Step |
Hyp |
Ref |
Expression |
1 |
|
evl1gsummon.q |
|- Q = ( eval1 ` R ) |
2 |
|
evl1gsummon.k |
|- K = ( Base ` R ) |
3 |
|
evl1gsummon.w |
|- W = ( Poly1 ` R ) |
4 |
|
evl1gsummon.b |
|- B = ( Base ` W ) |
5 |
|
evl1gsummon.x |
|- X = ( var1 ` R ) |
6 |
|
evl1gsummon.h |
|- H = ( mulGrp ` R ) |
7 |
|
evl1gsummon.e |
|- E = ( .g ` H ) |
8 |
|
evl1gsummon.g |
|- G = ( mulGrp ` W ) |
9 |
|
evl1gsummon.p |
|- .^ = ( .g ` G ) |
10 |
|
evl1gsummon.t1 |
|- .X. = ( .s ` W ) |
11 |
|
evl1gsummon.t2 |
|- .x. = ( .r ` R ) |
12 |
|
evl1gsummon.r |
|- ( ph -> R e. CRing ) |
13 |
|
evl1gsummon.a |
|- ( ph -> A. x e. M A e. K ) |
14 |
|
evl1gsummon.m |
|- ( ph -> M C_ NN0 ) |
15 |
|
evl1gsummon.f |
|- ( ph -> M e. Fin ) |
16 |
|
evl1gsummon.n |
|- ( ph -> A. x e. M N e. NN0 ) |
17 |
|
evl1gsummon.c |
|- ( ph -> C e. K ) |
18 |
|
eqid |
|- ( R ^s K ) = ( R ^s K ) |
19 |
|
crngring |
|- ( R e. CRing -> R e. Ring ) |
20 |
12 19
|
syl |
|- ( ph -> R e. Ring ) |
21 |
3
|
ply1lmod |
|- ( R e. Ring -> W e. LMod ) |
22 |
20 21
|
syl |
|- ( ph -> W e. LMod ) |
23 |
22
|
adantr |
|- ( ( ph /\ x e. M ) -> W e. LMod ) |
24 |
13
|
r19.21bi |
|- ( ( ph /\ x e. M ) -> A e. K ) |
25 |
3
|
ply1sca |
|- ( R e. CRing -> R = ( Scalar ` W ) ) |
26 |
12 25
|
syl |
|- ( ph -> R = ( Scalar ` W ) ) |
27 |
26
|
fveq2d |
|- ( ph -> ( Base ` R ) = ( Base ` ( Scalar ` W ) ) ) |
28 |
2 27
|
eqtrid |
|- ( ph -> K = ( Base ` ( Scalar ` W ) ) ) |
29 |
28
|
adantr |
|- ( ( ph /\ x e. M ) -> K = ( Base ` ( Scalar ` W ) ) ) |
30 |
24 29
|
eleqtrd |
|- ( ( ph /\ x e. M ) -> A e. ( Base ` ( Scalar ` W ) ) ) |
31 |
3
|
ply1ring |
|- ( R e. Ring -> W e. Ring ) |
32 |
20 31
|
syl |
|- ( ph -> W e. Ring ) |
33 |
8
|
ringmgp |
|- ( W e. Ring -> G e. Mnd ) |
34 |
32 33
|
syl |
|- ( ph -> G e. Mnd ) |
35 |
34
|
adantr |
|- ( ( ph /\ x e. M ) -> G e. Mnd ) |
36 |
16
|
r19.21bi |
|- ( ( ph /\ x e. M ) -> N e. NN0 ) |
37 |
20
|
adantr |
|- ( ( ph /\ x e. M ) -> R e. Ring ) |
38 |
5 3 4
|
vr1cl |
|- ( R e. Ring -> X e. B ) |
39 |
37 38
|
syl |
|- ( ( ph /\ x e. M ) -> X e. B ) |
40 |
8 4
|
mgpbas |
|- B = ( Base ` G ) |
41 |
40 9
|
mulgnn0cl |
|- ( ( G e. Mnd /\ N e. NN0 /\ X e. B ) -> ( N .^ X ) e. B ) |
42 |
35 36 39 41
|
syl3anc |
|- ( ( ph /\ x e. M ) -> ( N .^ X ) e. B ) |
43 |
|
eqid |
|- ( Scalar ` W ) = ( Scalar ` W ) |
44 |
|
eqid |
|- ( Base ` ( Scalar ` W ) ) = ( Base ` ( Scalar ` W ) ) |
45 |
4 43 10 44
|
lmodvscl |
|- ( ( W e. LMod /\ A e. ( Base ` ( Scalar ` W ) ) /\ ( N .^ X ) e. B ) -> ( A .X. ( N .^ X ) ) e. B ) |
46 |
23 30 42 45
|
syl3anc |
|- ( ( ph /\ x e. M ) -> ( A .X. ( N .^ X ) ) e. B ) |
47 |
1 2 3 18 4 12 46 14 15 17
|
evl1gsumaddval |
|- ( ph -> ( ( Q ` ( W gsum ( x e. M |-> ( A .X. ( N .^ X ) ) ) ) ) ` C ) = ( R gsum ( x e. M |-> ( ( Q ` ( A .X. ( N .^ X ) ) ) ` C ) ) ) ) |
48 |
12
|
adantr |
|- ( ( ph /\ x e. M ) -> R e. CRing ) |
49 |
17
|
adantr |
|- ( ( ph /\ x e. M ) -> C e. K ) |
50 |
1 3 8 5 2 9 48 36 10 24 49 6 7 11
|
evl1scvarpwval |
|- ( ( ph /\ x e. M ) -> ( ( Q ` ( A .X. ( N .^ X ) ) ) ` C ) = ( A .x. ( N E C ) ) ) |
51 |
50
|
mpteq2dva |
|- ( ph -> ( x e. M |-> ( ( Q ` ( A .X. ( N .^ X ) ) ) ` C ) ) = ( x e. M |-> ( A .x. ( N E C ) ) ) ) |
52 |
51
|
oveq2d |
|- ( ph -> ( R gsum ( x e. M |-> ( ( Q ` ( A .X. ( N .^ X ) ) ) ` C ) ) ) = ( R gsum ( x e. M |-> ( A .x. ( N E C ) ) ) ) ) |
53 |
47 52
|
eqtrd |
|- ( ph -> ( ( Q ` ( W gsum ( x e. M |-> ( A .X. ( N .^ X ) ) ) ) ) ` C ) = ( R gsum ( x e. M |-> ( A .x. ( N E C ) ) ) ) ) |