Step |
Hyp |
Ref |
Expression |
1 |
|
plypf1.r |
|- R = ( CCfld |`s S ) |
2 |
|
plypf1.p |
|- P = ( Poly1 ` R ) |
3 |
|
plypf1.a |
|- A = ( Base ` P ) |
4 |
|
plypf1.e |
|- E = ( eval1 ` CCfld ) |
5 |
|
elply |
|- ( 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 ) ) ) ) ) |
6 |
5
|
simprbi |
|- ( 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 ) ) ) ) |
7 |
|
eqid |
|- ( CCfld ^s CC ) = ( CCfld ^s CC ) |
8 |
|
cnfldbas |
|- CC = ( Base ` CCfld ) |
9 |
|
eqid |
|- ( 0g ` ( CCfld ^s CC ) ) = ( 0g ` ( CCfld ^s CC ) ) |
10 |
|
cnex |
|- CC e. _V |
11 |
10
|
a1i |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> CC e. _V ) |
12 |
|
fzfid |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( 0 ... n ) e. Fin ) |
13 |
|
cnring |
|- CCfld e. Ring |
14 |
|
ringcmn |
|- ( CCfld e. Ring -> CCfld e. CMnd ) |
15 |
13 14
|
mp1i |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> CCfld e. CMnd ) |
16 |
8
|
subrgss |
|- ( S e. ( SubRing ` CCfld ) -> S C_ CC ) |
17 |
16
|
ad2antrr |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> S C_ CC ) |
18 |
|
elmapi |
|- ( a e. ( ( S u. { 0 } ) ^m NN0 ) -> a : NN0 --> ( S u. { 0 } ) ) |
19 |
18
|
ad2antll |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> a : NN0 --> ( S u. { 0 } ) ) |
20 |
|
subrgsubg |
|- ( S e. ( SubRing ` CCfld ) -> S e. ( SubGrp ` CCfld ) ) |
21 |
|
cnfld0 |
|- 0 = ( 0g ` CCfld ) |
22 |
21
|
subg0cl |
|- ( S e. ( SubGrp ` CCfld ) -> 0 e. S ) |
23 |
20 22
|
syl |
|- ( S e. ( SubRing ` CCfld ) -> 0 e. S ) |
24 |
23
|
adantr |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> 0 e. S ) |
25 |
24
|
snssd |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> { 0 } C_ S ) |
26 |
|
ssequn2 |
|- ( { 0 } C_ S <-> ( S u. { 0 } ) = S ) |
27 |
25 26
|
sylib |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( S u. { 0 } ) = S ) |
28 |
27
|
feq3d |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( a : NN0 --> ( S u. { 0 } ) <-> a : NN0 --> S ) ) |
29 |
19 28
|
mpbid |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> a : NN0 --> S ) |
30 |
|
elfznn0 |
|- ( k e. ( 0 ... n ) -> k e. NN0 ) |
31 |
|
ffvelrn |
|- ( ( a : NN0 --> S /\ k e. NN0 ) -> ( a ` k ) e. S ) |
32 |
29 30 31
|
syl2an |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( a ` k ) e. S ) |
33 |
17 32
|
sseldd |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( a ` k ) e. CC ) |
34 |
33
|
adantrl |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( z e. CC /\ k e. ( 0 ... n ) ) ) -> ( a ` k ) e. CC ) |
35 |
|
simprl |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( z e. CC /\ k e. ( 0 ... n ) ) ) -> z e. CC ) |
36 |
30
|
ad2antll |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( z e. CC /\ k e. ( 0 ... n ) ) ) -> k e. NN0 ) |
37 |
|
expcl |
|- ( ( z e. CC /\ k e. NN0 ) -> ( z ^ k ) e. CC ) |
38 |
35 36 37
|
syl2anc |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( z e. CC /\ k e. ( 0 ... n ) ) ) -> ( z ^ k ) e. CC ) |
39 |
34 38
|
mulcld |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ ( z e. CC /\ k e. ( 0 ... n ) ) ) -> ( ( a ` k ) x. ( z ^ k ) ) e. CC ) |
40 |
|
eqid |
|- ( k e. ( 0 ... n ) |-> ( z e. CC |-> ( ( a ` k ) x. ( z ^ k ) ) ) ) = ( k e. ( 0 ... n ) |-> ( z e. CC |-> ( ( a ` k ) x. ( z ^ k ) ) ) ) |
41 |
10
|
mptex |
|- ( z e. CC |-> ( ( a ` k ) x. ( z ^ k ) ) ) e. _V |
42 |
41
|
a1i |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( z e. CC |-> ( ( a ` k ) x. ( z ^ k ) ) ) e. _V ) |
43 |
|
fvex |
|- ( 0g ` ( CCfld ^s CC ) ) e. _V |
44 |
43
|
a1i |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( 0g ` ( CCfld ^s CC ) ) e. _V ) |
45 |
40 12 42 44
|
fsuppmptdm |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( k e. ( 0 ... n ) |-> ( z e. CC |-> ( ( a ` k ) x. ( z ^ k ) ) ) ) finSupp ( 0g ` ( CCfld ^s CC ) ) ) |
46 |
7 8 9 11 12 15 39 45
|
pwsgsum |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( ( CCfld ^s CC ) gsum ( k e. ( 0 ... n ) |-> ( z e. CC |-> ( ( a ` k ) x. ( z ^ k ) ) ) ) ) = ( z e. CC |-> ( CCfld gsum ( k e. ( 0 ... n ) |-> ( ( a ` k ) x. ( z ^ k ) ) ) ) ) ) |
47 |
|
fzfid |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ z e. CC ) -> ( 0 ... n ) e. Fin ) |
48 |
39
|
anassrs |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ z e. CC ) /\ k e. ( 0 ... n ) ) -> ( ( a ` k ) x. ( z ^ k ) ) e. CC ) |
49 |
47 48
|
gsumfsum |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ z e. CC ) -> ( CCfld gsum ( k e. ( 0 ... n ) |-> ( ( a ` k ) x. ( z ^ k ) ) ) ) = sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) |
50 |
49
|
mpteq2dva |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( z e. CC |-> ( CCfld gsum ( k e. ( 0 ... n ) |-> ( ( a ` k ) x. ( z ^ k ) ) ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) |
51 |
46 50
|
eqtrd |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( ( CCfld ^s CC ) gsum ( 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 ) ) ) ) |
52 |
7
|
pwsring |
|- ( ( CCfld e. Ring /\ CC e. _V ) -> ( CCfld ^s CC ) e. Ring ) |
53 |
13 10 52
|
mp2an |
|- ( CCfld ^s CC ) e. Ring |
54 |
|
ringcmn |
|- ( ( CCfld ^s CC ) e. Ring -> ( CCfld ^s CC ) e. CMnd ) |
55 |
53 54
|
mp1i |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( CCfld ^s CC ) e. CMnd ) |
56 |
|
cncrng |
|- CCfld e. CRing |
57 |
|
eqid |
|- ( Poly1 ` CCfld ) = ( Poly1 ` CCfld ) |
58 |
4 57 7 8
|
evl1rhm |
|- ( CCfld e. CRing -> E e. ( ( Poly1 ` CCfld ) RingHom ( CCfld ^s CC ) ) ) |
59 |
56 58
|
ax-mp |
|- E e. ( ( Poly1 ` CCfld ) RingHom ( CCfld ^s CC ) ) |
60 |
57 1 2 3
|
subrgply1 |
|- ( S e. ( SubRing ` CCfld ) -> A e. ( SubRing ` ( Poly1 ` CCfld ) ) ) |
61 |
60
|
adantr |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> A e. ( SubRing ` ( Poly1 ` CCfld ) ) ) |
62 |
|
rhmima |
|- ( ( E e. ( ( Poly1 ` CCfld ) RingHom ( CCfld ^s CC ) ) /\ A e. ( SubRing ` ( Poly1 ` CCfld ) ) ) -> ( E " A ) e. ( SubRing ` ( CCfld ^s CC ) ) ) |
63 |
59 61 62
|
sylancr |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( E " A ) e. ( SubRing ` ( CCfld ^s CC ) ) ) |
64 |
|
subrgsubg |
|- ( ( E " A ) e. ( SubRing ` ( CCfld ^s CC ) ) -> ( E " A ) e. ( SubGrp ` ( CCfld ^s CC ) ) ) |
65 |
|
subgsubm |
|- ( ( E " A ) e. ( SubGrp ` ( CCfld ^s CC ) ) -> ( E " A ) e. ( SubMnd ` ( CCfld ^s CC ) ) ) |
66 |
63 64 65
|
3syl |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( E " A ) e. ( SubMnd ` ( CCfld ^s CC ) ) ) |
67 |
|
eqid |
|- ( Base ` ( CCfld ^s CC ) ) = ( Base ` ( CCfld ^s CC ) ) |
68 |
13
|
a1i |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> CCfld e. Ring ) |
69 |
10
|
a1i |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> CC e. _V ) |
70 |
|
fconst6g |
|- ( ( a ` k ) e. CC -> ( CC X. { ( a ` k ) } ) : CC --> CC ) |
71 |
33 70
|
syl |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( CC X. { ( a ` k ) } ) : CC --> CC ) |
72 |
7 8 67
|
pwselbasb |
|- ( ( CCfld e. Ring /\ CC e. _V ) -> ( ( CC X. { ( a ` k ) } ) e. ( Base ` ( CCfld ^s CC ) ) <-> ( CC X. { ( a ` k ) } ) : CC --> CC ) ) |
73 |
13 10 72
|
mp2an |
|- ( ( CC X. { ( a ` k ) } ) e. ( Base ` ( CCfld ^s CC ) ) <-> ( CC X. { ( a ` k ) } ) : CC --> CC ) |
74 |
71 73
|
sylibr |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( CC X. { ( a ` k ) } ) e. ( Base ` ( CCfld ^s CC ) ) ) |
75 |
38
|
anass1rs |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) /\ z e. CC ) -> ( z ^ k ) e. CC ) |
76 |
75
|
fmpttd |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( z e. CC |-> ( z ^ k ) ) : CC --> CC ) |
77 |
7 8 67
|
pwselbasb |
|- ( ( CCfld e. Ring /\ CC e. _V ) -> ( ( z e. CC |-> ( z ^ k ) ) e. ( Base ` ( CCfld ^s CC ) ) <-> ( z e. CC |-> ( z ^ k ) ) : CC --> CC ) ) |
78 |
13 10 77
|
mp2an |
|- ( ( z e. CC |-> ( z ^ k ) ) e. ( Base ` ( CCfld ^s CC ) ) <-> ( z e. CC |-> ( z ^ k ) ) : CC --> CC ) |
79 |
76 78
|
sylibr |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( z e. CC |-> ( z ^ k ) ) e. ( Base ` ( CCfld ^s CC ) ) ) |
80 |
|
cnfldmul |
|- x. = ( .r ` CCfld ) |
81 |
|
eqid |
|- ( .r ` ( CCfld ^s CC ) ) = ( .r ` ( CCfld ^s CC ) ) |
82 |
7 67 68 69 74 79 80 81
|
pwsmulrval |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( ( CC X. { ( a ` k ) } ) ( .r ` ( CCfld ^s CC ) ) ( z e. CC |-> ( z ^ k ) ) ) = ( ( CC X. { ( a ` k ) } ) oF x. ( z e. CC |-> ( z ^ k ) ) ) ) |
83 |
33
|
adantr |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) /\ z e. CC ) -> ( a ` k ) e. CC ) |
84 |
|
fconstmpt |
|- ( CC X. { ( a ` k ) } ) = ( z e. CC |-> ( a ` k ) ) |
85 |
84
|
a1i |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( CC X. { ( a ` k ) } ) = ( z e. CC |-> ( a ` k ) ) ) |
86 |
|
eqidd |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( z e. CC |-> ( z ^ k ) ) = ( z e. CC |-> ( z ^ k ) ) ) |
87 |
69 83 75 85 86
|
offval2 |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( 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 ) ) ) ) |
88 |
82 87
|
eqtrd |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( ( CC X. { ( a ` k ) } ) ( .r ` ( CCfld ^s CC ) ) ( z e. CC |-> ( z ^ k ) ) ) = ( z e. CC |-> ( ( a ` k ) x. ( z ^ k ) ) ) ) |
89 |
63
|
adantr |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( E " A ) e. ( SubRing ` ( CCfld ^s CC ) ) ) |
90 |
|
eqid |
|- ( algSc ` ( Poly1 ` CCfld ) ) = ( algSc ` ( Poly1 ` CCfld ) ) |
91 |
4 57 8 90
|
evl1sca |
|- ( ( CCfld e. CRing /\ ( a ` k ) e. CC ) -> ( E ` ( ( algSc ` ( Poly1 ` CCfld ) ) ` ( a ` k ) ) ) = ( CC X. { ( a ` k ) } ) ) |
92 |
56 33 91
|
sylancr |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( E ` ( ( algSc ` ( Poly1 ` CCfld ) ) ` ( a ` k ) ) ) = ( CC X. { ( a ` k ) } ) ) |
93 |
|
eqid |
|- ( Base ` ( Poly1 ` CCfld ) ) = ( Base ` ( Poly1 ` CCfld ) ) |
94 |
93 67
|
rhmf |
|- ( E e. ( ( Poly1 ` CCfld ) RingHom ( CCfld ^s CC ) ) -> E : ( Base ` ( Poly1 ` CCfld ) ) --> ( Base ` ( CCfld ^s CC ) ) ) |
95 |
59 94
|
ax-mp |
|- E : ( Base ` ( Poly1 ` CCfld ) ) --> ( Base ` ( CCfld ^s CC ) ) |
96 |
|
ffn |
|- ( E : ( Base ` ( Poly1 ` CCfld ) ) --> ( Base ` ( CCfld ^s CC ) ) -> E Fn ( Base ` ( Poly1 ` CCfld ) ) ) |
97 |
95 96
|
mp1i |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> E Fn ( Base ` ( Poly1 ` CCfld ) ) ) |
98 |
93
|
subrgss |
|- ( A e. ( SubRing ` ( Poly1 ` CCfld ) ) -> A C_ ( Base ` ( Poly1 ` CCfld ) ) ) |
99 |
60 98
|
syl |
|- ( S e. ( SubRing ` CCfld ) -> A C_ ( Base ` ( Poly1 ` CCfld ) ) ) |
100 |
99
|
ad2antrr |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> A C_ ( Base ` ( Poly1 ` CCfld ) ) ) |
101 |
|
simpll |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> S e. ( SubRing ` CCfld ) ) |
102 |
57 90 1 2 101 3 8 33
|
subrg1asclcl |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( ( ( algSc ` ( Poly1 ` CCfld ) ) ` ( a ` k ) ) e. A <-> ( a ` k ) e. S ) ) |
103 |
32 102
|
mpbird |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( ( algSc ` ( Poly1 ` CCfld ) ) ` ( a ` k ) ) e. A ) |
104 |
|
fnfvima |
|- ( ( E Fn ( Base ` ( Poly1 ` CCfld ) ) /\ A C_ ( Base ` ( Poly1 ` CCfld ) ) /\ ( ( algSc ` ( Poly1 ` CCfld ) ) ` ( a ` k ) ) e. A ) -> ( E ` ( ( algSc ` ( Poly1 ` CCfld ) ) ` ( a ` k ) ) ) e. ( E " A ) ) |
105 |
97 100 103 104
|
syl3anc |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( E ` ( ( algSc ` ( Poly1 ` CCfld ) ) ` ( a ` k ) ) ) e. ( E " A ) ) |
106 |
92 105
|
eqeltrrd |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( CC X. { ( a ` k ) } ) e. ( E " A ) ) |
107 |
67
|
subrgss |
|- ( ( E " A ) e. ( SubRing ` ( CCfld ^s CC ) ) -> ( E " A ) C_ ( Base ` ( CCfld ^s CC ) ) ) |
108 |
89 107
|
syl |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( E " A ) C_ ( Base ` ( CCfld ^s CC ) ) ) |
109 |
60
|
ad2antrr |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> A e. ( SubRing ` ( Poly1 ` CCfld ) ) ) |
110 |
|
eqid |
|- ( mulGrp ` ( Poly1 ` CCfld ) ) = ( mulGrp ` ( Poly1 ` CCfld ) ) |
111 |
110
|
subrgsubm |
|- ( A e. ( SubRing ` ( Poly1 ` CCfld ) ) -> A e. ( SubMnd ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ) |
112 |
109 111
|
syl |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> A e. ( SubMnd ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ) |
113 |
30
|
adantl |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> k e. NN0 ) |
114 |
|
eqid |
|- ( var1 ` CCfld ) = ( var1 ` CCfld ) |
115 |
114 101 1 2 3
|
subrgvr1cl |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( var1 ` CCfld ) e. A ) |
116 |
|
eqid |
|- ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) = ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) |
117 |
116
|
submmulgcl |
|- ( ( A e. ( SubMnd ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) /\ k e. NN0 /\ ( var1 ` CCfld ) e. A ) -> ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) e. A ) |
118 |
112 113 115 117
|
syl3anc |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) e. A ) |
119 |
|
fnfvima |
|- ( ( E Fn ( Base ` ( Poly1 ` CCfld ) ) /\ A C_ ( Base ` ( Poly1 ` CCfld ) ) /\ ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) e. A ) -> ( E ` ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) e. ( E " A ) ) |
120 |
97 100 118 119
|
syl3anc |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( E ` ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) e. ( E " A ) ) |
121 |
108 120
|
sseldd |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( E ` ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) e. ( Base ` ( CCfld ^s CC ) ) ) |
122 |
7 8 67 68 69 121
|
pwselbas |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( E ` ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) : CC --> CC ) |
123 |
122
|
feqmptd |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( E ` ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) = ( z e. CC |-> ( ( E ` ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ` z ) ) ) |
124 |
56
|
a1i |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) /\ z e. CC ) -> CCfld e. CRing ) |
125 |
|
simpr |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) /\ z e. CC ) -> z e. CC ) |
126 |
4 114 8 57 93 124 125
|
evl1vard |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) /\ z e. CC ) -> ( ( var1 ` CCfld ) e. ( Base ` ( Poly1 ` CCfld ) ) /\ ( ( E ` ( var1 ` CCfld ) ) ` z ) = z ) ) |
127 |
|
eqid |
|- ( .g ` ( mulGrp ` CCfld ) ) = ( .g ` ( mulGrp ` CCfld ) ) |
128 |
113
|
adantr |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) /\ z e. CC ) -> k e. NN0 ) |
129 |
4 57 8 93 124 125 126 116 127 128
|
evl1expd |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) /\ z e. CC ) -> ( ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) e. ( Base ` ( Poly1 ` CCfld ) ) /\ ( ( E ` ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ` z ) = ( k ( .g ` ( mulGrp ` CCfld ) ) z ) ) ) |
130 |
129
|
simprd |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) /\ z e. CC ) -> ( ( E ` ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ` z ) = ( k ( .g ` ( mulGrp ` CCfld ) ) z ) ) |
131 |
|
cnfldexp |
|- ( ( z e. CC /\ k e. NN0 ) -> ( k ( .g ` ( mulGrp ` CCfld ) ) z ) = ( z ^ k ) ) |
132 |
125 128 131
|
syl2anc |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) /\ z e. CC ) -> ( k ( .g ` ( mulGrp ` CCfld ) ) z ) = ( z ^ k ) ) |
133 |
130 132
|
eqtrd |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) /\ z e. CC ) -> ( ( E ` ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ` z ) = ( z ^ k ) ) |
134 |
133
|
mpteq2dva |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( z e. CC |-> ( ( E ` ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ` z ) ) = ( z e. CC |-> ( z ^ k ) ) ) |
135 |
123 134
|
eqtrd |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( E ` ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) = ( z e. CC |-> ( z ^ k ) ) ) |
136 |
135 120
|
eqeltrrd |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( z e. CC |-> ( z ^ k ) ) e. ( E " A ) ) |
137 |
81
|
subrgmcl |
|- ( ( ( E " A ) e. ( SubRing ` ( CCfld ^s CC ) ) /\ ( CC X. { ( a ` k ) } ) e. ( E " A ) /\ ( z e. CC |-> ( z ^ k ) ) e. ( E " A ) ) -> ( ( CC X. { ( a ` k ) } ) ( .r ` ( CCfld ^s CC ) ) ( z e. CC |-> ( z ^ k ) ) ) e. ( E " A ) ) |
138 |
89 106 136 137
|
syl3anc |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... n ) ) -> ( ( CC X. { ( a ` k ) } ) ( .r ` ( CCfld ^s CC ) ) ( z e. CC |-> ( z ^ k ) ) ) e. ( E " A ) ) |
139 |
88 138
|
eqeltrrd |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ ( 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 ) ) |
140 |
139
|
fmpttd |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( 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 ) ) |
141 |
40 12 139 44
|
fsuppmptdm |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( k e. ( 0 ... n ) |-> ( z e. CC |-> ( ( a ` k ) x. ( z ^ k ) ) ) ) finSupp ( 0g ` ( CCfld ^s CC ) ) ) |
142 |
9 55 12 66 140 141
|
gsumsubmcl |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( n e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( ( CCfld ^s CC ) gsum ( k e. ( 0 ... n ) |-> ( z e. CC |-> ( ( a ` k ) x. ( z ^ k ) ) ) ) ) e. ( E " A ) ) |
143 |
51 142
|
eqeltrrd |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( 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 ) ) |
144 |
|
eleq1 |
|- ( 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 ) ) ) |
145 |
143 144
|
syl5ibrcom |
|- ( ( S e. ( SubRing ` CCfld ) /\ ( 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 ) ) ) |
146 |
145
|
rexlimdvva |
|- ( S e. ( SubRing ` CCfld ) -> ( 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 ) ) ) |
147 |
6 146
|
syl5 |
|- ( S e. ( SubRing ` CCfld ) -> ( f e. ( Poly ` S ) -> f e. ( E " A ) ) ) |
148 |
|
ffun |
|- ( E : ( Base ` ( Poly1 ` CCfld ) ) --> ( Base ` ( CCfld ^s CC ) ) -> Fun E ) |
149 |
95 148
|
ax-mp |
|- Fun E |
150 |
|
fvelima |
|- ( ( Fun E /\ f e. ( E " A ) ) -> E. a e. A ( E ` a ) = f ) |
151 |
149 150
|
mpan |
|- ( f e. ( E " A ) -> E. a e. A ( E ` a ) = f ) |
152 |
99
|
sselda |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> a e. ( Base ` ( Poly1 ` CCfld ) ) ) |
153 |
|
eqid |
|- ( .s ` ( Poly1 ` CCfld ) ) = ( .s ` ( Poly1 ` CCfld ) ) |
154 |
|
eqid |
|- ( coe1 ` a ) = ( coe1 ` a ) |
155 |
57 114 93 153 110 116 154
|
ply1coe |
|- ( ( CCfld e. Ring /\ a e. ( Base ` ( Poly1 ` CCfld ) ) ) -> a = ( ( Poly1 ` CCfld ) gsum ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) ) ) |
156 |
13 152 155
|
sylancr |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> a = ( ( Poly1 ` CCfld ) gsum ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) ) ) |
157 |
156
|
fveq2d |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( E ` a ) = ( E ` ( ( Poly1 ` CCfld ) gsum ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) ) ) ) |
158 |
|
eqid |
|- ( 0g ` ( Poly1 ` CCfld ) ) = ( 0g ` ( Poly1 ` CCfld ) ) |
159 |
57
|
ply1ring |
|- ( CCfld e. Ring -> ( Poly1 ` CCfld ) e. Ring ) |
160 |
13 159
|
ax-mp |
|- ( Poly1 ` CCfld ) e. Ring |
161 |
|
ringcmn |
|- ( ( Poly1 ` CCfld ) e. Ring -> ( Poly1 ` CCfld ) e. CMnd ) |
162 |
160 161
|
mp1i |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( Poly1 ` CCfld ) e. CMnd ) |
163 |
|
ringmnd |
|- ( ( CCfld ^s CC ) e. Ring -> ( CCfld ^s CC ) e. Mnd ) |
164 |
53 163
|
mp1i |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( CCfld ^s CC ) e. Mnd ) |
165 |
|
nn0ex |
|- NN0 e. _V |
166 |
165
|
a1i |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> NN0 e. _V ) |
167 |
|
rhmghm |
|- ( E e. ( ( Poly1 ` CCfld ) RingHom ( CCfld ^s CC ) ) -> E e. ( ( Poly1 ` CCfld ) GrpHom ( CCfld ^s CC ) ) ) |
168 |
59 167
|
ax-mp |
|- E e. ( ( Poly1 ` CCfld ) GrpHom ( CCfld ^s CC ) ) |
169 |
|
ghmmhm |
|- ( E e. ( ( Poly1 ` CCfld ) GrpHom ( CCfld ^s CC ) ) -> E e. ( ( Poly1 ` CCfld ) MndHom ( CCfld ^s CC ) ) ) |
170 |
168 169
|
mp1i |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> E e. ( ( Poly1 ` CCfld ) MndHom ( CCfld ^s CC ) ) ) |
171 |
57
|
ply1lmod |
|- ( CCfld e. Ring -> ( Poly1 ` CCfld ) e. LMod ) |
172 |
13 171
|
mp1i |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) -> ( Poly1 ` CCfld ) e. LMod ) |
173 |
16
|
ad2antrr |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) -> S C_ CC ) |
174 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
175 |
154 3 2 174
|
coe1f |
|- ( a e. A -> ( coe1 ` a ) : NN0 --> ( Base ` R ) ) |
176 |
1
|
subrgbas |
|- ( S e. ( SubRing ` CCfld ) -> S = ( Base ` R ) ) |
177 |
176
|
feq3d |
|- ( S e. ( SubRing ` CCfld ) -> ( ( coe1 ` a ) : NN0 --> S <-> ( coe1 ` a ) : NN0 --> ( Base ` R ) ) ) |
178 |
175 177
|
syl5ibr |
|- ( S e. ( SubRing ` CCfld ) -> ( a e. A -> ( coe1 ` a ) : NN0 --> S ) ) |
179 |
178
|
imp |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( coe1 ` a ) : NN0 --> S ) |
180 |
179
|
ffvelrnda |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) -> ( ( coe1 ` a ) ` k ) e. S ) |
181 |
173 180
|
sseldd |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) -> ( ( coe1 ` a ) ` k ) e. CC ) |
182 |
110
|
ringmgp |
|- ( ( Poly1 ` CCfld ) e. Ring -> ( mulGrp ` ( Poly1 ` CCfld ) ) e. Mnd ) |
183 |
160 182
|
mp1i |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) -> ( mulGrp ` ( Poly1 ` CCfld ) ) e. Mnd ) |
184 |
|
simpr |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) -> k e. NN0 ) |
185 |
114 57 93
|
vr1cl |
|- ( CCfld e. Ring -> ( var1 ` CCfld ) e. ( Base ` ( Poly1 ` CCfld ) ) ) |
186 |
13 185
|
mp1i |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) -> ( var1 ` CCfld ) e. ( Base ` ( Poly1 ` CCfld ) ) ) |
187 |
110 93
|
mgpbas |
|- ( Base ` ( Poly1 ` CCfld ) ) = ( Base ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) |
188 |
187 116
|
mulgnn0cl |
|- ( ( ( mulGrp ` ( Poly1 ` CCfld ) ) e. Mnd /\ k e. NN0 /\ ( var1 ` CCfld ) e. ( Base ` ( Poly1 ` CCfld ) ) ) -> ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) e. ( Base ` ( Poly1 ` CCfld ) ) ) |
189 |
183 184 186 188
|
syl3anc |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) -> ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) e. ( Base ` ( Poly1 ` CCfld ) ) ) |
190 |
57
|
ply1sca |
|- ( CCfld e. Ring -> CCfld = ( Scalar ` ( Poly1 ` CCfld ) ) ) |
191 |
13 190
|
ax-mp |
|- CCfld = ( Scalar ` ( Poly1 ` CCfld ) ) |
192 |
93 191 153 8
|
lmodvscl |
|- ( ( ( Poly1 ` CCfld ) e. LMod /\ ( ( coe1 ` a ) ` k ) e. CC /\ ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) e. ( Base ` ( Poly1 ` CCfld ) ) ) -> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) e. ( Base ` ( Poly1 ` CCfld ) ) ) |
193 |
172 181 189 192
|
syl3anc |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) -> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) e. ( Base ` ( Poly1 ` CCfld ) ) ) |
194 |
193
|
fmpttd |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) : NN0 --> ( Base ` ( Poly1 ` CCfld ) ) ) |
195 |
165
|
mptex |
|- ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) e. _V |
196 |
|
funmpt |
|- Fun ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) |
197 |
|
fvex |
|- ( 0g ` ( Poly1 ` CCfld ) ) e. _V |
198 |
195 196 197
|
3pm3.2i |
|- ( ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) e. _V /\ Fun ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) /\ ( 0g ` ( Poly1 ` CCfld ) ) e. _V ) |
199 |
198
|
a1i |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) e. _V /\ Fun ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) /\ ( 0g ` ( Poly1 ` CCfld ) ) e. _V ) ) |
200 |
154 93 57 21
|
coe1sfi |
|- ( a e. ( Base ` ( Poly1 ` CCfld ) ) -> ( coe1 ` a ) finSupp 0 ) |
201 |
152 200
|
syl |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( coe1 ` a ) finSupp 0 ) |
202 |
201
|
fsuppimpd |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( ( coe1 ` a ) supp 0 ) e. Fin ) |
203 |
179
|
feqmptd |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( coe1 ` a ) = ( k e. NN0 |-> ( ( coe1 ` a ) ` k ) ) ) |
204 |
203
|
oveq1d |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( ( coe1 ` a ) supp 0 ) = ( ( k e. NN0 |-> ( ( coe1 ` a ) ` k ) ) supp 0 ) ) |
205 |
|
eqimss2 |
|- ( ( ( 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 ) ) |
206 |
204 205
|
syl |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( ( k e. NN0 |-> ( ( coe1 ` a ) ` k ) ) supp 0 ) C_ ( ( coe1 ` a ) supp 0 ) ) |
207 |
13 171
|
mp1i |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( Poly1 ` CCfld ) e. LMod ) |
208 |
93 191 153 21 158
|
lmod0vs |
|- ( ( ( Poly1 ` CCfld ) e. LMod /\ x e. ( Base ` ( Poly1 ` CCfld ) ) ) -> ( 0 ( .s ` ( Poly1 ` CCfld ) ) x ) = ( 0g ` ( Poly1 ` CCfld ) ) ) |
209 |
207 208
|
sylan |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ x e. ( Base ` ( Poly1 ` CCfld ) ) ) -> ( 0 ( .s ` ( Poly1 ` CCfld ) ) x ) = ( 0g ` ( Poly1 ` CCfld ) ) ) |
210 |
|
c0ex |
|- 0 e. _V |
211 |
210
|
a1i |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> 0 e. _V ) |
212 |
206 209 180 189 211
|
suppssov1 |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) supp ( 0g ` ( Poly1 ` CCfld ) ) ) C_ ( ( coe1 ` a ) supp 0 ) ) |
213 |
|
suppssfifsupp |
|- ( ( ( ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) e. _V /\ Fun ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) /\ ( 0g ` ( Poly1 ` CCfld ) ) e. _V ) /\ ( ( ( coe1 ` a ) supp 0 ) e. Fin /\ ( ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) supp ( 0g ` ( Poly1 ` CCfld ) ) ) C_ ( ( coe1 ` a ) supp 0 ) ) ) -> ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) finSupp ( 0g ` ( Poly1 ` CCfld ) ) ) |
214 |
199 202 212 213
|
syl12anc |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) finSupp ( 0g ` ( Poly1 ` CCfld ) ) ) |
215 |
93 158 162 164 166 170 194 214
|
gsummhm |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( ( CCfld ^s CC ) gsum ( E o. ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) ) ) = ( E ` ( ( Poly1 ` CCfld ) gsum ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) ) ) ) |
216 |
95
|
a1i |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> E : ( Base ` ( Poly1 ` CCfld ) ) --> ( Base ` ( CCfld ^s CC ) ) ) |
217 |
216 193
|
cofmpt |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( E o. ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) ) = ( k e. NN0 |-> ( E ` ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) ) ) |
218 |
13
|
a1i |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) -> CCfld e. Ring ) |
219 |
10
|
a1i |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) -> CC e. _V ) |
220 |
95
|
ffvelrni |
|- ( ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) e. ( Base ` ( Poly1 ` CCfld ) ) -> ( E ` ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) e. ( Base ` ( CCfld ^s CC ) ) ) |
221 |
193 220
|
syl |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) -> ( E ` ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) e. ( Base ` ( CCfld ^s CC ) ) ) |
222 |
7 8 67 218 219 221
|
pwselbas |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) -> ( E ` ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) : CC --> CC ) |
223 |
222
|
feqmptd |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) -> ( E ` ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) = ( z e. CC |-> ( ( E ` ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) ` z ) ) ) |
224 |
56
|
a1i |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) /\ z e. CC ) -> CCfld e. CRing ) |
225 |
|
simpr |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) /\ z e. CC ) -> z e. CC ) |
226 |
4 114 8 57 93 224 225
|
evl1vard |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) /\ z e. CC ) -> ( ( var1 ` CCfld ) e. ( Base ` ( Poly1 ` CCfld ) ) /\ ( ( E ` ( var1 ` CCfld ) ) ` z ) = z ) ) |
227 |
184
|
adantr |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) /\ z e. CC ) -> k e. NN0 ) |
228 |
4 57 8 93 224 225 226 116 127 227
|
evl1expd |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) /\ z e. CC ) -> ( ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) e. ( Base ` ( Poly1 ` CCfld ) ) /\ ( ( E ` ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ` z ) = ( k ( .g ` ( mulGrp ` CCfld ) ) z ) ) ) |
229 |
225 227 131
|
syl2anc |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) /\ z e. CC ) -> ( k ( .g ` ( mulGrp ` CCfld ) ) z ) = ( z ^ k ) ) |
230 |
229
|
eqeq2d |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) /\ z e. CC ) -> ( ( ( E ` ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ` z ) = ( k ( .g ` ( mulGrp ` CCfld ) ) z ) <-> ( ( E ` ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ` z ) = ( z ^ k ) ) ) |
231 |
230
|
anbi2d |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) /\ z e. CC ) -> ( ( ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) e. ( Base ` ( Poly1 ` CCfld ) ) /\ ( ( E ` ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ` z ) = ( k ( .g ` ( mulGrp ` CCfld ) ) z ) ) <-> ( ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) e. ( Base ` ( Poly1 ` CCfld ) ) /\ ( ( E ` ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ` z ) = ( z ^ k ) ) ) ) |
232 |
228 231
|
mpbid |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) /\ z e. CC ) -> ( ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) e. ( Base ` ( Poly1 ` CCfld ) ) /\ ( ( E ` ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ` z ) = ( z ^ k ) ) ) |
233 |
181
|
adantr |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) /\ z e. CC ) -> ( ( coe1 ` a ) ` k ) e. CC ) |
234 |
4 57 8 93 224 225 232 233 153 80
|
evl1vsd |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) /\ z e. CC ) -> ( ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) e. ( Base ` ( Poly1 ` CCfld ) ) /\ ( ( E ` ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) ` z ) = ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) |
235 |
234
|
simprd |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) /\ z e. CC ) -> ( ( E ` ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) ` z ) = ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) |
236 |
235
|
mpteq2dva |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) -> ( z e. CC |-> ( ( E ` ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) ` z ) ) = ( z e. CC |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) |
237 |
223 236
|
eqtrd |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. NN0 ) -> ( E ` ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) = ( z e. CC |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) |
238 |
237
|
mpteq2dva |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( k e. NN0 |-> ( E ` ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) ) = ( k e. NN0 |-> ( z e. CC |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) ) |
239 |
217 238
|
eqtrd |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( E o. ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) ) = ( k e. NN0 |-> ( z e. CC |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) ) |
240 |
239
|
oveq2d |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( ( CCfld ^s CC ) gsum ( E o. ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) ( .s ` ( Poly1 ` CCfld ) ) ( k ( .g ` ( mulGrp ` ( Poly1 ` CCfld ) ) ) ( var1 ` CCfld ) ) ) ) ) ) = ( ( CCfld ^s CC ) gsum ( k e. NN0 |-> ( z e. CC |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) ) ) |
241 |
157 215 240
|
3eqtr2d |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( E ` a ) = ( ( CCfld ^s CC ) gsum ( k e. NN0 |-> ( z e. CC |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) ) ) |
242 |
10
|
a1i |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> CC e. _V ) |
243 |
13 14
|
mp1i |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> CCfld e. CMnd ) |
244 |
181
|
adantlr |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. NN0 ) -> ( ( coe1 ` a ) ` k ) e. CC ) |
245 |
37
|
adantll |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. NN0 ) -> ( z ^ k ) e. CC ) |
246 |
244 245
|
mulcld |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. NN0 ) -> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) e. CC ) |
247 |
246
|
anasss |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ ( z e. CC /\ k e. NN0 ) ) -> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) e. CC ) |
248 |
165
|
mptex |
|- ( k e. NN0 |-> ( z e. CC |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) e. _V |
249 |
|
funmpt |
|- Fun ( k e. NN0 |-> ( z e. CC |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) |
250 |
248 249 43
|
3pm3.2i |
|- ( ( 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 ` ( CCfld ^s CC ) ) e. _V ) |
251 |
250
|
a1i |
|- ( ( S e. ( SubRing ` CCfld ) /\ 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 ` ( CCfld ^s CC ) ) e. _V ) ) |
252 |
|
fzfid |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) e. Fin ) |
253 |
|
eldifn |
|- ( k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) -> -. k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) |
254 |
253
|
adantl |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> -. k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) |
255 |
152
|
ad2antrr |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> a e. ( Base ` ( Poly1 ` CCfld ) ) ) |
256 |
|
eldifi |
|- ( k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) -> k e. NN0 ) |
257 |
256
|
adantl |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> k e. NN0 ) |
258 |
|
eqid |
|- ( deg1 ` CCfld ) = ( deg1 ` CCfld ) |
259 |
258 57 93 21 154
|
deg1ge |
|- ( ( a e. ( Base ` ( Poly1 ` CCfld ) ) /\ k e. NN0 /\ ( ( coe1 ` a ) ` k ) =/= 0 ) -> k <_ ( ( deg1 ` CCfld ) ` a ) ) |
260 |
259
|
3expia |
|- ( ( a e. ( Base ` ( Poly1 ` CCfld ) ) /\ k e. NN0 ) -> ( ( ( coe1 ` a ) ` k ) =/= 0 -> k <_ ( ( deg1 ` CCfld ) ` a ) ) ) |
261 |
255 257 260
|
syl2anc |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( ( ( coe1 ` a ) ` k ) =/= 0 -> k <_ ( ( deg1 ` CCfld ) ` a ) ) ) |
262 |
|
0xr |
|- 0 e. RR* |
263 |
258 57 93
|
deg1xrcl |
|- ( a e. ( Base ` ( Poly1 ` CCfld ) ) -> ( ( deg1 ` CCfld ) ` a ) e. RR* ) |
264 |
152 263
|
syl |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( ( deg1 ` CCfld ) ` a ) e. RR* ) |
265 |
264
|
ad2antrr |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( ( deg1 ` CCfld ) ` a ) e. RR* ) |
266 |
|
xrmax2 |
|- ( ( 0 e. RR* /\ ( ( deg1 ` CCfld ) ` a ) e. RR* ) -> ( ( deg1 ` CCfld ) ` a ) <_ if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) |
267 |
262 265 266
|
sylancr |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( ( deg1 ` CCfld ) ` a ) <_ if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) |
268 |
257
|
nn0red |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> k e. RR ) |
269 |
268
|
rexrd |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> k e. RR* ) |
270 |
|
ifcl |
|- ( ( ( ( deg1 ` CCfld ) ` a ) e. RR* /\ 0 e. RR* ) -> if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) e. RR* ) |
271 |
265 262 270
|
sylancl |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) e. RR* ) |
272 |
|
xrletr |
|- ( ( k e. RR* /\ ( ( deg1 ` CCfld ) ` a ) e. RR* /\ if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) e. RR* ) -> ( ( k <_ ( ( deg1 ` CCfld ) ` a ) /\ ( ( deg1 ` CCfld ) ` a ) <_ if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) -> k <_ if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) |
273 |
269 265 271 272
|
syl3anc |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( ( k <_ ( ( deg1 ` CCfld ) ` a ) /\ ( ( deg1 ` CCfld ) ` a ) <_ if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) -> k <_ if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) |
274 |
267 273
|
mpan2d |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( k <_ ( ( deg1 ` CCfld ) ` a ) -> k <_ if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) |
275 |
261 274
|
syld |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( ( ( coe1 ` a ) ` k ) =/= 0 -> k <_ if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) |
276 |
275 257
|
jctild |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( ( ( coe1 ` a ) ` k ) =/= 0 -> ( k e. NN0 /\ k <_ if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) |
277 |
258 57 93
|
deg1cl |
|- ( a e. ( Base ` ( Poly1 ` CCfld ) ) -> ( ( deg1 ` CCfld ) ` a ) e. ( NN0 u. { -oo } ) ) |
278 |
152 277
|
syl |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( ( deg1 ` CCfld ) ` a ) e. ( NN0 u. { -oo } ) ) |
279 |
|
elun |
|- ( ( ( deg1 ` CCfld ) ` a ) e. ( NN0 u. { -oo } ) <-> ( ( ( deg1 ` CCfld ) ` a ) e. NN0 \/ ( ( deg1 ` CCfld ) ` a ) e. { -oo } ) ) |
280 |
278 279
|
sylib |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( ( ( deg1 ` CCfld ) ` a ) e. NN0 \/ ( ( deg1 ` CCfld ) ` a ) e. { -oo } ) ) |
281 |
|
nn0ge0 |
|- ( ( ( deg1 ` CCfld ) ` a ) e. NN0 -> 0 <_ ( ( deg1 ` CCfld ) ` a ) ) |
282 |
281
|
iftrued |
|- ( ( ( deg1 ` CCfld ) ` a ) e. NN0 -> if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) = ( ( deg1 ` CCfld ) ` a ) ) |
283 |
|
id |
|- ( ( ( deg1 ` CCfld ) ` a ) e. NN0 -> ( ( deg1 ` CCfld ) ` a ) e. NN0 ) |
284 |
282 283
|
eqeltrd |
|- ( ( ( deg1 ` CCfld ) ` a ) e. NN0 -> if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) e. NN0 ) |
285 |
|
mnflt0 |
|- -oo < 0 |
286 |
|
mnfxr |
|- -oo e. RR* |
287 |
|
xrltnle |
|- ( ( -oo e. RR* /\ 0 e. RR* ) -> ( -oo < 0 <-> -. 0 <_ -oo ) ) |
288 |
286 262 287
|
mp2an |
|- ( -oo < 0 <-> -. 0 <_ -oo ) |
289 |
285 288
|
mpbi |
|- -. 0 <_ -oo |
290 |
|
elsni |
|- ( ( ( deg1 ` CCfld ) ` a ) e. { -oo } -> ( ( deg1 ` CCfld ) ` a ) = -oo ) |
291 |
290
|
breq2d |
|- ( ( ( deg1 ` CCfld ) ` a ) e. { -oo } -> ( 0 <_ ( ( deg1 ` CCfld ) ` a ) <-> 0 <_ -oo ) ) |
292 |
289 291
|
mtbiri |
|- ( ( ( deg1 ` CCfld ) ` a ) e. { -oo } -> -. 0 <_ ( ( deg1 ` CCfld ) ` a ) ) |
293 |
292
|
iffalsed |
|- ( ( ( deg1 ` CCfld ) ` a ) e. { -oo } -> if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) = 0 ) |
294 |
|
0nn0 |
|- 0 e. NN0 |
295 |
293 294
|
eqeltrdi |
|- ( ( ( deg1 ` CCfld ) ` a ) e. { -oo } -> if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) e. NN0 ) |
296 |
284 295
|
jaoi |
|- ( ( ( ( deg1 ` CCfld ) ` a ) e. NN0 \/ ( ( deg1 ` CCfld ) ` a ) e. { -oo } ) -> if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) e. NN0 ) |
297 |
280 296
|
syl |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) e. NN0 ) |
298 |
297
|
ad2antrr |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) e. NN0 ) |
299 |
|
fznn0 |
|- ( if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) e. NN0 -> ( k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) <-> ( k e. NN0 /\ k <_ if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) |
300 |
298 299
|
syl |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) <-> ( k e. NN0 /\ k <_ if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) |
301 |
276 300
|
sylibrd |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( ( ( coe1 ` a ) ` k ) =/= 0 -> k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) |
302 |
301
|
necon1bd |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( -. k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) -> ( ( coe1 ` a ) ` k ) = 0 ) ) |
303 |
254 302
|
mpd |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( ( coe1 ` a ) ` k ) = 0 ) |
304 |
303
|
oveq1d |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) = ( 0 x. ( z ^ k ) ) ) |
305 |
256 245
|
sylan2 |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( z ^ k ) e. CC ) |
306 |
305
|
mul02d |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( 0 x. ( z ^ k ) ) = 0 ) |
307 |
304 306
|
eqtrd |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) = 0 ) |
308 |
307
|
an32s |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) /\ z e. CC ) -> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) = 0 ) |
309 |
308
|
mpteq2dva |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( z e. CC |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> 0 ) ) |
310 |
|
fconstmpt |
|- ( CC X. { 0 } ) = ( z e. CC |-> 0 ) |
311 |
|
ringmnd |
|- ( CCfld e. Ring -> CCfld e. Mnd ) |
312 |
13 311
|
ax-mp |
|- CCfld e. Mnd |
313 |
7 21
|
pws0g |
|- ( ( CCfld e. Mnd /\ CC e. _V ) -> ( CC X. { 0 } ) = ( 0g ` ( CCfld ^s CC ) ) ) |
314 |
312 10 313
|
mp2an |
|- ( CC X. { 0 } ) = ( 0g ` ( CCfld ^s CC ) ) |
315 |
310 314
|
eqtr3i |
|- ( z e. CC |-> 0 ) = ( 0g ` ( CCfld ^s CC ) ) |
316 |
309 315
|
eqtrdi |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ k e. ( NN0 \ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( z e. CC |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) = ( 0g ` ( CCfld ^s CC ) ) ) |
317 |
316 166
|
suppss2 |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( ( k e. NN0 |-> ( z e. CC |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) supp ( 0g ` ( CCfld ^s CC ) ) ) C_ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) |
318 |
|
suppssfifsupp |
|- ( ( ( ( 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 ` ( CCfld ^s CC ) ) e. _V ) /\ ( ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) e. Fin /\ ( ( k e. NN0 |-> ( z e. CC |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) supp ( 0g ` ( CCfld ^s CC ) ) ) C_ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( k e. NN0 |-> ( z e. CC |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) finSupp ( 0g ` ( CCfld ^s CC ) ) ) |
319 |
251 252 317 318
|
syl12anc |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( k e. NN0 |-> ( z e. CC |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) finSupp ( 0g ` ( CCfld ^s CC ) ) ) |
320 |
7 8 9 242 166 243 247 319
|
pwsgsum |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( ( CCfld ^s CC ) gsum ( k e. NN0 |-> ( z e. CC |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) ) = ( z e. CC |-> ( CCfld gsum ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) ) ) |
321 |
|
fz0ssnn0 |
|- ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) C_ NN0 |
322 |
|
resmpt |
|- ( ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) C_ NN0 -> ( ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) |` ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) = ( k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) |
323 |
321 322
|
ax-mp |
|- ( ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) |` ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) = ( k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) |
324 |
323
|
oveq2i |
|- ( CCfld gsum ( ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) |` ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) = ( CCfld gsum ( k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) |
325 |
13 14
|
mp1i |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) -> CCfld e. CMnd ) |
326 |
165
|
a1i |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) -> NN0 e. _V ) |
327 |
246
|
fmpttd |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) -> ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) : NN0 --> CC ) |
328 |
307 326
|
suppss2 |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) -> ( ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) supp 0 ) C_ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) |
329 |
165
|
mptex |
|- ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) e. _V |
330 |
|
funmpt |
|- Fun ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) |
331 |
329 330 210
|
3pm3.2i |
|- ( ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) e. _V /\ Fun ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) /\ 0 e. _V ) |
332 |
331
|
a1i |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) -> ( ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) e. _V /\ Fun ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) /\ 0 e. _V ) ) |
333 |
|
fzfid |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) -> ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) e. Fin ) |
334 |
|
suppssfifsupp |
|- ( ( ( ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) e. _V /\ Fun ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) /\ 0 e. _V ) /\ ( ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) e. Fin /\ ( ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) supp 0 ) C_ ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) -> ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) finSupp 0 ) |
335 |
332 333 328 334
|
syl12anc |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) -> ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) finSupp 0 ) |
336 |
8 21 325 326 327 328 335
|
gsumres |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) -> ( CCfld gsum ( ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) |` ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) ) = ( CCfld gsum ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) ) |
337 |
|
elfznn0 |
|- ( k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) -> k e. NN0 ) |
338 |
337 246
|
sylan2 |
|- ( ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) /\ k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ) -> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) e. CC ) |
339 |
333 338
|
gsumfsum |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) -> ( CCfld gsum ( k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) = sum_ k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) |
340 |
324 336 339
|
3eqtr3a |
|- ( ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) /\ z e. CC ) -> ( CCfld gsum ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) = sum_ k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) |
341 |
340
|
mpteq2dva |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( z e. CC |-> ( CCfld gsum ( k e. NN0 |-> ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) |
342 |
241 320 341
|
3eqtrd |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( E ` a ) = ( z e. CC |-> sum_ k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) ) |
343 |
16
|
adantr |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> S C_ CC ) |
344 |
|
elplyr |
|- ( ( S C_ CC /\ if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) e. NN0 /\ ( coe1 ` a ) : NN0 --> S ) -> ( z e. CC |-> sum_ k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) e. ( Poly ` S ) ) |
345 |
343 297 179 344
|
syl3anc |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( z e. CC |-> sum_ k e. ( 0 ... if ( 0 <_ ( ( deg1 ` CCfld ) ` a ) , ( ( deg1 ` CCfld ) ` a ) , 0 ) ) ( ( ( coe1 ` a ) ` k ) x. ( z ^ k ) ) ) e. ( Poly ` S ) ) |
346 |
342 345
|
eqeltrd |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( E ` a ) e. ( Poly ` S ) ) |
347 |
|
eleq1 |
|- ( ( E ` a ) = f -> ( ( E ` a ) e. ( Poly ` S ) <-> f e. ( Poly ` S ) ) ) |
348 |
346 347
|
syl5ibcom |
|- ( ( S e. ( SubRing ` CCfld ) /\ a e. A ) -> ( ( E ` a ) = f -> f e. ( Poly ` S ) ) ) |
349 |
348
|
rexlimdva |
|- ( S e. ( SubRing ` CCfld ) -> ( E. a e. A ( E ` a ) = f -> f e. ( Poly ` S ) ) ) |
350 |
151 349
|
syl5 |
|- ( S e. ( SubRing ` CCfld ) -> ( f e. ( E " A ) -> f e. ( Poly ` S ) ) ) |
351 |
147 350
|
impbid |
|- ( S e. ( SubRing ` CCfld ) -> ( f e. ( Poly ` S ) <-> f e. ( E " A ) ) ) |
352 |
351
|
eqrdv |
|- ( S e. ( SubRing ` CCfld ) -> ( Poly ` S ) = ( E " A ) ) |