  { 0 } )  =  ( 0g
`  (fld 
^s 
CC ) ) )
323321, 6, 322mp2an 672 . . . . . . . . . . . . 13  |-  ( CC 
X.  { 0 } )  =  ( 0g
`  (fld 
^s 
CC ) )
324319, 323eqtr3i 2465 . . . . . . . . . . . 12  |-  ( z  e.  CC  |->  0 )  =  ( 0g `  (fld  ^s  CC ) )
325318, 324syl6eq 2491 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubRing ` fld )  /\  a  e.  A
)  /\  k  e.  ( NN0  \  ( 0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  (
z  e.  CC  |->  ( ( (coe1 `  a ) `  k )  x.  (
z ^ k ) ) )  =  ( 0g `  (fld  ^s  CC ) ) )
326325, 168suppss2 6726 . . . . . . . . . 10  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
( k  e.  NN0  |->  ( z  e.  CC  |->  ( ( (coe1 `  a
) `  k )  x.  ( z ^ k
) ) ) ) supp  ( 0g `  (fld  ^s  CC ) ) )  C_  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) )
327 suppssfifsupp 7638 . . . . . . . . . 10  |-  ( ( ( ( k  e. 
NN0  |->  ( z  e.  CC  |->  ( ( (coe1 `  a ) `  k
)  x.  ( z ^ k ) ) ) )  e.  _V  /\ 
Fun  ( k  e. 
NN0  |->  ( z  e.  CC  |->  ( ( (coe1 `  a ) `  k
)  x.  ( z ^ k ) ) ) )  /\  ( 0g `  (fld 
^s 
CC ) )  e. 
_V )  /\  (
( 0 ... if ( 0  <_  (
( deg1  `
fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) )  e.  Fin  /\  (
( k  e.  NN0  |->  ( z  e.  CC  |->  ( ( (coe1 `  a
) `  k )  x.  ( z ^ k
) ) ) ) supp  ( 0g `  (fld  ^s  CC ) ) )  C_  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) ) )  ->  (
k  e.  NN0  |->  ( z  e.  CC  |->  ( ( (coe1 `  a ) `  k )  x.  (
z ^ k ) ) ) ) finSupp  ( 0g `  (fld 
^s 
CC ) ) )
328258, 259, 326, 327syl12anc 1216 . . . . . . . . 9  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
k  e.  NN0  |->  ( z  e.  CC  |->  ( ( (coe1 `  a ) `  k )  x.  (
z ^ k ) ) ) ) finSupp  ( 0g `  (fld 
^s 
CC ) ) )
3293, 4, 5, 249, 168, 250, 254, 328pwsgsum 16476 . . . . . . . 8  |-  ( ( S  e.  (SubRing ` fld )  /\  a  e.  A )  ->  (
(fld  ^s  CC )  gsumg  ( k  e.  NN0  |->  ( z  e.  CC  |->  ( ( (coe1 `  a
) `  k )  x.  ( z ^ k
) ) ) ) )  =  ( z  e.  CC  |->  (fld  gsumg  ( k  e.  NN0  |->  ( ( (coe1 `  a
) `  k )  x.  ( z ^ k
) ) ) ) ) )
330 elfznn0 11484 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0 ...
if ( 0  <_ 
( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) )  ->  k  e.  NN0 )
331330ssriv 3363 . . . . . . . . . . . 12  |-  ( 0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) 
C_  NN0
332 resmpt 5159 . . . . . . . . . . . 12  |-  ( ( 0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) 
C_  NN0  ->  ( ( k  e.  NN0  |->  ( ( (coe1 `  a ) `  k )  x.  (
z ^ k ) ) )  |`  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) )  =  ( k  e.  ( 0 ...
if ( 0  <_ 
( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) 
|->  ( ( (coe1 `  a
) `  k )  x.  ( z ^ k
) ) ) )
333331, 332ax-mp 5 . . . . . . . . . . 11  |-  ( ( k  e.  NN0  |->  ( ( (coe1 `  a ) `  k )  x.  (
z ^ k ) ) )  |`  (
0 ... if ( 0  <_  ( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0 ) ) )  =  ( k  e.  ( 0 ...
if ( 0  <_ 
( ( deg1  ` fld ) `  a ) ,  ( ( deg1  ` fld ) `  a ) ,  0