Step | Hyp | Ref
| Expression |
1 | | elply 23951 |
. . . . 5
Poly |
2 | 1 | simprbi 480 |
. . . 4
Poly
|
3 | | eqid 2622 |
. . . . . . . . 9
ℂfld s ℂfld s |
4 | | cnfldbas 19750 |
. . . . . . . . 9
ℂfld |
5 | | eqid 2622 |
. . . . . . . . 9
ℂfld s ℂfld s |
6 | | cnex 10017 |
. . . . . . . . . 10
|
7 | 6 | a1i 11 |
. . . . . . . . 9
SubRingℂfld
|
8 | | fzfid 12772 |
. . . . . . . . 9
SubRingℂfld
|
9 | | cnring 19768 |
. . . . . . . . . 10
ℂfld |
10 | | ringcmn 18581 |
. . . . . . . . . 10
ℂfld ℂfld CMnd |
11 | 9, 10 | mp1i 13 |
. . . . . . . . 9
SubRingℂfld
ℂfld CMnd |
12 | 4 | subrgss 18781 |
. . . . . . . . . . . . 13
SubRingℂfld
|
13 | 12 | ad2antrr 762 |
. . . . . . . . . . . 12
SubRingℂfld
|
14 | | elmapi 7879 |
. . . . . . . . . . . . . . 15
|
15 | 14 | ad2antll 765 |
. . . . . . . . . . . . . 14
SubRingℂfld
|
16 | | subrgsubg 18786 |
. . . . . . . . . . . . . . . . . . 19
SubRingℂfld
SubGrpℂfld |
17 | | cnfld0 19770 |
. . . . . . . . . . . . . . . . . . . 20
ℂfld |
18 | 17 | subg0cl 17602 |
. . . . . . . . . . . . . . . . . . 19
SubGrpℂfld
|
19 | 16, 18 | syl 17 |
. . . . . . . . . . . . . . . . . 18
SubRingℂfld
|
20 | 19 | adantr 481 |
. . . . . . . . . . . . . . . . 17
SubRingℂfld
|
21 | 20 | snssd 4340 |
. . . . . . . . . . . . . . . 16
SubRingℂfld
|
22 | | ssequn2 3786 |
. . . . . . . . . . . . . . . 16
|
23 | 21, 22 | sylib 208 |
. . . . . . . . . . . . . . 15
SubRingℂfld
|
24 | 23 | feq3d 6032 |
. . . . . . . . . . . . . 14
SubRingℂfld
|
25 | 15, 24 | mpbid 222 |
. . . . . . . . . . . . 13
SubRingℂfld
|
26 | | elfznn0 12433 |
. . . . . . . . . . . . 13
|
27 | | ffvelrn 6357 |
. . . . . . . . . . . . 13
|
28 | 25, 26, 27 | syl2an 494 |
. . . . . . . . . . . 12
SubRingℂfld
|
29 | 13, 28 | sseldd 3604 |
. . . . . . . . . . 11
SubRingℂfld
|
30 | 29 | adantrl 752 |
. . . . . . . . . 10
SubRingℂfld
|
31 | | simprl 794 |
. . . . . . . . . . 11
SubRingℂfld
|
32 | 26 | ad2antll 765 |
. . . . . . . . . . 11
SubRingℂfld
|
33 | | expcl 12878 |
. . . . . . . . . . 11
|
34 | 31, 32, 33 | syl2anc 693 |
. . . . . . . . . 10
SubRingℂfld
|
35 | 30, 34 | mulcld 10060 |
. . . . . . . . 9
SubRingℂfld
|
36 | | eqid 2622 |
. . . . . . . . . 10
|
37 | 6 | mptex 6486 |
. . . . . . . . . . 11
|
38 | 37 | a1i 11 |
. . . . . . . . . 10
SubRingℂfld
|
39 | | fvex 6201 |
. . . . . . . . . . 11
ℂfld s |
40 | 39 | a1i 11 |
. . . . . . . . . 10
SubRingℂfld
ℂfld s |
41 | 36, 8, 38, 40 | fsuppmptdm 8286 |
. . . . . . . . 9
SubRingℂfld
finSupp ℂfld s |
42 | 3, 4, 5, 7, 8, 11,
35, 41 | pwsgsum 18378 |
. . . . . . . 8
SubRingℂfld
ℂfld s g ℂfld g |
43 | | fzfid 12772 |
. . . . . . . . . 10
SubRingℂfld
|
44 | 35 | anassrs 680 |
. . . . . . . . . 10
SubRingℂfld
|
45 | 43, 44 | gsumfsum 19813 |
. . . . . . . . 9
SubRingℂfld
ℂfld g |
46 | 45 | mpteq2dva 4744 |
. . . . . . . 8
SubRingℂfld
ℂfld g |
47 | 42, 46 | eqtrd 2656 |
. . . . . . 7
SubRingℂfld
ℂfld s g |
48 | 3 | pwsring 18615 |
. . . . . . . . . 10
ℂfld ℂfld s |
49 | 9, 6, 48 | mp2an 708 |
. . . . . . . . 9
ℂfld s |
50 | | ringcmn 18581 |
. . . . . . . . 9
ℂfld s
ℂfld s CMnd |
51 | 49, 50 | mp1i 13 |
. . . . . . . 8
SubRingℂfld
ℂfld s
CMnd |
52 | | cncrng 19767 |
. . . . . . . . . . 11
ℂfld |
53 | | plypf1.e |
. . . . . . . . . . . 12
eval1ℂfld |
54 | | eqid 2622 |
. . . . . . . . . . . 12
Poly1ℂfld
Poly1ℂfld |
55 | 53, 54, 3, 4 | evl1rhm 19696 |
. . . . . . . . . . 11
ℂfld
Poly1ℂfld RingHom ℂfld s |
56 | 52, 55 | ax-mp 5 |
. . . . . . . . . 10
Poly1ℂfld RingHom ℂfld s |
57 | | plypf1.r |
. . . . . . . . . . . 12
ℂfld ↾s |
58 | | plypf1.p |
. . . . . . . . . . . 12
Poly1 |
59 | | plypf1.a |
. . . . . . . . . . . 12
|
60 | 54, 57, 58, 59 | subrgply1 19603 |
. . . . . . . . . . 11
SubRingℂfld
SubRingPoly1ℂfld |
61 | 60 | adantr 481 |
. . . . . . . . . 10
SubRingℂfld
SubRingPoly1ℂfld |
62 | | rhmima 18811 |
. . . . . . . . . 10
Poly1ℂfld RingHom ℂfld s SubRingPoly1ℂfld SubRingℂfld s |
63 | 56, 61, 62 | sylancr 695 |
. . . . . . . . 9
SubRingℂfld
SubRingℂfld s |
64 | | subrgsubg 18786 |
. . . . . . . . 9
SubRingℂfld s
SubGrpℂfld s |
65 | | subgsubm 17616 |
. . . . . . . . 9
SubGrpℂfld s
SubMndℂfld s |
66 | 63, 64, 65 | 3syl 18 |
. . . . . . . 8
SubRingℂfld
SubMndℂfld s |
67 | | eqid 2622 |
. . . . . . . . . . . 12
ℂfld s ℂfld s |
68 | 9 | a1i 11 |
. . . . . . . . . . . 12
SubRingℂfld
ℂfld |
69 | 6 | a1i 11 |
. . . . . . . . . . . 12
SubRingℂfld
|
70 | | fconst6g 6094 |
. . . . . . . . . . . . . 14
|
71 | 29, 70 | syl 17 |
. . . . . . . . . . . . 13
SubRingℂfld
|
72 | 3, 4, 67 | pwselbasb 16148 |
. . . . . . . . . . . . . 14
ℂfld ℂfld s
|
73 | 9, 6, 72 | mp2an 708 |
. . . . . . . . . . . . 13
ℂfld
s
|
74 | 71, 73 | sylibr 224 |
. . . . . . . . . . . 12
SubRingℂfld
ℂfld s |
75 | 34 | anass1rs 849 |
. . . . . . . . . . . . . 14
SubRingℂfld
|
76 | | eqid 2622 |
. . . . . . . . . . . . . 14
|
77 | 75, 76 | fmptd 6385 |
. . . . . . . . . . . . 13
SubRingℂfld
|
78 | 3, 4, 67 | pwselbasb 16148 |
. . . . . . . . . . . . . 14
ℂfld ℂfld s |
79 | 9, 6, 78 | mp2an 708 |
. . . . . . . . . . . . 13
ℂfld s
|
80 | 77, 79 | sylibr 224 |
. . . . . . . . . . . 12
SubRingℂfld
ℂfld s |
81 | | cnfldmul 19752 |
. . . . . . . . . . . 12
ℂfld |
82 | | eqid 2622 |
. . . . . . . . . . . 12
ℂfld s ℂfld s |
83 | 3, 67, 68, 69, 74, 80, 81, 82 | pwsmulrval 16151 |
. . . . . . . . . . 11
SubRingℂfld
ℂfld s |
84 | 29 | adantr 481 |
. . . . . . . . . . . 12
SubRingℂfld
|
85 | | fconstmpt 5163 |
. . . . . . . . . . . . 13
|
86 | 85 | a1i 11 |
. . . . . . . . . . . 12
SubRingℂfld
|
87 | | eqidd 2623 |
. . . . . . . . . . . 12
SubRingℂfld
|
88 | 69, 84, 75, 86, 87 | offval2 6914 |
. . . . . . . . . . 11
SubRingℂfld
|
89 | 83, 88 | eqtrd 2656 |
. . . . . . . . . 10
SubRingℂfld
ℂfld s |
90 | 63 | adantr 481 |
. . . . . . . . . . 11
SubRingℂfld
SubRingℂfld s |
91 | | eqid 2622 |
. . . . . . . . . . . . . 14
algScPoly1ℂfld algScPoly1ℂfld |
92 | 53, 54, 4, 91 | evl1sca 19698 |
. . . . . . . . . . . . 13
ℂfld algScPoly1ℂfld |
93 | 52, 29, 92 | sylancr 695 |
. . . . . . . . . . . 12
SubRingℂfld
algScPoly1ℂfld |
94 | | eqid 2622 |
. . . . . . . . . . . . . . . 16
Poly1ℂfld Poly1ℂfld |
95 | 94, 67 | rhmf 18726 |
. . . . . . . . . . . . . . 15
Poly1ℂfld
RingHom ℂfld s Poly1ℂfldℂfld s |
96 | 56, 95 | ax-mp 5 |
. . . . . . . . . . . . . 14
Poly1ℂfldℂfld s |
97 | | ffn 6045 |
. . . . . . . . . . . . . 14
Poly1ℂfldℂfld s
Poly1ℂfld |
98 | 96, 97 | mp1i 13 |
. . . . . . . . . . . . 13
SubRingℂfld
Poly1ℂfld |
99 | 94 | subrgss 18781 |
. . . . . . . . . . . . . . 15
SubRingPoly1ℂfld
Poly1ℂfld |
100 | 60, 99 | syl 17 |
. . . . . . . . . . . . . 14
SubRingℂfld
Poly1ℂfld |
101 | 100 | ad2antrr 762 |
. . . . . . . . . . . . 13
SubRingℂfld
Poly1ℂfld |
102 | | simpll 790 |
. . . . . . . . . . . . . . 15
SubRingℂfld
SubRingℂfld |
103 | 54, 91, 57, 58, 102, 59, 4, 29 | subrg1asclcl 19630 |
. . . . . . . . . . . . . 14
SubRingℂfld
algScPoly1ℂfld
|
104 | 28, 103 | mpbird 247 |
. . . . . . . . . . . . 13
SubRingℂfld
algScPoly1ℂfld |
105 | | fnfvima 6496 |
. . . . . . . . . . . . 13
Poly1ℂfld Poly1ℂfld
algScPoly1ℂfld algScPoly1ℂfld |
106 | 98, 101, 104, 105 | syl3anc 1326 |
. . . . . . . . . . . 12
SubRingℂfld
algScPoly1ℂfld |
107 | 93, 106 | eqeltrrd 2702 |
. . . . . . . . . . 11
SubRingℂfld
|
108 | 67 | subrgss 18781 |
. . . . . . . . . . . . . . . . 17
SubRingℂfld s
ℂfld s |
109 | 90, 108 | syl 17 |
. . . . . . . . . . . . . . . 16
SubRingℂfld
ℂfld s |
110 | 60 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . 19
SubRingℂfld
SubRingPoly1ℂfld |
111 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . 20
mulGrpPoly1ℂfld mulGrpPoly1ℂfld |
112 | 111 | subrgsubm 18793 |
. . . . . . . . . . . . . . . . . . 19
SubRingPoly1ℂfld
SubMndmulGrpPoly1ℂfld |
113 | 110, 112 | syl 17 |
. . . . . . . . . . . . . . . . . 18
SubRingℂfld
SubMndmulGrpPoly1ℂfld |
114 | 26 | adantl 482 |
. . . . . . . . . . . . . . . . . 18
SubRingℂfld
|
115 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . 19
var1ℂfld
var1ℂfld |
116 | 115, 102,
57, 58, 59 | subrgvr1cl 19632 |
. . . . . . . . . . . . . . . . . 18
SubRingℂfld
var1ℂfld |
117 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . 19
.gmulGrpPoly1ℂfld .gmulGrpPoly1ℂfld |
118 | 117 | submmulgcl 17585 |
. . . . . . . . . . . . . . . . . 18
SubMndmulGrpPoly1ℂfld
var1ℂfld .gmulGrpPoly1ℂfldvar1ℂfld |
119 | 113, 114,
116, 118 | syl3anc 1326 |
. . . . . . . . . . . . . . . . 17
SubRingℂfld
.gmulGrpPoly1ℂfldvar1ℂfld |
120 | | fnfvima 6496 |
. . . . . . . . . . . . . . . . 17
Poly1ℂfld Poly1ℂfld
.gmulGrpPoly1ℂfldvar1ℂfld
.gmulGrpPoly1ℂfldvar1ℂfld |
121 | 98, 101, 119, 120 | syl3anc 1326 |
. . . . . . . . . . . . . . . 16
SubRingℂfld
.gmulGrpPoly1ℂfldvar1ℂfld |
122 | 109, 121 | sseldd 3604 |
. . . . . . . . . . . . . . 15
SubRingℂfld
.gmulGrpPoly1ℂfldvar1ℂfld ℂfld
s
|
123 | 3, 4, 67, 68, 69, 122 | pwselbas 16149 |
. . . . . . . . . . . . . 14
SubRingℂfld
.gmulGrpPoly1ℂfldvar1ℂfld |
124 | 123 | feqmptd 6249 |
. . . . . . . . . . . . 13
SubRingℂfld
.gmulGrpPoly1ℂfldvar1ℂfld .gmulGrpPoly1ℂfldvar1ℂfld |
125 | 52 | a1i 11 |
. . . . . . . . . . . . . . . . 17
SubRingℂfld
ℂfld |
126 | | simpr 477 |
. . . . . . . . . . . . . . . . 17
SubRingℂfld
|
127 | 53, 115, 4, 54, 94, 125, 126 | evl1vard 19701 |
. . . . . . . . . . . . . . . . 17
SubRingℂfld
var1ℂfld
Poly1ℂfld var1ℂfld |
128 | | eqid 2622 |
. . . . . . . . . . . . . . . . 17
.gmulGrpℂfld .gmulGrpℂfld |
129 | 114 | adantr 481 |
. . . . . . . . . . . . . . . . 17
SubRingℂfld
|
130 | 53, 54, 4, 94, 125, 126, 127, 117, 128, 129 | evl1expd 19709 |
. . . . . . . . . . . . . . . 16
SubRingℂfld
.gmulGrpPoly1ℂfldvar1ℂfld Poly1ℂfld .gmulGrpPoly1ℂfldvar1ℂfld .gmulGrpℂfld |
131 | 130 | simprd 479 |
. . . . . . . . . . . . . . 15
SubRingℂfld
.gmulGrpPoly1ℂfldvar1ℂfld .gmulGrpℂfld |
132 | | cnfldexp 19779 |
. . . . . . . . . . . . . . . 16
.gmulGrpℂfld |
133 | 126, 129,
132 | syl2anc 693 |
. . . . . . . . . . . . . . 15
SubRingℂfld
.gmulGrpℂfld |
134 | 131, 133 | eqtrd 2656 |
. . . . . . . . . . . . . 14
SubRingℂfld
.gmulGrpPoly1ℂfldvar1ℂfld |
135 | 134 | mpteq2dva 4744 |
. . . . . . . . . . . . 13
SubRingℂfld
.gmulGrpPoly1ℂfldvar1ℂfld |
136 | 124, 135 | eqtrd 2656 |
. . . . . . . . . . . 12
SubRingℂfld
.gmulGrpPoly1ℂfldvar1ℂfld |
137 | 136, 121 | eqeltrrd 2702 |
. . . . . . . . . . 11
SubRingℂfld
|
138 | 82 | subrgmcl 18792 |
. . . . . . . . . . 11
SubRingℂfld s
ℂfld s |
139 | 90, 107, 137, 138 | syl3anc 1326 |
. . . . . . . . . 10
SubRingℂfld
ℂfld s |
140 | 89, 139 | eqeltrrd 2702 |
. . . . . . . . 9
SubRingℂfld
|
141 | 140, 36 | fmptd 6385 |
. . . . . . . 8
SubRingℂfld
|
142 | 36, 8, 140, 40 | fsuppmptdm 8286 |
. . . . . . . 8
SubRingℂfld
finSupp ℂfld s |
143 | 5, 51, 8, 66, 141, 142 | gsumsubmcl 18319 |
. . . . . . 7
SubRingℂfld
ℂfld s g |
144 | 47, 143 | eqeltrrd 2702 |
. . . . . 6
SubRingℂfld
|
145 | | eleq1 2689 |
. . . . . 6
|
146 | 144, 145 | syl5ibrcom 237 |
. . . . 5
SubRingℂfld
|
147 | 146 | rexlimdvva 3038 |
. . . 4
SubRingℂfld
|
148 | 2, 147 | syl5 34 |
. . 3
SubRingℂfld
Poly
|
149 | | ffun 6048 |
. . . . . 6
Poly1ℂfldℂfld s
|
150 | 96, 149 | ax-mp 5 |
. . . . 5
|
151 | | fvelima 6248 |
. . . . 5
|
152 | 150, 151 | mpan 706 |
. . . 4
|
153 | 100 | sselda 3603 |
. . . . . . . . . . 11
SubRingℂfld
Poly1ℂfld |
154 | | eqid 2622 |
. . . . . . . . . . . 12
Poly1ℂfld Poly1ℂfld |
155 | | eqid 2622 |
. . . . . . . . . . . 12
coe1 coe1 |
156 | 54, 115, 94, 154, 111, 117, 155 | ply1coe 19666 |
. . . . . . . . . . 11
ℂfld Poly1ℂfld Poly1ℂfld g coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld |
157 | 9, 153, 156 | sylancr 695 |
. . . . . . . . . 10
SubRingℂfld
Poly1ℂfld
g coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld |
158 | 157 | fveq2d 6195 |
. . . . . . . . 9
SubRingℂfld
Poly1ℂfld g coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld |
159 | | eqid 2622 |
. . . . . . . . . 10
Poly1ℂfld Poly1ℂfld |
160 | 54 | ply1ring 19618 |
. . . . . . . . . . . 12
ℂfld Poly1ℂfld |
161 | 9, 160 | ax-mp 5 |
. . . . . . . . . . 11
Poly1ℂfld
|
162 | | ringcmn 18581 |
. . . . . . . . . . 11
Poly1ℂfld
Poly1ℂfld
CMnd |
163 | 161, 162 | mp1i 13 |
. . . . . . . . . 10
SubRingℂfld
Poly1ℂfld CMnd |
164 | | ringmnd 18556 |
. . . . . . . . . . 11
ℂfld s
ℂfld s |
165 | 49, 164 | mp1i 13 |
. . . . . . . . . 10
SubRingℂfld
ℂfld s |
166 | | nn0ex 11298 |
. . . . . . . . . . 11
|
167 | 166 | a1i 11 |
. . . . . . . . . 10
SubRingℂfld
|
168 | | rhmghm 18725 |
. . . . . . . . . . . 12
Poly1ℂfld
RingHom ℂfld s
Poly1ℂfld ℂfld s |
169 | 56, 168 | ax-mp 5 |
. . . . . . . . . . 11
Poly1ℂfld ℂfld s |
170 | | ghmmhm 17670 |
. . . . . . . . . . 11
Poly1ℂfld
ℂfld s
Poly1ℂfld MndHom ℂfld s |
171 | 169, 170 | mp1i 13 |
. . . . . . . . . 10
SubRingℂfld
Poly1ℂfld MndHom ℂfld s |
172 | 54 | ply1lmod 19622 |
. . . . . . . . . . . . 13
ℂfld Poly1ℂfld |
173 | 9, 172 | mp1i 13 |
. . . . . . . . . . . 12
SubRingℂfld
Poly1ℂfld
|
174 | 12 | ad2antrr 762 |
. . . . . . . . . . . . 13
SubRingℂfld
|
175 | | eqid 2622 |
. . . . . . . . . . . . . . . . 17
|
176 | 155, 59, 58, 175 | coe1f 19581 |
. . . . . . . . . . . . . . . 16
coe1 |
177 | 57 | subrgbas 18789 |
. . . . . . . . . . . . . . . . 17
SubRingℂfld
|
178 | 177 | feq3d 6032 |
. . . . . . . . . . . . . . . 16
SubRingℂfld
coe1 coe1 |
179 | 176, 178 | syl5ibr 236 |
. . . . . . . . . . . . . . 15
SubRingℂfld
coe1 |
180 | 179 | imp 445 |
. . . . . . . . . . . . . 14
SubRingℂfld
coe1 |
181 | 180 | ffvelrnda 6359 |
. . . . . . . . . . . . 13
SubRingℂfld
coe1 |
182 | 174, 181 | sseldd 3604 |
. . . . . . . . . . . 12
SubRingℂfld
coe1 |
183 | 111 | ringmgp 18553 |
. . . . . . . . . . . . . 14
Poly1ℂfld
mulGrpPoly1ℂfld |
184 | 161, 183 | mp1i 13 |
. . . . . . . . . . . . 13
SubRingℂfld
mulGrpPoly1ℂfld |
185 | | simpr 477 |
. . . . . . . . . . . . 13
SubRingℂfld
|
186 | 115, 54, 94 | vr1cl 19587 |
. . . . . . . . . . . . . 14
ℂfld var1ℂfld Poly1ℂfld |
187 | 9, 186 | mp1i 13 |
. . . . . . . . . . . . 13
SubRingℂfld
var1ℂfld
Poly1ℂfld |
188 | 111, 94 | mgpbas 18495 |
. . . . . . . . . . . . . 14
Poly1ℂfld mulGrpPoly1ℂfld |
189 | 188, 117 | mulgnn0cl 17558 |
. . . . . . . . . . . . 13
mulGrpPoly1ℂfld var1ℂfld
Poly1ℂfld .gmulGrpPoly1ℂfldvar1ℂfld Poly1ℂfld |
190 | 184, 185,
187, 189 | syl3anc 1326 |
. . . . . . . . . . . 12
SubRingℂfld
.gmulGrpPoly1ℂfldvar1ℂfld Poly1ℂfld |
191 | 54 | ply1sca 19623 |
. . . . . . . . . . . . . 14
ℂfld ℂfld ScalarPoly1ℂfld |
192 | 9, 191 | ax-mp 5 |
. . . . . . . . . . . . 13
ℂfld
ScalarPoly1ℂfld |
193 | 94, 192, 154, 4 | lmodvscl 18880 |
. . . . . . . . . . . 12
Poly1ℂfld coe1 .gmulGrpPoly1ℂfldvar1ℂfld Poly1ℂfld coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld Poly1ℂfld |
194 | 173, 182,
190, 193 | syl3anc 1326 |
. . . . . . . . . . 11
SubRingℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld Poly1ℂfld |
195 | | eqid 2622 |
. . . . . . . . . . 11
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld |
196 | 194, 195 | fmptd 6385 |
. . . . . . . . . 10
SubRingℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfldPoly1ℂfld |
197 | 166 | mptex 6486 |
. . . . . . . . . . . . 13
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld |
198 | | funmpt 5926 |
. . . . . . . . . . . . 13
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld |
199 | | fvex 6201 |
. . . . . . . . . . . . 13
Poly1ℂfld |
200 | 197, 198,
199 | 3pm3.2i 1239 |
. . . . . . . . . . . 12
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld Poly1ℂfld |
201 | 200 | a1i 11 |
. . . . . . . . . . 11
SubRingℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld Poly1ℂfld |
202 | 155, 94, 54, 17 | coe1sfi 19583 |
. . . . . . . . . . . . 13
Poly1ℂfld
coe1 finSupp |
203 | 153, 202 | syl 17 |
. . . . . . . . . . . 12
SubRingℂfld
coe1
finSupp |
204 | 203 | fsuppimpd 8282 |
. . . . . . . . . . 11
SubRingℂfld
coe1 supp |
205 | 180 | feqmptd 6249 |
. . . . . . . . . . . . . 14
SubRingℂfld
coe1 coe1 |
206 | 205 | oveq1d 6665 |
. . . . . . . . . . . . 13
SubRingℂfld
coe1 supp coe1 supp |
207 | | eqimss2 3658 |
. . . . . . . . . . . . 13
coe1 supp coe1 supp
coe1 supp coe1 supp |
208 | 206, 207 | syl 17 |
. . . . . . . . . . . 12
SubRingℂfld
coe1 supp
coe1 supp
|
209 | 9, 172 | mp1i 13 |
. . . . . . . . . . . . 13
SubRingℂfld
Poly1ℂfld |
210 | 94, 192, 154, 17, 159 | lmod0vs 18896 |
. . . . . . . . . . . . 13
Poly1ℂfld Poly1ℂfld Poly1ℂfld Poly1ℂfld |
211 | 209, 210 | sylan 488 |
. . . . . . . . . . . 12
SubRingℂfld
Poly1ℂfld Poly1ℂfld Poly1ℂfld |
212 | | c0ex 10034 |
. . . . . . . . . . . . 13
|
213 | 212 | a1i 11 |
. . . . . . . . . . . 12
SubRingℂfld
|
214 | 208, 211,
181, 190, 213 | suppssov1 7327 |
. . . . . . . . . . 11
SubRingℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld supp Poly1ℂfld coe1 supp |
215 | | suppssfifsupp 8290 |
. . . . . . . . . . 11
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld Poly1ℂfld coe1 supp coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld supp Poly1ℂfld coe1 supp coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld finSupp Poly1ℂfld |
216 | 201, 204,
214, 215 | syl12anc 1324 |
. . . . . . . . . 10
SubRingℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld finSupp Poly1ℂfld |
217 | 94, 159, 163, 165, 167, 171, 196, 216 | gsummhm 18338 |
. . . . . . . . 9
SubRingℂfld
ℂfld s
g coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld Poly1ℂfld g coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld |
218 | | eqidd 2623 |
. . . . . . . . . . . 12
SubRingℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld |
219 | 96 | a1i 11 |
. . . . . . . . . . . . 13
SubRingℂfld
Poly1ℂfldℂfld s |
220 | 219 | feqmptd 6249 |
. . . . . . . . . . . 12
SubRingℂfld
Poly1ℂfld |
221 | | fveq2 6191 |
. . . . . . . . . . . 12
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld |
222 | 194, 218,
220, 221 | fmptco 6396 |
. . . . . . . . . . 11
SubRingℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld |
223 | 9 | a1i 11 |
. . . . . . . . . . . . . . 15
SubRingℂfld
ℂfld |
224 | 6 | a1i 11 |
. . . . . . . . . . . . . . 15
SubRingℂfld
|
225 | 96 | ffvelrni 6358 |
. . . . . . . . . . . . . . . 16
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld Poly1ℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld ℂfld s |
226 | 194, 225 | syl 17 |
. . . . . . . . . . . . . . 15
SubRingℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld ℂfld s |
227 | 3, 4, 67, 223, 224, 226 | pwselbas 16149 |
. . . . . . . . . . . . . 14
SubRingℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld |
228 | 227 | feqmptd 6249 |
. . . . . . . . . . . . 13
SubRingℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld |
229 | 52 | a1i 11 |
. . . . . . . . . . . . . . . 16
SubRingℂfld
ℂfld |
230 | | simpr 477 |
. . . . . . . . . . . . . . . 16
SubRingℂfld
|
231 | 53, 115, 4, 54, 94, 229, 230 | evl1vard 19701 |
. . . . . . . . . . . . . . . . . 18
SubRingℂfld
var1ℂfld Poly1ℂfld var1ℂfld |
232 | 185 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
SubRingℂfld
|
233 | 53, 54, 4, 94, 229, 230, 231, 117, 128, 232 | evl1expd 19709 |
. . . . . . . . . . . . . . . . 17
SubRingℂfld
.gmulGrpPoly1ℂfldvar1ℂfld Poly1ℂfld .gmulGrpPoly1ℂfldvar1ℂfld .gmulGrpℂfld |
234 | 230, 232,
132 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . 19
SubRingℂfld
.gmulGrpℂfld |
235 | 234 | eqeq2d 2632 |
. . . . . . . . . . . . . . . . . 18
SubRingℂfld
.gmulGrpPoly1ℂfldvar1ℂfld .gmulGrpℂfld .gmulGrpPoly1ℂfldvar1ℂfld |
236 | 235 | anbi2d 740 |
. . . . . . . . . . . . . . . . 17
SubRingℂfld
.gmulGrpPoly1ℂfldvar1ℂfld Poly1ℂfld .gmulGrpPoly1ℂfldvar1ℂfld .gmulGrpℂfld .gmulGrpPoly1ℂfldvar1ℂfld Poly1ℂfld .gmulGrpPoly1ℂfldvar1ℂfld |
237 | 233, 236 | mpbid 222 |
. . . . . . . . . . . . . . . 16
SubRingℂfld
.gmulGrpPoly1ℂfldvar1ℂfld Poly1ℂfld .gmulGrpPoly1ℂfldvar1ℂfld |
238 | 182 | adantr 481 |
. . . . . . . . . . . . . . . 16
SubRingℂfld
coe1 |
239 | 53, 54, 4, 94, 229, 230, 237, 238, 154, 81 | evl1vsd 19708 |
. . . . . . . . . . . . . . 15
SubRingℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld Poly1ℂfld coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld coe1 |
240 | 239 | simprd 479 |
. . . . . . . . . . . . . 14
SubRingℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld coe1 |
241 | 240 | mpteq2dva 4744 |
. . . . . . . . . . . . 13
SubRingℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld coe1 |
242 | 228, 241 | eqtrd 2656 |
. . . . . . . . . . . 12
SubRingℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld coe1 |
243 | 242 | mpteq2dva 4744 |
. . . . . . . . . . 11
SubRingℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld
coe1 |
244 | 222, 243 | eqtrd 2656 |
. . . . . . . . . 10
SubRingℂfld
coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld
coe1 |
245 | 244 | oveq2d 6666 |
. . . . . . . . 9
SubRingℂfld
ℂfld s
g coe1Poly1ℂfld.gmulGrpPoly1ℂfldvar1ℂfld ℂfld s g coe1 |
246 | 158, 217,
245 | 3eqtr2d 2662 |
. . . . . . . 8
SubRingℂfld
ℂfld s g coe1 |
247 | 6 | a1i 11 |
. . . . . . . . 9
SubRingℂfld
|
248 | 9, 10 | mp1i 13 |
. . . . . . . . 9
SubRingℂfld
ℂfld CMnd |
249 | 182 | adantlr 751 |
. . . . . . . . . . 11
SubRingℂfld
coe1 |
250 | 33 | adantll 750 |
. . . . . . . . . . 11
SubRingℂfld
|
251 | 249, 250 | mulcld 10060 |
. . . . . . . . . 10
SubRingℂfld
coe1 |
252 | 251 | anasss 679 |
. . . . . . . . 9
SubRingℂfld
coe1 |
253 | 166 | mptex 6486 |
. . . . . . . . . . . 12
coe1 |
254 | | funmpt 5926 |
. . . . . . . . . . . 12
coe1 |
255 | 253, 254,
39 | 3pm3.2i 1239 |
. . . . . . . . . . 11
coe1
coe1 ℂfld s |
256 | 255 | a1i 11 |
. . . . . . . . . 10
SubRingℂfld
coe1 coe1 ℂfld s |
257 | | fzfid 12772 |
. . . . . . . . . 10
SubRingℂfld
deg1 ℂfld deg1 ℂfld |
258 | | eldifn 3733 |
. . . . . . . . . . . . . . . . . 18
deg1 ℂfld deg1 ℂfld
deg1 ℂfld deg1 ℂfld |
259 | 258 | adantl 482 |
. . . . . . . . . . . . . . . . 17
SubRingℂfld
deg1 ℂfld deg1 ℂfld
deg1 ℂfld deg1 ℂfld |
260 | 153 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . 22
SubRingℂfld
deg1 ℂfld deg1 ℂfld Poly1ℂfld |
261 | | eldifi 3732 |
. . . . . . . . . . . . . . . . . . . . . . 23
deg1 ℂfld deg1 ℂfld
|
262 | 261 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . 22
SubRingℂfld
deg1 ℂfld deg1 ℂfld |
263 | | eqid 2622 |
. . . . . . . . . . . . . . . . . . . . . . . 24
deg1 ℂfld
deg1 ℂfld |
264 | 263, 54, 94, 17, 155 | deg1ge 23858 |
. . . . . . . . . . . . . . . . . . . . . . 23
Poly1ℂfld coe1 deg1 ℂfld |
265 | 264 | 3expia 1267 |
. . . . . . . . . . . . . . . . . . . . . 22
Poly1ℂfld coe1
deg1 ℂfld |
266 | 260, 262,
265 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . . 21
SubRingℂfld
deg1 ℂfld deg1 ℂfld coe1
deg1 ℂfld |
267 | | 0xr 10086 |
. . . . . . . . . . . . . . . . . . . . . . 23
|
268 | 263, 54, 94 | deg1xrcl 23842 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
Poly1ℂfld
deg1 ℂfld |
269 | 153, 268 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
SubRingℂfld
deg1 ℂfld |
270 | 269 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . . 23
SubRingℂfld
deg1 ℂfld deg1 ℂfld deg1 ℂfld |
271 | | xrmax2 12007 |
. . . . . . . . . . . . . . . . . . . . . . 23
deg1 ℂfld deg1 ℂfld deg1 ℂfld deg1 ℂfld |
272 | 267, 270,
271 | sylancr 695 |
. . . . . . . . . . . . . . . . . . . . . 22
SubRingℂfld
deg1 ℂfld deg1 ℂfld deg1 ℂfld deg1 ℂfld deg1 ℂfld |
273 | 262 | nn0red 11352 |
. . . . . . . . . . . . . . . . . . . . . . . 24
SubRingℂfld
deg1 ℂfld deg1 ℂfld |
274 | 273 | rexrd 10089 |
. . . . . . . . . . . . . . . . . . . . . . 23
SubRingℂfld
deg1 ℂfld deg1 ℂfld |
275 | | ifcl 4130 |
. . . . . . . . . . . . . . . . . . . . . . . 24
deg1 ℂfld
deg1 ℂfld deg1 ℂfld |
276 | 270, 267,
275 | sylancl 694 |
. . . . . . . . . . . . . . . . . . . . . . 23
SubRingℂfld
deg1 ℂfld deg1 ℂfld deg1 ℂfld deg1 ℂfld |
277 | | xrletr 11989 |
. . . . . . . . . . . . . . . . . . . . . . 23
deg1 ℂfld
deg1 ℂfld deg1 ℂfld deg1 ℂfld deg1 ℂfld deg1 ℂfld deg1 ℂfld deg1 ℂfld deg1 ℂfld |
278 | 274, 270,
276, 277 | syl3anc 1326 |
. . . . . . . . . . . . . . . . . . . . . 22
SubRingℂfld
deg1 ℂfld deg1 ℂfld deg1 ℂfld deg1 ℂfld deg1 ℂfld deg1 ℂfld deg1 ℂfld deg1 ℂfld |
279 | 272, 278 | mpan2d 710 |
. . . . . . . . . . . . . . . . . . . . 21
SubRingℂfld
deg1 ℂfld deg1 ℂfld
deg1 ℂfld
deg1 ℂfld deg1 ℂfld |
280 | 266, 279 | syld 47 |
. . . . . . . . . . . . . . . . . . . 20
SubRingℂfld
deg1 ℂfld deg1 ℂfld coe1
deg1 ℂfld deg1 ℂfld |
281 | 280, 262 | jctild 566 |
. . . . . . . . . . . . . . . . . . 19
SubRingℂfld
deg1 ℂfld deg1 ℂfld coe1
deg1 ℂfld deg1 ℂfld |
282 | 263, 54, 94 | deg1cl 23843 |
. . . . . . . . . . . . . . . . . . . . . . . 24
Poly1ℂfld
deg1 ℂfld |
283 | 153, 282 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
SubRingℂfld
deg1 ℂfld |
284 | | elun 3753 |
. . . . . . . . . . . . . . . . . . . . . . 23
deg1 ℂfld
deg1 ℂfld
deg1 ℂfld |
285 | 283, 284 | sylib 208 |
. . . . . . . . . . . . . . . . . . . . . 22
SubRingℂfld
deg1 ℂfld
deg1 ℂfld |
286 | | nn0ge0 11318 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
deg1 ℂfld
deg1 ℂfld |
287 | 286 | iftrued 4094 |
. . . . . . . . . . . . . . . . . . . . . . . 24
deg1 ℂfld
deg1 ℂfld deg1 ℂfld
deg1 ℂfld |
288 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . . 24
deg1 ℂfld
deg1 ℂfld |
289 | 287, 288 | eqeltrd 2701 |
. . . . . . . . . . . . . . . . . . . . . . 23
deg1 ℂfld
deg1 ℂfld deg1 ℂfld |
290 | | mnflt0 11959 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
|
291 | | mnfxr 10096 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
|
292 | | xrltnle 10105 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
|
293 | 291, 267,
292 | mp2an 708 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
|
294 | 290, 293 | mpbi 220 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
|
295 | | elsni 4194 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
deg1 ℂfld
deg1 ℂfld |
296 | 295 | breq2d 4665 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
deg1 ℂfld deg1 ℂfld
|
297 | 294, 296 | mtbiri 317 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
deg1 ℂfld deg1 ℂfld |
298 | 297 | iffalsed 4097 |
. . . . . . . . . . . . . . . . . . . . . . . 24
deg1 ℂfld
deg1 ℂfld deg1 ℂfld |
299 | | 0nn0 11307 |
. . . . . . . . . . . . . . . . . . . . . . . 24
|
300 | 298, 299 | syl6eqel 2709 |
. . . . . . . . . . . . . . . . . . . . . . 23
deg1 ℂfld
deg1 ℂfld deg1 ℂfld |
301 | 289, 300 | jaoi 394 |
. . . . . . . . . . . . . . . . . . . . . 22
deg1 ℂfld
deg1 ℂfld
deg1 ℂfld deg1 ℂfld |
302 | 285, 301 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
SubRingℂfld
deg1 ℂfld deg1 ℂfld |
303 | 302 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . 20
SubRingℂfld
deg1 ℂfld deg1 ℂfld deg1 ℂfld deg1 ℂfld |
304 | | fznn0 12432 |
. . . . . . . . . . . . . . . . . . . 20
deg1 ℂfld deg1 ℂfld deg1 ℂfld deg1 ℂfld
deg1 ℂfld deg1 ℂfld |
305 | 303, 304 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
SubRingℂfld
deg1 ℂfld deg1 ℂfld deg1 ℂfld deg1 ℂfld
deg1 ℂfld deg1 ℂfld |
306 | 281, 305 | sylibrd 249 |
. . . . . . . . . . . . . . . . . 18
SubRingℂfld
deg1 ℂfld deg1 ℂfld coe1 deg1 ℂfld deg1 ℂfld |
307 | 306 | necon1bd 2812 |
. . . . . . . . . . . . . . . . 17
SubRingℂfld
deg1 ℂfld deg1 ℂfld
deg1 ℂfld deg1 ℂfld coe1 |
308 | 259, 307 | mpd 15 |
. . . . . . . . . . . . . . . 16
SubRingℂfld
deg1 ℂfld deg1 ℂfld coe1 |
309 | 308 | oveq1d 6665 |
. . . . . . . . . . . . . . 15
SubRingℂfld
deg1 ℂfld deg1 ℂfld coe1 |
310 | 261, 250 | sylan2 491 |
. . . . . . . . . . . . . . . 16
SubRingℂfld
deg1 ℂfld deg1 ℂfld |
311 | 310 | mul02d 10234 |
. . . . . . . . . . . . . . 15
SubRingℂfld
deg1 ℂfld deg1 ℂfld |
312 | 309, 311 | eqtrd 2656 |
. . . . . . . . . . . . . 14
SubRingℂfld
deg1 ℂfld deg1 ℂfld coe1 |
313 | 312 | an32s 846 |
. . . . . . . . . . . . 13
SubRingℂfld
deg1 ℂfld deg1 ℂfld coe1 |
314 | 313 | mpteq2dva 4744 |
. . . . . . . . . . . 12
SubRingℂfld
deg1 ℂfld deg1 ℂfld coe1 |
315 | | fconstmpt 5163 |
. . . . . . . . . . . . 13
|
316 | | ringmnd 18556 |
. . . . . . . . . . . . . . 15
ℂfld ℂfld |
317 | 9, 316 | ax-mp 5 |
. . . . . . . . . . . . . 14
ℂfld |
318 | 3, 17 | pws0g 17326 |
. . . . . . . . . . . . . 14
ℂfld
ℂfld s |
319 | 317, 6, 318 | mp2an 708 |
. . . . . . . . . . . . 13
ℂfld s |
320 | 315, 319 | eqtr3i 2646 |
. . . . . . . . . . . 12
ℂfld s |
321 | 314, 320 | syl6eq 2672 |
. . . . . . . . . . 11
SubRingℂfld
deg1 ℂfld deg1 ℂfld coe1 ℂfld s |
322 | 321, 167 | suppss2 7329 |
. . . . . . . . . 10
SubRingℂfld
coe1 supp ℂfld s deg1 ℂfld deg1 ℂfld |
323 | | suppssfifsupp 8290 |
. . . . . . . . . 10
coe1 coe1 ℂfld s deg1 ℂfld deg1 ℂfld coe1 supp ℂfld s deg1 ℂfld deg1 ℂfld coe1 finSupp ℂfld s |
324 | 256, 257,
322, 323 | syl12anc 1324 |
. . . . . . . . 9
SubRingℂfld
coe1 finSupp ℂfld s |
325 | 3, 4, 5, 247, 167, 248, 252, 324 | pwsgsum 18378 |
. . . . . . . 8
SubRingℂfld
ℂfld s
g coe1 ℂfld g coe1 |
326 | | fz0ssnn0 12435 |
. . . . . . . . . . . 12
deg1 ℂfld deg1 ℂfld |
327 | | resmpt 5449 |
. . . . . . . . . . . 12
deg1 ℂfld deg1 ℂfld
coe1 deg1 ℂfld deg1 ℂfld
deg1 ℂfld deg1 ℂfld coe1 |
328 | 326, 327 | ax-mp 5 |
. . . . . . . . . . 11
coe1 deg1 ℂfld deg1 ℂfld
deg1 ℂfld deg1 ℂfld coe1 |
329 | 328 | oveq2i 6661 |
. . . . . . . . . 10
ℂfld g coe1
deg1 ℂfld deg1 ℂfld ℂfld g
deg1 ℂfld deg1 ℂfld coe1 |
330 | 9, 10 | mp1i 13 |
. . . . . . . . . . 11
SubRingℂfld
ℂfld CMnd |
331 | 166 | a1i 11 |
. . . . . . . . . . 11
SubRingℂfld
|
332 | | eqid 2622 |
. . . . . . . . . . . 12
coe1 coe1 |
333 | 251, 332 | fmptd 6385 |
. . . . . . . . . . 11
SubRingℂfld
coe1 |
334 | 312, 331 | suppss2 7329 |
. . . . . . . . . . 11
SubRingℂfld
coe1 supp
deg1 ℂfld deg1 ℂfld |
335 | 166 | mptex 6486 |
. . . . . . . . . . . . . 14
coe1 |
336 | | funmpt 5926 |
. . . . . . . . . . . . . 14
coe1 |
337 | 335, 336,
212 | 3pm3.2i 1239 |
. . . . . . . . . . . . 13
coe1 coe1 |
338 | 337 | a1i 11 |
. . . . . . . . . . . 12
SubRingℂfld
coe1 coe1 |
339 | | fzfid 12772 |
. . . . . . . . . . . 12
SubRingℂfld
deg1 ℂfld deg1 ℂfld |
340 | | suppssfifsupp 8290 |
. . . . . . . . . . . 12
coe1 coe1
deg1 ℂfld deg1 ℂfld coe1 supp
deg1 ℂfld deg1 ℂfld coe1 finSupp |
341 | 338, 339,
334, 340 | syl12anc 1324 |
. . . . . . . . . . 11
SubRingℂfld
coe1 finSupp |
342 | 4, 17, 330, 331, 333, 334, 341 | gsumres 18314 |
. . . . . . . . . 10
SubRingℂfld
ℂfld g coe1
deg1 ℂfld deg1 ℂfld ℂfld g
coe1 |
343 | | elfznn0 12433 |
. . . . . . . . . . . 12
deg1 ℂfld deg1 ℂfld |
344 | 343, 251 | sylan2 491 |
. . . . . . . . . . 11
SubRingℂfld
deg1 ℂfld deg1 ℂfld
coe1 |
345 | 339, 344 | gsumfsum 19813 |
. . . . . . . . . 10
SubRingℂfld
ℂfld g
deg1 ℂfld deg1 ℂfld coe1
deg1 ℂfld deg1 ℂfld coe1 |
346 | 329, 342,
345 | 3eqtr3a 2680 |
. . . . . . . . 9
SubRingℂfld
ℂfld g coe1
deg1 ℂfld deg1 ℂfld coe1 |
347 | 346 | mpteq2dva 4744 |
. . . . . . . 8
SubRingℂfld
ℂfld g
coe1
deg1 ℂfld deg1 ℂfld coe1 |
348 | 246, 325,
347 | 3eqtrd 2660 |
. . . . . . 7
SubRingℂfld
deg1 ℂfld deg1 ℂfld coe1 |
349 | 12 | adantr 481 |
. . . . . . . 8
SubRingℂfld
|
350 | | elplyr 23957 |
. . . . . . . 8
deg1 ℂfld deg1 ℂfld
coe1
deg1 ℂfld deg1 ℂfld coe1 Poly |
351 | 349, 302,
180, 350 | syl3anc 1326 |
. . . . . . 7
SubRingℂfld
deg1 ℂfld deg1 ℂfld coe1 Poly |
352 | 348, 351 | eqeltrd 2701 |
. . . . . 6
SubRingℂfld
Poly |
353 | | eleq1 2689 |
. . . . . 6
Poly
Poly |
354 | 352, 353 | syl5ibcom 235 |
. . . . 5
SubRingℂfld
Poly |
355 | 354 | rexlimdva 3031 |
. . . 4
SubRingℂfld
Poly |
356 | 152, 355 | syl5 34 |
. . 3
SubRingℂfld
Poly |
357 | 148, 356 | impbid 202 |
. 2
SubRingℂfld
Poly |
358 | 357 | eqrdv 2620 |
1
SubRingℂfld
Poly |