Step |
Hyp |
Ref |
Expression |
1 |
|
coe1mul3.s |
|- Y = ( Poly1 ` R ) |
2 |
|
coe1mul3.t |
|- .xb = ( .r ` Y ) |
3 |
|
coe1mul3.u |
|- .x. = ( .r ` R ) |
4 |
|
coe1mul3.b |
|- B = ( Base ` Y ) |
5 |
|
coe1mul3.d |
|- D = ( deg1 ` R ) |
6 |
|
coe1mul3.r |
|- ( ph -> R e. Ring ) |
7 |
|
coe1mul3.f1 |
|- ( ph -> F e. B ) |
8 |
|
coe1mul3.f2 |
|- ( ph -> I e. NN0 ) |
9 |
|
coe1mul3.f3 |
|- ( ph -> ( D ` F ) <_ I ) |
10 |
|
coe1mul3.g1 |
|- ( ph -> G e. B ) |
11 |
|
coe1mul3.g2 |
|- ( ph -> J e. NN0 ) |
12 |
|
coe1mul3.g3 |
|- ( ph -> ( D ` G ) <_ J ) |
13 |
1 2 3 4
|
coe1mul |
|- ( ( R e. Ring /\ F e. B /\ G e. B ) -> ( coe1 ` ( F .xb G ) ) = ( x e. NN0 |-> ( R gsum ( y e. ( 0 ... x ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( x - y ) ) ) ) ) ) ) |
14 |
6 7 10 13
|
syl3anc |
|- ( ph -> ( coe1 ` ( F .xb G ) ) = ( x e. NN0 |-> ( R gsum ( y e. ( 0 ... x ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( x - y ) ) ) ) ) ) ) |
15 |
14
|
fveq1d |
|- ( ph -> ( ( coe1 ` ( F .xb G ) ) ` ( I + J ) ) = ( ( x e. NN0 |-> ( R gsum ( y e. ( 0 ... x ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( x - y ) ) ) ) ) ) ` ( I + J ) ) ) |
16 |
8 11
|
nn0addcld |
|- ( ph -> ( I + J ) e. NN0 ) |
17 |
|
oveq2 |
|- ( x = ( I + J ) -> ( 0 ... x ) = ( 0 ... ( I + J ) ) ) |
18 |
|
fvoveq1 |
|- ( x = ( I + J ) -> ( ( coe1 ` G ) ` ( x - y ) ) = ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) |
19 |
18
|
oveq2d |
|- ( x = ( I + J ) -> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( x - y ) ) ) = ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) |
20 |
17 19
|
mpteq12dv |
|- ( x = ( I + J ) -> ( y e. ( 0 ... x ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( x - y ) ) ) ) = ( y e. ( 0 ... ( I + J ) ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ) |
21 |
20
|
oveq2d |
|- ( x = ( I + J ) -> ( R gsum ( y e. ( 0 ... x ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( x - y ) ) ) ) ) = ( R gsum ( y e. ( 0 ... ( I + J ) ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ) ) |
22 |
|
eqid |
|- ( x e. NN0 |-> ( R gsum ( y e. ( 0 ... x ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( x - y ) ) ) ) ) ) = ( x e. NN0 |-> ( R gsum ( y e. ( 0 ... x ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( x - y ) ) ) ) ) ) |
23 |
|
ovex |
|- ( R gsum ( y e. ( 0 ... ( I + J ) ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ) e. _V |
24 |
21 22 23
|
fvmpt |
|- ( ( I + J ) e. NN0 -> ( ( x e. NN0 |-> ( R gsum ( y e. ( 0 ... x ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( x - y ) ) ) ) ) ) ` ( I + J ) ) = ( R gsum ( y e. ( 0 ... ( I + J ) ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ) ) |
25 |
16 24
|
syl |
|- ( ph -> ( ( x e. NN0 |-> ( R gsum ( y e. ( 0 ... x ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( x - y ) ) ) ) ) ) ` ( I + J ) ) = ( R gsum ( y e. ( 0 ... ( I + J ) ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ) ) |
26 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
27 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
28 |
|
ringmnd |
|- ( R e. Ring -> R e. Mnd ) |
29 |
6 28
|
syl |
|- ( ph -> R e. Mnd ) |
30 |
|
ovexd |
|- ( ph -> ( 0 ... ( I + J ) ) e. _V ) |
31 |
8
|
nn0red |
|- ( ph -> I e. RR ) |
32 |
|
nn0addge1 |
|- ( ( I e. RR /\ J e. NN0 ) -> I <_ ( I + J ) ) |
33 |
31 11 32
|
syl2anc |
|- ( ph -> I <_ ( I + J ) ) |
34 |
|
fznn0 |
|- ( ( I + J ) e. NN0 -> ( I e. ( 0 ... ( I + J ) ) <-> ( I e. NN0 /\ I <_ ( I + J ) ) ) ) |
35 |
16 34
|
syl |
|- ( ph -> ( I e. ( 0 ... ( I + J ) ) <-> ( I e. NN0 /\ I <_ ( I + J ) ) ) ) |
36 |
8 33 35
|
mpbir2and |
|- ( ph -> I e. ( 0 ... ( I + J ) ) ) |
37 |
6
|
adantr |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> R e. Ring ) |
38 |
|
eqid |
|- ( coe1 ` F ) = ( coe1 ` F ) |
39 |
38 4 1 26
|
coe1f |
|- ( F e. B -> ( coe1 ` F ) : NN0 --> ( Base ` R ) ) |
40 |
7 39
|
syl |
|- ( ph -> ( coe1 ` F ) : NN0 --> ( Base ` R ) ) |
41 |
|
elfznn0 |
|- ( y e. ( 0 ... ( I + J ) ) -> y e. NN0 ) |
42 |
|
ffvelrn |
|- ( ( ( coe1 ` F ) : NN0 --> ( Base ` R ) /\ y e. NN0 ) -> ( ( coe1 ` F ) ` y ) e. ( Base ` R ) ) |
43 |
40 41 42
|
syl2an |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( coe1 ` F ) ` y ) e. ( Base ` R ) ) |
44 |
|
eqid |
|- ( coe1 ` G ) = ( coe1 ` G ) |
45 |
44 4 1 26
|
coe1f |
|- ( G e. B -> ( coe1 ` G ) : NN0 --> ( Base ` R ) ) |
46 |
10 45
|
syl |
|- ( ph -> ( coe1 ` G ) : NN0 --> ( Base ` R ) ) |
47 |
|
fznn0sub |
|- ( y e. ( 0 ... ( I + J ) ) -> ( ( I + J ) - y ) e. NN0 ) |
48 |
|
ffvelrn |
|- ( ( ( coe1 ` G ) : NN0 --> ( Base ` R ) /\ ( ( I + J ) - y ) e. NN0 ) -> ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) e. ( Base ` R ) ) |
49 |
46 47 48
|
syl2an |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) e. ( Base ` R ) ) |
50 |
26 3
|
ringcl |
|- ( ( R e. Ring /\ ( ( coe1 ` F ) ` y ) e. ( Base ` R ) /\ ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) e. ( Base ` R ) ) -> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) e. ( Base ` R ) ) |
51 |
37 43 49 50
|
syl3anc |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) e. ( Base ` R ) ) |
52 |
51
|
fmpttd |
|- ( ph -> ( y e. ( 0 ... ( I + J ) ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) : ( 0 ... ( I + J ) ) --> ( Base ` R ) ) |
53 |
|
eldifsn |
|- ( y e. ( ( 0 ... ( I + J ) ) \ { I } ) <-> ( y e. ( 0 ... ( I + J ) ) /\ y =/= I ) ) |
54 |
41
|
adantl |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> y e. NN0 ) |
55 |
54
|
nn0red |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> y e. RR ) |
56 |
31
|
adantr |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> I e. RR ) |
57 |
55 56
|
lttri2d |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( y =/= I <-> ( y < I \/ I < y ) ) ) |
58 |
10
|
ad2antrr |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> G e. B ) |
59 |
47
|
adantl |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( I + J ) - y ) e. NN0 ) |
60 |
59
|
adantr |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( ( I + J ) - y ) e. NN0 ) |
61 |
5 1 4
|
deg1xrcl |
|- ( G e. B -> ( D ` G ) e. RR* ) |
62 |
10 61
|
syl |
|- ( ph -> ( D ` G ) e. RR* ) |
63 |
62
|
ad2antrr |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( D ` G ) e. RR* ) |
64 |
11
|
nn0red |
|- ( ph -> J e. RR ) |
65 |
64
|
rexrd |
|- ( ph -> J e. RR* ) |
66 |
65
|
ad2antrr |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> J e. RR* ) |
67 |
16
|
nn0red |
|- ( ph -> ( I + J ) e. RR ) |
68 |
67
|
adantr |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( I + J ) e. RR ) |
69 |
68 55
|
resubcld |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( I + J ) - y ) e. RR ) |
70 |
69
|
rexrd |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( I + J ) - y ) e. RR* ) |
71 |
70
|
adantr |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( ( I + J ) - y ) e. RR* ) |
72 |
12
|
ad2antrr |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( D ` G ) <_ J ) |
73 |
64
|
adantr |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> J e. RR ) |
74 |
55 56 73
|
ltadd1d |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( y < I <-> ( y + J ) < ( I + J ) ) ) |
75 |
55 73 68
|
ltaddsub2d |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( y + J ) < ( I + J ) <-> J < ( ( I + J ) - y ) ) ) |
76 |
74 75
|
bitrd |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( y < I <-> J < ( ( I + J ) - y ) ) ) |
77 |
76
|
biimpa |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> J < ( ( I + J ) - y ) ) |
78 |
63 66 71 72 77
|
xrlelttrd |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( D ` G ) < ( ( I + J ) - y ) ) |
79 |
5 1 4 27 44
|
deg1lt |
|- ( ( G e. B /\ ( ( I + J ) - y ) e. NN0 /\ ( D ` G ) < ( ( I + J ) - y ) ) -> ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) = ( 0g ` R ) ) |
80 |
58 60 78 79
|
syl3anc |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) = ( 0g ` R ) ) |
81 |
80
|
oveq2d |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( ( ( coe1 ` F ) ` y ) .x. ( 0g ` R ) ) ) |
82 |
26 3 27
|
ringrz |
|- ( ( R e. Ring /\ ( ( coe1 ` F ) ` y ) e. ( Base ` R ) ) -> ( ( ( coe1 ` F ) ` y ) .x. ( 0g ` R ) ) = ( 0g ` R ) ) |
83 |
37 43 82
|
syl2anc |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( ( coe1 ` F ) ` y ) .x. ( 0g ` R ) ) = ( 0g ` R ) ) |
84 |
83
|
adantr |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( ( ( coe1 ` F ) ` y ) .x. ( 0g ` R ) ) = ( 0g ` R ) ) |
85 |
81 84
|
eqtrd |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ y < I ) -> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) ) |
86 |
7
|
ad2antrr |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> F e. B ) |
87 |
54
|
adantr |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> y e. NN0 ) |
88 |
5 1 4
|
deg1xrcl |
|- ( F e. B -> ( D ` F ) e. RR* ) |
89 |
7 88
|
syl |
|- ( ph -> ( D ` F ) e. RR* ) |
90 |
89
|
ad2antrr |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> ( D ` F ) e. RR* ) |
91 |
31
|
rexrd |
|- ( ph -> I e. RR* ) |
92 |
91
|
ad2antrr |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> I e. RR* ) |
93 |
55
|
rexrd |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> y e. RR* ) |
94 |
93
|
adantr |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> y e. RR* ) |
95 |
9
|
ad2antrr |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> ( D ` F ) <_ I ) |
96 |
|
simpr |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> I < y ) |
97 |
90 92 94 95 96
|
xrlelttrd |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> ( D ` F ) < y ) |
98 |
5 1 4 27 38
|
deg1lt |
|- ( ( F e. B /\ y e. NN0 /\ ( D ` F ) < y ) -> ( ( coe1 ` F ) ` y ) = ( 0g ` R ) ) |
99 |
86 87 97 98
|
syl3anc |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> ( ( coe1 ` F ) ` y ) = ( 0g ` R ) ) |
100 |
99
|
oveq1d |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( ( 0g ` R ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) |
101 |
26 3 27
|
ringlz |
|- ( ( R e. Ring /\ ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) e. ( Base ` R ) ) -> ( ( 0g ` R ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) ) |
102 |
37 49 101
|
syl2anc |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( 0g ` R ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) ) |
103 |
102
|
adantr |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> ( ( 0g ` R ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) ) |
104 |
100 103
|
eqtrd |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ I < y ) -> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) ) |
105 |
85 104
|
jaodan |
|- ( ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) /\ ( y < I \/ I < y ) ) -> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) ) |
106 |
105
|
ex |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( ( y < I \/ I < y ) -> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) ) ) |
107 |
57 106
|
sylbid |
|- ( ( ph /\ y e. ( 0 ... ( I + J ) ) ) -> ( y =/= I -> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) ) ) |
108 |
107
|
impr |
|- ( ( ph /\ ( y e. ( 0 ... ( I + J ) ) /\ y =/= I ) ) -> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) ) |
109 |
53 108
|
sylan2b |
|- ( ( ph /\ y e. ( ( 0 ... ( I + J ) ) \ { I } ) ) -> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( 0g ` R ) ) |
110 |
109 30
|
suppss2 |
|- ( ph -> ( ( y e. ( 0 ... ( I + J ) ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) supp ( 0g ` R ) ) C_ { I } ) |
111 |
26 27 29 30 36 52 110
|
gsumpt |
|- ( ph -> ( R gsum ( y e. ( 0 ... ( I + J ) ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ) = ( ( y e. ( 0 ... ( I + J ) ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ` I ) ) |
112 |
|
fveq2 |
|- ( y = I -> ( ( coe1 ` F ) ` y ) = ( ( coe1 ` F ) ` I ) ) |
113 |
|
oveq2 |
|- ( y = I -> ( ( I + J ) - y ) = ( ( I + J ) - I ) ) |
114 |
113
|
fveq2d |
|- ( y = I -> ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) = ( ( coe1 ` G ) ` ( ( I + J ) - I ) ) ) |
115 |
112 114
|
oveq12d |
|- ( y = I -> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) = ( ( ( coe1 ` F ) ` I ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - I ) ) ) ) |
116 |
|
eqid |
|- ( y e. ( 0 ... ( I + J ) ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) = ( y e. ( 0 ... ( I + J ) ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) |
117 |
|
ovex |
|- ( ( ( coe1 ` F ) ` I ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - I ) ) ) e. _V |
118 |
115 116 117
|
fvmpt |
|- ( I e. ( 0 ... ( I + J ) ) -> ( ( y e. ( 0 ... ( I + J ) ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ` I ) = ( ( ( coe1 ` F ) ` I ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - I ) ) ) ) |
119 |
36 118
|
syl |
|- ( ph -> ( ( y e. ( 0 ... ( I + J ) ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ` I ) = ( ( ( coe1 ` F ) ` I ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - I ) ) ) ) |
120 |
8
|
nn0cnd |
|- ( ph -> I e. CC ) |
121 |
11
|
nn0cnd |
|- ( ph -> J e. CC ) |
122 |
120 121
|
pncan2d |
|- ( ph -> ( ( I + J ) - I ) = J ) |
123 |
122
|
fveq2d |
|- ( ph -> ( ( coe1 ` G ) ` ( ( I + J ) - I ) ) = ( ( coe1 ` G ) ` J ) ) |
124 |
123
|
oveq2d |
|- ( ph -> ( ( ( coe1 ` F ) ` I ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - I ) ) ) = ( ( ( coe1 ` F ) ` I ) .x. ( ( coe1 ` G ) ` J ) ) ) |
125 |
111 119 124
|
3eqtrd |
|- ( ph -> ( R gsum ( y e. ( 0 ... ( I + J ) ) |-> ( ( ( coe1 ` F ) ` y ) .x. ( ( coe1 ` G ) ` ( ( I + J ) - y ) ) ) ) ) = ( ( ( coe1 ` F ) ` I ) .x. ( ( coe1 ` G ) ` J ) ) ) |
126 |
15 25 125
|
3eqtrd |
|- ( ph -> ( ( coe1 ` ( F .xb G ) ) ` ( I + J ) ) = ( ( ( coe1 ` F ) ` I ) .x. ( ( coe1 ` G ) ` J ) ) ) |